高等数学-多元微积分-曲线积分与曲面积分习题

高等数学-多元微积分-曲线积分与曲面积分习题

对弧长的曲线积分

平面上对弧长的曲线积分

定积分法计算曲线积分

例1
image-20200908161927973
image-20200908161943734
例2
image-20200909080136016
image-20200909080148446
image-20200909080204906

本题也可以利用奇偶对称性消去积分中的\(2xy\)项，并把曲线表示为$x=t , y = t $的参数方程来做。

例3
image-20200909104603162
image-20200909104617370

奇偶性对称性计算曲线积分

例1
image-20200908170857630
image-20200908170909878

替换法计算曲线积分

例1
image-20200909081222333
image-20200909081239333

空间中对弧长的曲线积分

奇偶性对称性计算曲线积分

例1
image-20200909082742684
image-20200909082815350

对坐标的曲线积分

平面上对坐标的曲线积分

定积分法计算曲线积分

例1
image-20200908173101430
image-20200908173046204
例2
image-20200908183019048
image-20200908183030996
例3
image-20200909084520590
image-20200909084540456

此题也可以用（格林公式法）将平面上的曲线积分转换为二重积分的方法来做。

例4
image-20200909104958408
image-20200909105010744
例5
image-20200909105231456
image-20200909105249755

此题也可通过直观判断\(\left(x y^{2}+y\right) d x+\left(x^{2} y+x\right) d y = d(\frac{1}{2} x^2 y^2 +xy)\)，能写成全微分形式，得到该曲线积分与路径无关的结论，然后利用与路径无关的曲线积分计算方式来做。

例6
image-20200909112108399
image-20200909112057285
例7
image-20200909132732006
image-20200909132750325
image-20200909132803198

平面上曲线积分转换为二重积分（格林公式法）

例1
image-20200908174735224
image-20200908174805172
例2
image-20200909023610788
image-20200909023623355
image-20200909023636228
例3
image-20200909025047400
image-20200909025102565
例4
image-20200909074346379
image-20200909074356443
例5
image-20200909101412634
image-20200909101426175
image-20200909101443444
例6
image-20200909112841068
image-20200909112850410
image-20200909112903715
例7
image-20200909113826193
image-20200909113845372
例8
image-20200909115237702
image-20200909115251068
例9
image-20200909140859731
image-20200909140918590
image-20200909140940876
例10
image-20200909142744640
image-20200909142807551

计算与积分路径无关的曲线积分

例1
image-20200908175347683
image-20200908175406442
image-20200908175418177
例2
image-20200909032236871
image-20200909032251090
例3
image-20200909084914602
image-20200909084928338
例4
image-20200909100125200
image-20200909100137193
例5
image-20200909133819047
image-20200909133833288
例6
image-20200909135629890
image-20200909135702576

对面积的曲面积分

转换为二重积分法

例1
image-20200908182101058
image-20200908182050248
例2
image-20200909030202414
image-20200909030222779
例3
image-20200909030635094
image-20200909030702416
例4
image-20200909090022760
image-20200909090036343

提示：该平面写成截距式，易得在3个坐标轴上的截距分别为\(1,-\frac{1}{2} ,1\)

例5
image-20200909103702066
image-20200909103716653
例6
image-20200909144054963
image-20200909144110884
例7
image-20200909144534743
image-20200909144520686
例8
image-20200909145240912
image-20200909145301082
例9
image-20200909150143341
image-20200909150108497
image-20200909150122149
例10
image-20200909151954480
image-20200909152005537
例11
image-20200909153622789
image-20200909153636070

奇偶性对称性计算曲面积分

例1
image-20200909102748323
image-20200909102759017

提示：\(\iint_{\Sigma} z \mathrm{d} S = \iint_{\Sigma} \sqrt{1-x^2-y^2} \mathrm{d} S= 4 \iint_{\Sigma_1} \sqrt{1-x^2-y^2} \mathrm{d} S = 4 \iint_{\Sigma_1} z \mathrm{d} S = 4 \iint_{\Sigma_1} x \mathrm{d} S\)

对坐标的曲面积分

转化为二重积分法

例1
image-20200908213935545
image-20200908214009126
例2
image-20200909170854575
image-20200909170908903
image-20200909170919856

注意：\(\iint_{\Sigma} y^{2} \mathrm{d} z \mathrm{d} x = \iint_{\Sigma} y^{2} \cos\beta dS\)，显然\(\cos\beta\)是关于xOz是奇函数，可看出对称性

例3
image-20200909173318555
image-20200909173336222

转换为三重积分法（高斯公式/空间内域与界的关系）

例1
image-20200909032022005
image-20200909032038529
image-20200909032111867
例2
image-20200909154711784
image-20200909154722836
例3
image-20200909155041073
image-20200909155056096
例4 注意哈
image-20200909165410875
image-20200909165419290
image-20200909165432069
例5
image-20200909171604713
image-20200909171621318
例6
image-20200909171801259
image-20200909171816284
例7
image-20200909172253261
image-20200909172502013
image-20200909172516051
例8 嘿，脑细胞要死光了
image-20200909175733129
image-20200909175806000
image-20200909175826311
例9
image-20200909181514294
image-20200909181530845
例10
image-20200909203948135
image-20200909204004787
image-20200909204028731
例11
image-20200909210323065
image-20200909210342134
例12
image-20200909212045639
image-20200909212132309
例13
image-20200909212310868
image-20200909212323689
image-20200909212338770
例14
image-20200909213125688
image-20200909213143472
例15
image-20200909213457215
image-20200909213521231

转换为对面积的曲面积分

例1
image-20200909180943526
image-20200909181001097

物理应用

散度

例1
image-20200908214623383
image-20200908214644351

实际上\(\operatorname{div}(\operatorname{grad(f)})=\nabla \cdot(\nabla f)= \Delta f = \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2}\)，一般称为拉普拉斯算子

旋度

例1
image-20200909213649682
image-20200909213701616
image-20200909213709378