高數(shù)下冊(cè)知識(shí)點(diǎn)

第九章多元函數(shù)微分法及其應(yīng)用基本概念距離，鄰域，內(nèi)點(diǎn)，外點(diǎn)，邊界點(diǎn)，聚點(diǎn)，開(kāi)集，閉集，連通集，區(qū)域，閉區(qū)域，有界集，無(wú)界集。多元函數(shù)：SKIPIF1<0，圖形：極限：SKIPIF1<0連續(xù)：SKIPIF1<0偏導(dǎo)數(shù)：SKIPIF1<0SKIPIF1<0方向?qū)?shù)：SKIPIF1<0其中SKIPIF1<0為SKIPIF1<0的方向角。梯度：SKIPIF1<0，則SKIPIF1<0。全微分：設(shè)SKIPIF1<0，則SKIPIF1<0性質(zhì)函數(shù)可微，偏導(dǎo)連續(xù)，偏導(dǎo)存在，函數(shù)連續(xù)等概念之間的關(guān)系：偏導(dǎo)數(shù)存在偏導(dǎo)數(shù)存在函數(shù)可微函數(shù)連續(xù)偏導(dǎo)數(shù)連續(xù)充分條件必要條件定義12234閉區(qū)域上連續(xù)函數(shù)的性質(zhì)（有界性定理，最大最小值定理，介值定理）微分法定義：SKIPIF1<0SKIPIF1<0復(fù)合函數(shù)求導(dǎo)：鏈?zhǔn)椒▌tSKIPIF1<0若SKIPIF1<0，則SKIPIF1<0SKIPIF1<0SKIPIF1<0，SKIPIF1<0隱函數(shù)求導(dǎo)：兩邊求偏導(dǎo)，然后解方程（組）應(yīng)用極值無(wú)條件極值：求函數(shù)SKIPIF1<0的極值解方程組SKIPIF1<0求出所有駐點(diǎn)，對(duì)于每一個(gè)駐點(diǎn)SKIPIF1<0，令SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，函數(shù)有極小值，若SKIPIF1<0，SKIPIF1<0，函數(shù)有極大值；若SKIPIF1<0，函數(shù)沒(méi)有極值；若SKIPIF1<0，不定。條件極值：求函數(shù)SKIPIF1<0在條件SKIPIF1<0下的極值令：SKIPIF1<0———Lagrange函數(shù)解方程組SKIPIF1<0幾何應(yīng)用曲線的切線與法平面曲線SKIPIF1<0，則SKIPIF1<0上一點(diǎn)SKIPIF1<0（對(duì)應(yīng)參數(shù)為SKIPIF1<0）處的切線方程為：SKIPIF1<0法平面方程為：SKIPIF1<0曲面的切平面與法線曲面SKIPIF1<0，則SKIPIF1<0上一點(diǎn)SKIPIF1<0處的切平面方程為：SKIPIF1<0法線方程為：SKIPIF1<0第十章重積分二重積分定義：SKIPIF1<0性質(zhì)：（6條）幾何意義：曲頂柱體的體積。計(jì)算：直角坐標(biāo)SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0極坐標(biāo)SKIPIF1<0SKIPIF1<0三重積分定義：SKIPIF1<0性質(zhì)：計(jì)算：直角坐標(biāo)SKIPIF1<0-------------“先一后二”SKIPIF1<0-------------“先二后一”柱面坐標(biāo)SKIPIF1<0，SKIPIF1<0球面坐標(biāo)SKIPIF1<0SKIPIF1<0應(yīng)用曲面SKIPIF1<0的面積：SKIPIF1<0求立體的體積和質(zhì)量，SKIPIF1<0平面薄片的質(zhì)量SKIPIF1<0第十二章無(wú)窮級(jí)數(shù)常數(shù)項(xiàng)級(jí)數(shù)定義：1）無(wú)窮級(jí)數(shù)：SKIPIF1<0部分和：SKIPIF1<0，正項(xiàng)級(jí)數(shù)：SKIPIF1<0，SKIPIF1<0交錯(cuò)級(jí)數(shù)：SKIPIF1<0，SKIPIF1<02）級(jí)數(shù)收斂：若SKIPIF1<0存在，則稱級(jí)數(shù)SKIPIF1<0收斂，否則稱級(jí)數(shù)SKIPIF1<0發(fā)散3）條件收斂：SKIPIF1<0收斂，而SKIPIF1<0發(fā)散；絕對(duì)收斂：SKIPIF1<0收斂。性質(zhì)：改變有限項(xiàng)不影響級(jí)數(shù)的收斂性；級(jí)數(shù)SKIPIF1<0，SKIPIF1<0收斂，則SKIPIF1<0收斂；級(jí)數(shù)SKIPIF1<0收斂，則任意加括號(hào)后仍然收斂；必要條件：級(jí)數(shù)SKIPIF1<0收斂SKIPIF1<0SKIPIF1<0.（注意：不是充分條件?。彅糠ㄕ?xiàng)級(jí)數(shù)：SKIPIF1<0，SKIPIF1<0定義：SKIPIF1<0存在；SKIPIF1<0收斂SKIPIF1<0SKIPIF1<0有界；比較審斂法：SKIPIF1<0，SKIPIF1<0為正項(xiàng)級(jí)數(shù)，且SKIPIF1<0若SKIPIF1<0收斂，則SKIPIF1<0收斂；若SKIPIF1<0發(fā)散，則SKIPIF1<0發(fā)散.比較法的推論：SKIPIF1<0，SKIPIF1<0為正項(xiàng)級(jí)數(shù)，若存在正整數(shù)SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，而SKIPIF1<0收斂，則SKIPIF1<0收斂；若存在正整數(shù)SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，而SKIPIF1<0發(fā)散，則SKIPIF1<0發(fā)散.比較法的極限形式：SKIPIF1<0，SKIPIF1<0為正項(xiàng)級(jí)數(shù)，若SKIPIF1<0，而SKIPIF1<0收斂，則SKIPIF1<0收斂；若SKIPIF1<0或SKIPIF1<0，而SKIPIF1<0發(fā)散，則SKIPIF1<0發(fā)散.比值法：SKIPIF1<0為正項(xiàng)級(jí)數(shù)，設(shè)SKIPIF1<0，則當(dāng)SKIPIF1<0時(shí)，級(jí)數(shù)SKIPIF1<0收斂；則當(dāng)SKIPIF1<0時(shí)，級(jí)數(shù)SKIPIF1<0發(fā)散；當(dāng)SKIPIF1<0時(shí)，級(jí)數(shù)SKIPIF1<0可能收斂也可能發(fā)散.根值法：SKIPIF1<0為正項(xiàng)級(jí)數(shù)，設(shè)SKIPIF1<0，則當(dāng)SKIPIF1<0時(shí)，級(jí)數(shù)SKIPIF1<0收斂；則當(dāng)SKIPIF1<0時(shí)，級(jí)數(shù)SKIPIF1<0發(fā)散；當(dāng)SKIPIF1<0時(shí)，級(jí)數(shù)SKIPIF1<0可能收斂也可能發(fā)散.極限審斂法：SKIPIF1<0為正項(xiàng)級(jí)數(shù)，若SKIPIF1<0或SKIPIF1<0，則級(jí)數(shù)SKIPIF1<0發(fā)散；若存在SKIPIF1<0，使得SKIPIF1<0，則級(jí)數(shù)SKIPIF1<0收斂.交錯(cuò)級(jí)數(shù)：萊布尼茨審斂法：交錯(cuò)級(jí)數(shù)：SKIPIF1<0，SKIPIF1<0滿足：SKIPIF1<0，且SKIPIF1<0，則級(jí)數(shù)SKIPIF1<0收斂。任意項(xiàng)級(jí)數(shù)：SKIPIF1<0絕對(duì)收斂，則SKIPIF1<0收斂。常見(jiàn)典型級(jí)數(shù)：幾何級(jí)數(shù)：SKIPIF1<0p-級(jí)數(shù)：SKIPIF1<0函數(shù)項(xiàng)級(jí)數(shù)定義：函數(shù)項(xiàng)級(jí)數(shù)SKIPIF1<0，收斂域，收斂半徑，和函數(shù)；冪級(jí)數(shù)：SKIPIF1<0收斂半徑的求法：SKIPIF1<0，則收斂半徑SKIPIF1<0泰勒級(jí)數(shù)SKIPIF1<0SKIPIF1<0SKIPIF1<0展開(kāi)步驟：（直接展開(kāi)法）求出SKIPIF1<0；求出SKIPIF1<0；寫出SKIPIF1<