高數(shù)第十章曲線積分與曲面積分

/第十章曲線積分與曲面積分一、對(duì)弧長(zhǎng)的曲線積分（又稱第一類曲線積分）1、定義SKIPIF1<0，SKIPIF1<02、物理意義線密度為SKIPIF1<0的曲線SKIPIF1<0質(zhì)量為SKIPIF1<0線密度為SKIPIF1<0的曲線SKIPIF1<0質(zhì)量為SKIPIF1<03、幾何意義曲線SKIPIF1<0的弧長(zhǎng)SKIPIF1<0SKIPIF1<0，曲線SKIPIF1<0的弧長(zhǎng)SKIPIF1<04、若SKIPIF1<0：SKIPIF1<0（常數(shù)），則SKIPIF1<05、計(jì)算（上限大于下限）（1）SKIPIF1<0SKIPIF1<0，則SKIPIF1<0（2）SKIPIF1<0：SKIPIF1<0，則SKIPIF1<0（3）SKIPIF1<0：SKIPIF1<0，則SKIPIF1<0（4）SKIPIF1<0，則SKIPIF1<0二、對(duì)坐標(biāo)的曲線積分1、定義SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<02、計(jì)算（下限對(duì)應(yīng)起點(diǎn)，上限對(duì)應(yīng)終點(diǎn)）（1）SKIPIF1<0SKIPIF1<0，則SKIPIF1<0（2）SKIPIF1<0：SKIPIF1<0SKIPIF1<0，則SKIPIF1<0（3）SKIPIF1<0：SKIPIF1<0SKIPIF1<0，則SKIPIF1<0（4）SKIPIF1<0，則SKIPIF1<0SKIPIF1<03、兩類曲線積分之間的聯(lián)系SKIPIF1<0其中，SKIPIF1<0為有向曲線弧SKIPIF1<0上點(diǎn)SKIPIF1<0處的切線向量的方向角。SKIPIF1<0，其中SKIPIF1<0為有向曲線弧SKIPIF1<0上點(diǎn)SKIPIF1<0處切向量的方向角。三、格林公式及其應(yīng)用1、格林公式SKIPIF1<0其中SKIPIF1<0是SKIPIF1<0的取正向的整個(gè)邊界曲線2、平面上曲線積分與路徑無(wú)關(guān)的條件（SKIPIF1<0為單連通區(qū)域）定理設(shè)SKIPIF1<0是單連通閉區(qū)域，若SKIPIF1<0在SKIPIF1<0內(nèi)連續(xù)，且具有一階連續(xù)偏導(dǎo)數(shù)，則以下四個(gè)條件等價(jià)：(i)沿SKIPIF1<0內(nèi)任一按段光滑封閉曲線SKIPIF1<0，有SKIPIF1<0；(ii)對(duì)SKIPIF1<0內(nèi)任一光滑曲線SKIPIF1<0，曲線積分SKIPIF1<0與路徑無(wú)關(guān)，只與SKIPIF1<0的起點(diǎn)和終點(diǎn)有關(guān)；(iii)SKIPIF1<0是SKIPIF1<0內(nèi)某一函數(shù)SKIPIF1<0的全微分，即在SKIPIF1<0內(nèi)有SKIPIF1<0；(iv)在SKIPIF1<0內(nèi)處處成立SKIPIF1<0注若SKIPIF1<0則SKIPIF1<0的全微分SKIPIF1<0：SKIPIF1<0或SKIPIF1<0四、對(duì)面積的曲面積分1、定義SKIPIF1<0SKIPIF1<02、物理意義：SKIPIF1<0表示面密度為SKIPIF1<0的光滑曲面SKIPIF1<0的質(zhì)量。3、幾何意義曲面SKIPIF1<0的面積SKIPIF1<04、若SKIPIF1<0：SKIPIF1<0（常數(shù)），則SKIPIF1<0=SKIPIF1<0=SKIPIF1<0=SKIPIF1<05、計(jì)算（一投、二代、三換元）（1）SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（2）SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0SKIPIF1<0（3）SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0SKIPIF1<0。五、對(duì)坐標(biāo)的曲面積分1、定義SKIPIF1<0SKIPIF1<0SKIPIF1<02、物理意義流量SKIPIF1<0。SKIPIF1<0SKIPIF1<03、計(jì)算（一投、二代、三定號(hào)）（1）SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（上側(cè)取正，下側(cè)取負(fù)）（2）SKIPIF1<0:SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（前側(cè)取正，后側(cè)取負(fù)）（3）SKIPIF1<0:SKIPIF1<0SKIPIF1<0，則SKIPIF1<0（右側(cè)取正，左側(cè)取負(fù)）4、兩類曲面積分之間的聯(lián)系SKIPIF1<0，SKIPIF1<0其中SKIPIF1<0為有向曲面Σ上點(diǎn)SKIPIF1<0處的法向量的方向余弦六、高斯公式1、高斯公式SKIPIF1<0其中SKIPIF1<0為SKIPIF1<0的整個(gè)邊界曲面的外側(cè)，SKIPIF1<0是SKIPIF1<0上點(diǎn)SKIPIF1<0處的法向量的方向角。2、通量向量場(chǎng)SKIPIF1<0，沿場(chǎng)中有向曲面ΣSKIPIF1<0稱為向量場(chǎng)SKIPIF1<0向正側(cè)穿過(guò)曲面Σ的通量3、散度設(shè)SKIPIF1<0，則SKIPIF1<0七、斯托克斯公式1、Stokes公式SKIPIF1<0SKIPIF1<0=SKIPIF1<0=SKIPIF1<0SKIPIF1<0其中有向曲線SKIPIF1<0是有向曲面SKIPIF1<0的整個(gè)邊界，且滿足右手系法則2、環(huán)流量向量場(chǎng)SKIPIF1<0沿場(chǎng)SKIPIF1<0中某一封閉的有向曲線SKIPIF1<0上的曲線積分SKIPIF1<0稱為向量場(chǎng)SKIPIF1<0沿曲線SKIPIF1<0按所取方向的環(huán)流量。SKIPIF1<03、旋度向量SKIPIF1<0為向量場(chǎng)SKIPIF1<0的旋度SKIPIF1<0。旋度SKIPIF1<0SKIPIF1<0典型例題1.曲線積分1計(jì)算SKIPIF1<0其中SKIPIF1<0為圓周SKIPIF1<0SKIPIF1<0SKIPIF1<0。解：(方法一)根據(jù)公式將曲線積分化為定積分SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0(方法二)由于在曲線SKIPIF1<0上SKIPIF1<0,且SKIPIF1<0為曲線段SKIPIF1<0的長(zhǎng),所以SKIPIF1<02計(jì)算SKIPIF1<0,其中SKIPIF1<0為圓周SKIPIF1<0,直線SKIPIF1<0及SKIPIF1<0軸在第一象限內(nèi)所圍成的扇形的整個(gè)邊界。分析由于曲線SKIPIF1<0分段光滑，所以先將SKIPIF1<0分為若干光滑曲線段之和，再利用曲線積分的可加性計(jì)算曲線積分。解：SKIPIF1<0圖10-2SKIPIF1<0的方程為SKIPIF1<0，SKIPIF1<0圖10-2SKIPIF1<0SKIPIF1<0SKIPIF1<0的方程為：SKIPIF1<0SKIPIF1<0所以SKIPIF1<0SKIPIF1<0的方程為SKIPIF1<0，SKIPIF1<0.所以SKIPIF1<0SKIPIF1<0SKIPIF1<03計(jì)算SKIPIF1<0其中SKIPIF1<0為折線段SKIPIF1<0,這里SKIPIF1<0,SKIPIF1<0。分析求本曲線積分的關(guān)鍵是求直線SKIPIF1<0的參數(shù)方程.空間過(guò)點(diǎn)SKIPIF1<0的直線的對(duì)稱式方程SKIPIF1<0令該比式等于SKIPIF1<0,可得到直線的參數(shù)方程。圖10－3解：SKIPIF1<0圖10－3線段SKIPIF1<0：SKIPIF1<0SKIPIF1<0SKIPIF1<0線段SKIPIF1<0：SKIPIF1<0SKIPIF1<0SKIPIF1<0線段SKIPIF1<0：SKIPIF1<0SKIPIF1<0，SKIPIF1<0SKIPIF1<0所以SKIPIF1<04計(jì)算SKIPIF1<0,其中SKIPIF1<0為折線段SKIPIF1<0所圍成區(qū)域的整個(gè)邊界。解：(方法一)如圖10-4SKIPIF1<0SKIPIF1<0圖10-4SKIPIF1<0的方程為:SKIPIF1<0圖10-4SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0的方程為:SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0的方程為:SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0的方程為:SKIPIF1<0SKIPIF1<0SKIPIF1<0所以SKIPIF1<0(方法二)由于曲線SKIPIF1<0關(guān)于SKIPIF1<0軸對(duì)稱,而SKIPIF1<0是關(guān)于SKIPIF1<0的奇函數(shù),故SKIPIF1<0。又SKIPIF1<0關(guān)于SKIPIF1<0軸對(duì)稱,而SKIPIF1<0是關(guān)于SKIPIF1<0的奇函數(shù),故SKIPIF1<0所以SKIPIF1<0。注意一般地,若曲線SKIPIF1<0關(guān)于SKIPIF1<0軸對(duì)稱,則有SKIPIF1<0其中SKIPIF1<0是SKIPIF1<0在SKIPIF1<0的部分。若曲線SKIPIF1<0關(guān)于SKIPIF1<0軸對(duì)稱,則有SKIPIF1<0其中SKIPIF1<0是SKIPIF1<0在SKIPIF1<0的部分。5計(jì)算SKIPIF1<0,其中SKIPIF1<0為圓周SKIPIF1<0SKIPIF1<0。解：(方法一)如圖10-5（a）,SKIPIF1<0的參數(shù)方程為SKIPIF1<0圖10-5(a)SKIPIF1<0SKIPIF1<0SKIPIF1<0圖10-5(a)SKIPIF1<0SKIPIF1<0圖10-5(b)圖10-5(b)SKIPIF1<0SKIPIF1<0SKIPIF1<0。(方法二)如圖10－5（b）SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0，由直角坐標(biāo)與極坐標(biāo)的關(guān)系,則SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0故SKIPIF1<0。注意①在方法一中,參數(shù)SKIPIF1<0表示圓心角,而在方法二中,參數(shù)SKIPIF1<0表示極坐標(biāo)系下的極角,參數(shù)的意義不同,一般取值范圍也不相同。②若曲線在極坐標(biāo)系下的方程為SKIPIF1<0,則SKIPIF1<0,可直接引用此式。③該例也可以先利用對(duì)稱性化簡(jiǎn),再化為定積分計(jì)算。6計(jì)算SKIPIF1<0,其中SKIPIF1<0為SKIPIF1<0。分析計(jì)算這個(gè)曲線積分的關(guān)鍵,是正確的寫(xiě)出SKIPIF1<0的參數(shù)方程。一般地,如果SKIPIF1<0的方程形式為SKIPIF1<0時(shí),先求出SKIPIF1<0關(guān)于SKIPIF1<0的投影柱面SKIPIF1<0,即利用兩個(gè)曲面方程消去SKIPIF1<0,再求出平面曲線SKIPIF1<0的參數(shù)方程SKIPIF1<0,并將其代入其中一個(gè)曲面方程解出SKIPIF1<0,即得SKIPIF1<0的參數(shù)方程。解：(方法一)由于SKIPIF1<0是平面SKIPIF1<0上過(guò)球SKIPIF1<0的中心的大圓．兩個(gè)曲面方程聯(lián)立消去SKIPIF1<0,得SKIPIF1<0①在①式中,令SKIPIF1<0②SKIPIF1<0③將②,③代入平面SKIPIF1<0,得SKIPIF1<0,故SKIPIF1<0的參數(shù)方程為SKIPIF1<0，SKIPIF1<0SKIPIF1<0SKIPIF1<0所以SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0(方法二)由于積分曲線方程中的變量SKIPIF1<0具有輪換性,即三個(gè)變量輪換位置方程不變,且對(duì)弧長(zhǎng)的曲線積分與積分曲線的方向無(wú)關(guān)。故有SKIPIF1<0SKIPIF1<0同理SKIPIF1<0所以SKIPIF1<0SKIPIF1<0。注意利用變量之間的輪換對(duì)稱性技巧來(lái)解對(duì)弧長(zhǎng)的曲線積分,往往有事半功倍之效。7計(jì)算SKIPIF1<0,其中SKIPIF1<0是擺線SKIPIF1<0上對(duì)應(yīng)SKIPIF1<0從0到SKIPIF1<0的一段弧。解：根據(jù)公式SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<08計(jì)算SKIPIF1<0,其中SKIPIF1<0是曲線SKIPIF1<0對(duì)應(yīng)于SKIPIF1<0的點(diǎn)到SKIPIF1<0的點(diǎn)。解：如圖10-6，SKIPIF1<0圖10-6SKIPIF1<0SKIPIF1<0圖10-6(方法一)取SKIPIF1<0為參數(shù),SKIPIF1<0的方程為SKIPIF1<0SKIPIF1<0SKIPIF1<0的方程為SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0所以SKIPIF1<0。(法二)取SKIPIF1<0為參數(shù),SKIPIF1<0的方程為SKIPIF1<0SKIPIF1<0SKIPIF1<0的方程為SKIPIF1<0始點(diǎn)對(duì)應(yīng)的參數(shù)值為1,終點(diǎn)對(duì)應(yīng)的參數(shù)值為0。由于SKIPIF1<0,故有SKIPIF1<0所以SKIPIF1<09SKIPIF1<0,其中SKIPIF1<0是用平面SKIPIF1<0截球面SKIPIF1<0SKIPIF1<0所得的截痕,從SKIPIF1<0軸的正向看去,沿逆時(shí)針?lè)较?。解：將SKIPIF1<0代入球面方程SKIPIF1<0消去SKIPIF1<0得，SKIPIF1<0,令SKIPIF1<0,并將其代入SKIPIF1<0得,SKIPIF1<0。SKIPIF1<0的參數(shù)方程為:SKIPIF1<0SKIPIF1<0始點(diǎn)對(duì)應(yīng)的參數(shù)值為0,終點(diǎn)對(duì)應(yīng)的參數(shù)值為SKIPIF1<0。SKIPIF1<0SKIPIF1<0SKIPIF1<010利用格林公式計(jì)算SKIPIF1<0其中SKIPIF1<0為圓周SKIPIF1<0,沿逆時(shí)針?lè)较?。解：SKIPIF1<0由格林公式SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0常見(jiàn)錯(cuò)解設(shè)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0錯(cuò)誤原因在曲線積分中SKIPIF1<0的方程可以直接代入曲線積分中,但在二重積分中SKIPIF1<0所以把SKIPIF1<0的方程代入二重積分的被積函數(shù)中是錯(cuò)誤的。注意①利用格林公式計(jì)算對(duì)坐標(biāo)的曲線積分時(shí),SKIPIF1<0不要顛倒了。②計(jì)算沿閉曲線對(duì)坐標(biāo)的曲線積分時(shí),常利用格林公式簡(jiǎn)化計(jì)算。圖10-711計(jì)算SKIPIF1<0,其中SKIPIF1<0為上半圓周SKIPIF1<0,沿順時(shí)針?lè)较?。圖10-7解：SKIPIF1<0SKIPIF1<0SKIPIF1<0如圖10-7,由格林公式SKIPIF1<0SKIPIF1<0故有SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0所以SKIPIF1<0SKIPIF1<0。注意①利用格林公式計(jì)算沿非封閉曲線的積分時(shí)，常用坐標(biāo)軸上或平行于坐標(biāo)軸的直線段作為輔助線。圖10-8圖10-812計(jì)算SKIPIF1<0,其中SKIPIF1<0,沿逆時(shí)針?lè)较?。解：如圖10-8適當(dāng)選取SKIPIF1<0,作圓周SKIPIF1<0:SKIPIF1<0SKIPIF1<0，使SKIPIF1<0包含在SKIPIF1<0的內(nèi)部,并取SKIPIF1<0的方向?yàn)轫槙r(shí)針.則SKIPIF1<0在SKIPIF1<0所包圍的區(qū)域SKIPIF1<0內(nèi)有連續(xù)的一階偏導(dǎo)數(shù),且SKIPIF1<0構(gòu)成SKIPIF1<0的正邊界.由格林公式SKIPIF1<0SKIPIF1<0所以SKIPIF1<0SKIPIF1<0SKIPIF1<0。常見(jiàn)錯(cuò)解SKIPIF1<0錯(cuò)誤原因SKIPIF1<0及一階偏導(dǎo)數(shù)在SKIPIF1<0點(diǎn)沒(méi)有定義,故不能直接使用格林公式。注意在本例中，若把SKIPIF1<0換為不過(guò)原點(diǎn)的任意分段光滑且無(wú)重點(diǎn)的閉曲線，應(yīng)該分為原點(diǎn)在SKIPIF1<0所包圍的區(qū)域內(nèi)和原點(diǎn)不在這個(gè)區(qū)域內(nèi)兩種情況進(jìn)行討論。對(duì)前一種情況，曲線積分利用此例的方法就可以求出。圖10-913證明曲線積分SKIPIF1<0在整個(gè)坐標(biāo)面SKIPIF1<0上與路徑無(wú)關(guān),并計(jì)算積分值。圖10-9解：(方法一)SKIPIF1<0，因?yàn)镾KIPIF1<0且SKIPIF1<0在整個(gè)坐標(biāo)面SKIPIF1<0上有連續(xù)的一階偏導(dǎo)數(shù),所以曲線積分與路徑無(wú)關(guān)。SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0(方法二)由于被積表達(dá)式SKIPIF1<0SKIPIF1<0SKIPIF1<0所以曲線積分與路徑無(wú)關(guān).設(shè)SKIPIF1<0,則SKIPIF1<0SKIPIF1<0。14設(shè)SKIPIF1<0,求SKIPIF1<0。解：(方法一)設(shè)SKIPIF1<0由SKIPIF1<0所以SKIPIF1<0SKIPIF1<0SKIPIF1<0(方法二)由SKIPIF1<0,把SKIPIF1<0看作不變的,對(duì)SKIPIF1<0積分得SKIPIF1<0而SKIPIF1<0故有SKIPIF1<0所以SKIPIF1<0。注意①利用方法一求函數(shù)SKIPIF1<0時(shí)，選擇的起點(diǎn)不同求出的SKIPIF1<0可能相差一個(gè)常數(shù)。②此例還可以用例13中方法二來(lái)求。曲面積分例1計(jì)算SKIPIF1<0,其中SKIPIF1<0為平面SKIPIF1<0在第一卦限的部分。圖10-11解：設(shè)SKIPIF1<0SKIPIF1<0,SKIPIF1<0在坐標(biāo)面SKIPIF1<0上的投影區(qū)域SKIPIF1<0為：SKIPIF1<0.由于圖10-11SKIPIF1<0SKIPIF1<0SKIPIF1<0所以SKIPIF1<0。例2計(jì)算SKIPIF1<0SKIPIF1<0,其中SKIPIF1<0為錐面SKIPIF1<0被圓柱面SKIPIF1<0所截得的有限部分。解：(方法一)如圖10-12,SKIPIF1<0在坐標(biāo)面SKIPIF1<0上的投影區(qū)域SKIPIF1<0為:SKIPIF1<0。因?yàn)镾KIPIF1<0SKIPIF1<0SKIPIF1<0圖10-12所以SKIPIF1<0SKIPIF1<0圖10-12在極坐標(biāo)系下SKIPIF1<0為:SKIPIF1<0,故SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0。(注意奇函數(shù)在對(duì)稱區(qū)間上的積分等于0)(方法二)由于SKIPIF1<0關(guān)于SKIPIF1<0面對(duì)稱,且被積函數(shù)SKIPIF1<0及SKIPIF1<0是關(guān)于SKIPIF1<0的奇函數(shù),故SKIPIF1<0，于是SKIPIF1<0注意在計(jì)算對(duì)面積的曲面積分中,經(jīng)常用對(duì)稱性來(lái)化簡(jiǎn)運(yùn)算.但應(yīng)用這一性質(zhì)時(shí),不僅要考慮積分曲面的對(duì)稱性,同時(shí)要考慮被積函數(shù)的對(duì)稱性。例3計(jì)算SKIPIF1<0，其中SKIPIF1<0是界于平面SKIPIF1<0及SKIPIF1<0之間的圓柱面SKIPIF1<0。解：將SKIPIF1<0投影到坐標(biāo)面SKIPIF1<0上,其投影區(qū)域?yàn)镾KIPIF1<0SKIPIF1<0的方程為:SKIPIF1<0SKIPIF1<0記SKIPIF1<0,則SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0。常見(jiàn)錯(cuò)解因?yàn)镾KIPIF1<0在坐標(biāo)面SKIPIF1<0上的投影為圓周,其面積為0,于是SKIPIF1<0錯(cuò)誤原因此例中,說(shuō)“圓柱面SKIPIF1<0在坐標(biāo)面SKIPIF1<0上的投影為圓周,其面積為0”是對(duì)的,但據(jù)此確定曲面積分為0是錯(cuò)誤的。由于SKIPIF1<0的方程不能寫(xiě)成SKIPIF1<0的形式，所以應(yīng)將曲面投影到其它兩個(gè)坐標(biāo)面上。注意計(jì)算對(duì)面積的曲面積分時(shí),把積分曲面投影到哪個(gè)坐標(biāo)面上,要根據(jù)積分曲面方程的表達(dá)式來(lái)確定。一般地,把SKIPIF1<0投影到坐標(biāo)面SKIPIF1<0上,則SKIPIF1<0的方程應(yīng)寫(xiě)為SKIPIF1<0的形式；把SKIPIF1<0投影到SKIPIF1<0或SKIPIF1<0坐標(biāo)面上,SKIPIF1<0的方程應(yīng)寫(xiě)為SKIPIF1<0或SKIPIF1<0的形式。圖10-13例4計(jì)算積分SKIPIF1<0,其中SKIPIF1<0為平面SKIPIF1<0含在柱面SKIPIF1<0內(nèi)部分的上側(cè)。圖10-13解：如圖10-13,SKIPIF1<0在坐標(biāo)面SKIPIF1<0上的投影區(qū)域?yàn)镾KIPIF1<0。SKIPIF1<0SKIPIF1<0。常見(jiàn)錯(cuò)解SKIPIF1<0SKIPIF1<0其中SKIPIF1<0為SKIPIF1<0的面積。錯(cuò)誤原因這里把對(duì)坐標(biāo)的曲面積分與對(duì)面積的曲面積分混淆了,SKIPIF1<0是正確的,而在對(duì)坐標(biāo)的曲面積分中,微元SKIPIF1<0是SKIPIF1<0在坐標(biāo)面SKIPIF1<0上的投影與SKIPIF1<0不同,其正負(fù)由SKIPIF1<0的側(cè)來(lái)確定。圖10-14例5計(jì)算曲面積分SKIPIF1<0,其中SKIPIF1<0是拋物柱面SKIPIF1<0被平面SKIPIF1<0和SKIPIF1<0所截下的那部分的后側(cè)曲面。圖10-14解：如圖10-4,因?yàn)橹鍿KIPIF1<0在坐標(biāo)面SKIPIF1<0上的投影是一條曲線,由定義知SKIPIF1<0。SKIPIF1<0在坐標(biāo)面SKIPIF1<0上的投影區(qū)域記為SKIPIF1<0。由于SKIPIF1<0取后側(cè),故SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0注意將對(duì)坐標(biāo)的曲面積分投影到坐標(biāo)面上時(shí),不要忽視了SKIPIF1<0側(cè)。圖10-15例6計(jì)算曲面積分SKIPIF1<0，其中SKIPIF1<0是平面SKIPIF1<0所圍成的空間區(qū)域的整個(gè)邊界曲面的外側(cè)。圖10-15解：(方法一)將分片光滑曲面SKIPIF1<0化為若干片光滑曲面之和,即SKIPIF1<0,其中SKIPIF1<0；SKIPIF1<0；SKIPIF1<0；SKIPIF1<0,SKIPIF1<0。則在SKIPIF1<0上積分為0,而由輪換對(duì)稱性SKIPIF1<0，于是SKIPIF1<0SKIPIF1<0在SKIPIF1<0面上的投影記為SKIPIF1<0SKIPIF1<0，則SKIPIF1<0SKIPIF1<0所以SKIPIF1<0。(方法二)設(shè)SKIPIF1<0則SKIPIF1<0SKIPIF1<0在整個(gè)平面上有連續(xù)的一階偏導(dǎo)數(shù),由SKIPIF1<0所包圍的空間區(qū)域記為SKIPIF1<0,根據(jù)高斯公式SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0。注意如果認(rèn)為SKIPIF1<0,則是錯(cuò)誤的,因?yàn)樵谌胤e分中SKIPIF1<0。例7計(jì)算曲面積分SKIPIF1<0其中SKIPIF1<0是連續(xù)函數(shù),SKIPIF1<0是平面SKIPIF1<0在第四卦限部分的上側(cè)。圖10-16分析在被積函數(shù)中含有未知函數(shù)SKIPIF1<0,而根據(jù)已知條件不圖10-16能求出SKIPIF1<0,因此不能直接利用公式計(jì)算積分.雖然已知被積函數(shù)連續(xù),但沒(méi)有偏導(dǎo)數(shù)存在的條件,不能用高斯公式計(jì)算積分.在此題中,SKIPIF1<0上任意一點(diǎn)的法向量的方向余弦是常數(shù),化為對(duì)面積的曲面積分可以消去SKIPIF1<0。解：由于SKIPIF1<0取上側(cè),故SKIPIF1<0上任意一點(diǎn)的法向量SKIPIF1<0與SKIPIF1<0軸的夾角為銳角,其方向余弦為SKIPIF1<0于是SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0。例8SKIPIF1<0,其中SKIPIF1<0為有向曲面SKIPIF1<0,其法向量與SKIPIF1<0軸正向的夾角為銳角。分析直接利用公式計(jì)算這個(gè)積分,需要將SKIPIF1<0分別投影到SKIPIF1<0兩個(gè)坐標(biāo)面上,計(jì)算兩個(gè)二重積分,比較麻煩。雖然SKIPIF1<0不是閉曲面,不能直接用高斯公式,但可以通過(guò)添加輔助面化為沿封閉曲面的積分。圖10-17解：設(shè)SKIPIF1<0取下側(cè),SKIPIF1<0與SKIPIF1<0所包圍的空間區(qū)域SKIPIF1<0,SKIPIF1<0在SKIPIF1<0面上的投影為SKIPIF1<0。圖10-17SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0所以SKIPIF1<0SKIPIF1<0SKIPIF1<0。例9SKIPIF1<0,其中SKIPIF1<0閉曲面SKIPIF1<0包含原點(diǎn)且分片光滑,取其外側(cè)。解：設(shè)SKIPIF1<0是由SKIPIF1<0所圍成的空間區(qū)域,在SKIPIF1<0內(nèi)以原點(diǎn)為中心,作球面SKIPIF1<0,取其外側(cè)。SKIPIF1<0與SKIPIF1<0所圍成的閉區(qū)域記為SKIPIF1<0,SKIPIF1<0在SKIPIF1<0內(nèi)具有一階連續(xù)的偏導(dǎo)數(shù),由SKIPIF1<0SKIPIF1<0，SKIPIF1<0根據(jù)高斯公式,得SKIPIF1<0SKIPIF1<0SKIPIF1<0于是SKIPIF1<0SKIPIF1<0SKIPIF1<0由于在球面SKIPIF1<0上的任意點(diǎn)SKIPIF1<0的外法線向量的方向余弦為:SKIPIF1<0,所以SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0。常見(jiàn)錯(cuò)解設(shè)SKIPIF1<0是由SKIPIF1<0所圍成的空間區(qū)域,根據(jù)高斯公式SKIPIF1<0SKIPIF1<0SKIPIF1<0錯(cuò)誤原因因?yàn)镾KIPIF1<0及一階偏導(dǎo)數(shù)在SKIPIF1<0處無(wú)定義,不滿足高斯公式的條件,所以直接應(yīng)用高斯公式計(jì)算這個(gè)積分是錯(cuò)誤的。注意①由輪換對(duì)稱性,積分SKIPIF1<0，然后直接用公式計(jì)算該積分也較簡(jiǎn)單。②積分SKIPIF1<0,其中SKIPIF1<0為曲面SKIPIF1<0上在點(diǎn)SKIPIF1<0處的外法線向量,SKIPIF1<0為點(diǎn)SKIPIF1<0的向徑,是本題的另一種表達(dá)形式,這個(gè)積分也稱為高斯積分。例10計(jì)算SKIPIF1<0,其中SKIPIF1<0為下半球面SKIPIF1<0的上側(cè),a為大于零的常數(shù)。 分析本題可以根據(jù)公式分塊計(jì)算,也可以添加輔助平面SKIPIF1<0與SKIPIF1<0構(gòu)成封閉曲面，然后利用高斯公式計(jì)算。但應(yīng)注意,被積函數(shù)在（0,0）點(diǎn)沒(méi)有定義,所以應(yīng)先根據(jù)曲面積分的性質(zhì)處理,再添加輔助平面SKIPIF1<0。解：(方法一)將SKIPIF1<0代入被積函數(shù),SKIPIF1<0SKIPIF1<0分塊計(jì)算.分別設(shè)SKIPIF1<0、SKIPIF1<0是SKIPIF1<0在SKIPIF1<0及SKIPIF1<0面上的投影,即SKIPIF1<0:SKIPIF1<0;SKIPIF1<0:SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0所以,SKIPIF1<0。(方法二)設(shè)SKIPIF1<0,取其上側(cè),由高斯公式，得SKIPIF1<0SKIPIF1<0其中SKIPIF1<0為SKIPIF1<0所包圍的區(qū)域,SKIPIF1<0為SKIPIF1<0上的平面區(qū)域SKIPIF1<0，于是SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0。例11計(jì)算曲面積分SKIPIF1<0，其中Σ為錐面SKIPIF1<0介于平面SKIPIF1<0及SKIPIF1<