微積分問(wèn)題課件

第3章

微積分問(wèn)題的計(jì)算機(jī)求解薛定宇，陳陽(yáng)泉著.高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解.北京：清華大學(xué)出版社，2004CAI課件開(kāi)發(fā)：劉瑩瑩，薛定宇8/23/20231高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解第3章

微積分問(wèn)題的計(jì)算機(jī)求解薛定宇，陳陽(yáng)泉著.高等應(yīng)主要內(nèi)容微積分問(wèn)題的解析解函數(shù)的級(jí)數(shù)展開(kāi)與級(jí)數(shù)求和問(wèn)題求解數(shù)值微分?jǐn)?shù)值積分問(wèn)題曲線積分與曲面積分的計(jì)算本章要點(diǎn)簡(jiǎn)介8/23/20232高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解主要內(nèi)容微積分問(wèn)題的解析解8/2/20232高等應(yīng)用數(shù)學(xué)問(wèn)題3.1微積分問(wèn)題的解析解3.1.1極限問(wèn)題的解析解3.1.2函數(shù)導(dǎo)數(shù)的解析解3.1.3積分問(wèn)題的解析解8/23/20233高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1微積分問(wèn)題的解析解3.1.1極限問(wèn)題的解析解8/23.1.1極限問(wèn)題的解析解

3.1.1.1單變量函數(shù)的極限8/23/20234高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1.1極限問(wèn)題的解析解

3.1.1.1單變量函數(shù)的極限【例3-1】試求解極限問(wèn)題

8/23/20235高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-1】試求解極限問(wèn)題

8/2/20235高等應(yīng)用數(shù)學(xué)問(wèn)【例3-2】求解單邊極限問(wèn)題

8/23/20236高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-2】求解單邊極限問(wèn)題

8/2/20236高等應(yīng)用數(shù)學(xué)3.1.1.2多變量函數(shù)的極限8/23/20237高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1.1.2多變量函數(shù)的極限8/2/20237高等應(yīng)用數(shù)【例3-3】求出二元函數(shù)極限值

8/23/20238高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-3】求出二元函數(shù)極限值

8/2/20238高等應(yīng)用3.1.2函數(shù)導(dǎo)數(shù)的解析解

3.1.2.1函數(shù)的導(dǎo)數(shù)和高階導(dǎo)數(shù)8/23/20239高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1.2函數(shù)導(dǎo)數(shù)的解析解

3.1.2.1函數(shù)的導(dǎo)數(shù)和高【例3-4】

8/23/202310高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-4】

8/2/202310高等應(yīng)用數(shù)學(xué)問(wèn)題的M8/23/202311高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202311高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1.2.2多元函數(shù)的偏導(dǎo)8/23/202312高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1.2.2多元函數(shù)的偏導(dǎo)8/2/202312高等應(yīng)用數(shù)【例3-5】8/23/202313高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-5】8/2/202313高等應(yīng)用數(shù)學(xué)問(wèn)題的MATL三維曲面：引力線：8/23/202314高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解三維曲面：引力線：8/2/202314高等應(yīng)用數(shù)學(xué)問(wèn)題的M【例3-6】

8/23/202315高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-6】

8/2/202315高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT3.1.2.3多元函數(shù)的Jacobi矩陣

8/23/202316高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1.2.3多元函數(shù)的Jacobi矩陣

8/2/2023X是自變量構(gòu)成的向量，Y是由各個(gè)函數(shù)構(gòu)成的向量。8/23/202317高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解X是自變量構(gòu)成的向量，8/2/202317高等應(yīng)用數(shù)學(xué)問(wèn)題的【例3-7】試推導(dǎo)其Jacobi矩陣8/23/202318高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-7】試推導(dǎo)其Jacobi矩陣8/2/2023183.1.2.4隱函數(shù)的偏導(dǎo)數(shù)

8/23/202319高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1.2.4隱函數(shù)的偏導(dǎo)數(shù)

8/2/202319高等應(yīng)用【例3-8】

8/23/202320高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-8】

8/2/202320高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT3.1.2.5參數(shù)方程的導(dǎo)數(shù)已知參數(shù)方程，求【例3-9】8/23/202321高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1.2.5參數(shù)方程的導(dǎo)數(shù)已知參數(shù)方程3.1.3積分問(wèn)題的解析解

3.1.3.1不定積分的推導(dǎo)

8/23/202322高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1.3積分問(wèn)題的解析解

3.1.3.1不定積分的推導(dǎo)【例3-10】

用diff()函數(shù)求其一階導(dǎo)數(shù)，再積分，

檢驗(yàn)是否可以得出一致的結(jié)果。

8/23/202323高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-10】

用diff()對(duì)原函數(shù)求4階導(dǎo)數(shù)，再對(duì)結(jié)果進(jìn)行4次積分

8/23/202324高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解對(duì)原函數(shù)求4階導(dǎo)數(shù)，再對(duì)結(jié)果進(jìn)行4次積分

8/2/202【例3-11】證明

8/23/202325高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-11】證明

8/2/202325高等應(yīng)用數(shù)學(xué)問(wèn)題的【例3-12】?jī)蓚€(gè)不可積問(wèn)題

的積分問(wèn)題求解。8/23/202326高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-12】?jī)蓚€(gè)不可積問(wèn)題3.1.3.2定積分與無(wú)窮積分計(jì)算

8/23/202327高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.1.3.2定積分與無(wú)窮積分計(jì)算

8/2/202327高【例3-13】8/23/202328高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-13】8/2/202328高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT【例3-14】8/23/202329高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-14】8/2/202329高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT【例3-15】

3.1.3.3多重積分問(wèn)題的MATLAB求解

8/23/202330高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-15】

3.1.3.3多重積分問(wèn)題的MATLAB求解

8/23/202331高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解

8/2/202331高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/23/202332高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202332高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-16】

8/23/202333高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-16】

8/2/202333高等應(yīng)用數(shù)學(xué)問(wèn)題的MA3.2函數(shù)的級(jí)數(shù)展開(kāi)與

級(jí)數(shù)求和問(wèn)題求解3.2.1Taylor冪級(jí)數(shù)展開(kāi)3.2.2Fourier級(jí)數(shù)展開(kāi)3.2.3級(jí)數(shù)求和的計(jì)算8/23/202334高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.2函數(shù)的級(jí)數(shù)展開(kāi)與

級(jí)數(shù)求和問(wèn)題求解3.2.13.2.1Taylor冪級(jí)數(shù)展開(kāi)

3.2.1.1單變量函數(shù)的Taylor

冪級(jí)數(shù)展開(kāi)

8/23/202335高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.2.1Taylor冪級(jí)數(shù)展開(kāi)

3.2.1.1單8/23/202336高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202336高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-17】

8/23/202337高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-17】

8/2/202337高等應(yīng)用數(shù)學(xué)問(wèn)題的MA8/23/202338高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202338高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.2.1.2多變量函數(shù)的Taylor

冪級(jí)數(shù)展開(kāi)

8/23/202339高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.2.1.2多變量函數(shù)的Taylor

冪8/23/202340高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202340高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-18】8/23/202341高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-18】8/2/202341高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT8/23/202342高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202342高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/23/202343高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202343高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.2.2Fourier級(jí)數(shù)展開(kāi)

8/23/202344高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.2.2Fourier級(jí)數(shù)展開(kāi)

8/2/202344高8/23/202345高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202345高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/23/202346高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202346高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-19】8/23/202347高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-19】8/2/202347高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT【例3-20】8/23/202348高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-20】8/2/202348高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT8/23/202349高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202349高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.2.3級(jí)數(shù)求和的計(jì)算

8/23/202350高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.2.3級(jí)數(shù)求和的計(jì)算

8/2/202350高等應(yīng)用數(shù)學(xué)【例3-21】計(jì)算

數(shù)值計(jì)算方法8/23/202351高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-21】計(jì)算

數(shù)值計(jì)算方法8/2/202351高等應(yīng)用【例3-22】試求解無(wú)窮級(jí)數(shù)的和

8/23/202352高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-22】試求解無(wú)窮級(jí)數(shù)的和

8/2/202352高等應(yīng)【例3-23】求解

8/23/202353高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-23】求解

8/2/202353高等應(yīng)用數(shù)學(xué)問(wèn)題的【例3-24】求解8/23/202354高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-24】求解8/2/202354高等應(yīng)用數(shù)學(xué)問(wèn)題的M3.3數(shù)值微分3.3.1數(shù)值微分算法3.3.2中心差分方法及其MATLAB實(shí)現(xiàn)3.3.3二元函數(shù)的梯度計(jì)算

8/23/202355高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.3數(shù)值微分3.3.1數(shù)值微分算法8/2/2023553.3.1數(shù)值微分算法

8/23/202356高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.3.1數(shù)值微分算法

8/2/202356高等應(yīng)用數(shù)學(xué)問(wèn)兩種中心差分：8/23/202357高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解兩種中心差分：8/2/202357高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT8/23/202358高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202358高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/23/202359高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202359高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.3.2中心差分方法及其MATLAB實(shí)現(xiàn)

8/23/202360高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.3.2中心差分方法及其MATLAB實(shí)現(xiàn)

8/2/28/23/202361高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202361高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-25】求導(dǎo)數(shù)的解析解,再用數(shù)值微分求取原函數(shù)的1～4階導(dǎo)數(shù)，并和解析解比較精度。8/23/202362高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-25】求導(dǎo)數(shù)的解析解,再用數(shù)值微分求取原函數(shù)的1～48/23/202363高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202363高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.3.3二元函數(shù)的梯度計(jì)算8/23/202364高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.3.3二元函數(shù)的梯度計(jì)算8/2/202364高等應(yīng)用數(shù)【例3-26】計(jì)算梯度，繪制引力線圖：8/23/202365高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-26】計(jì)算梯度，繪制引力線圖：8/2/202365高繪制誤差曲面:8/23/202366高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解繪制誤差曲面:8/2/202366高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT將網(wǎng)格加密一倍:8/23/202367高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解將網(wǎng)格加密一倍:8/2/202367高等應(yīng)用數(shù)學(xué)問(wèn)題的MA8/23/202368高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202368高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.4數(shù)值積分問(wèn)題3.4.1由給定數(shù)據(jù)進(jìn)行梯形求積3.4.2單變量數(shù)值積分問(wèn)題求解3.4.3雙重積分問(wèn)題的數(shù)值解3.4.4三重定積分的數(shù)值求解8/23/202369高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.4數(shù)值積分問(wèn)題3.4.1由給定數(shù)據(jù)進(jìn)行梯形求積8/23.4.1由給定數(shù)據(jù)進(jìn)行梯形求積8/23/202370高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.4.1由給定數(shù)據(jù)進(jìn)行梯形求積8/2/202370高等應(yīng)8/23/202371高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202371高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-27】8/23/202372高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-27】8/2/202372高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT【例3-28】畫(huà)圖：8/23/202373高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-28】畫(huà)圖：8/2/202373高等應(yīng)用數(shù)學(xué)問(wèn)題的求理論值：不同步距：8/23/202374高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解求理論值：不同步距：8/2/202374高等應(yīng)用數(shù)學(xué)問(wèn)題的3.4.2單變量數(shù)值積分問(wèn)題求解8/23/202375高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.4.2單變量數(shù)值積分問(wèn)題求解8/2/202375高等應(yīng)【例3-29】第三種：匿名函數(shù)(MATLAB7.0)第二種：inline函數(shù)第一種，一般函數(shù)方法8/23/202376高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-29】第三種：匿名函數(shù)(MATLAB7.0)第二種用inline函數(shù)定義:8/23/202377高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解用inline函數(shù)定義:8/2/202377高等應(yīng)用數(shù)學(xué)問(wèn)題【例3-30】提高求解精度。8/23/202378高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-30】提高求解精度。8/2/202378高等應(yīng)用數(shù)學(xué)【例3-31】求解繪制函數(shù):8/23/202379高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-31】求解繪制函數(shù):8/2/202379高等應(yīng)用數(shù)學(xué)8/23/202380高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202380高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-32】采用默認(rèn)精度人為給定精度限制8/23/202381高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-32】采用默認(rèn)精度人為給定精度限制8/2/202383.4.3雙重積分問(wèn)題的數(shù)值解8/23/202382高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.4.3雙重積分問(wèn)題的數(shù)值解8/2/202382高等應(yīng)用【例3-33】求解8/23/202383高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-33】求解8/2/202383高等應(yīng)用數(shù)學(xué)問(wèn)題的M比較8/23/202384高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解比較8/2/202384高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求8/23/202385高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解8/2/202385高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-34】8/23/202386高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-34】8/2/202386高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT解析解方法：高精度數(shù)值解8/23/202387高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解解析解方法：高精度數(shù)值解8/2/202387高等應(yīng)用數(shù)學(xué)問(wèn)題數(shù)值解求解積分問(wèn)題變成8/23/202388高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解數(shù)值解求解積分問(wèn)題變成8/2/202388高等應(yīng)用數(shù)學(xué)問(wèn)題的3.4.4三重定積分的數(shù)值求解8/23/202389高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.4.4三重定積分的數(shù)值求解8/2/202389高等應(yīng)用【例3-35】8/23/202390高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-35】8/2/202390高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT3.5曲線積分與曲面積分的計(jì)算3.5.1曲線積分及MATLAB求解3.5.2曲面積分與MATLAB語(yǔ)言求解8/23/202391高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.5曲線積分與曲面積分的計(jì)算3.5.1曲線積分及MAT3.5.1曲線積分及MATLAB求解

3.5.1.1第一類曲線積分8/23/202392高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.5.1曲線積分及MATLAB求解

3.5.1.1第一【例3-36】8/23/202393高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-36】8/2/202393高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT【例3-37】繪制曲線8/23/202394高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-37】繪制曲線8/2/202394高等應(yīng)用數(shù)學(xué)問(wèn)題的3.5.1.2第二類曲線積分8/23/202395高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.5.1.2第二類曲線積分8/2/202395高等應(yīng)用數(shù)【例3-38】8/23/202396高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-38】8/2/202396高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT【例3-39】8/23/202397高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解【例3-39】8/2/202397高等應(yīng)用數(shù)學(xué)問(wèn)題的MAT3.5.2曲面積分與MATLAB語(yǔ)言求解

3.5.2.1第一類曲面積分8/23/202398高等應(yīng)用數(shù)學(xué)問(wèn)題的MATLAB求解3.5.2曲面積分與MATLAB語(yǔ)言求解

3.5.2.1第【例3