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第十六章 第二型曲线积分与第二型曲面积分
§1 第二型曲线积分
我们已经熟悉了“对弧长”的曲线积分——第一型曲线积分.这里再来讨论“对坐标”的曲线积分——第二型曲线积分.
l. a定义与性质
一条参数曲线
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0050_0340.jpg?sign=1734425262-6zxAjecQkz0RrGvvkceSOFPbbOsqxmyH-0-f2de134faa8f4c6764c3ca4209fc568a)
总是可以定向的.例如我们可以选择参数t增加的方向为曲线的正方向.指定了正方向的一条曲线被称为有向曲线.
设在空间某区域Ω中有一个力场
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0050_0341.jpg?sign=1734425262-oiyTMZPn4e1b8fPZzGFoilco6pF4qB2n-0-02277c92dfc8efac212e1001355f2b49)
设有一个单位质量的质点在这力场中沿一条曲线γ从A点移动到b点.我们来考查力场对这质点所做的功.请注意,在这样的问题中,应该把γ看作是从A到B的有向曲线.因为沿同一条曲线,从B移动到A所做的功,与从A移动到B所做的功,一般是不同的(符号正好相反).
设曲线γ的参数方程为
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0050_0342.jpg?sign=1734425262-GpMSeCM44cT1OXii20VFfmJnqwV1TlMV-0-b44a7164c09dfb412c812e19eeb4842f)
给参数区间一个分割
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0050_0343.jpg?sign=1734425262-B8L0MbuXeAazabbbAi2Z0F0JwjgIrD68-0-2aca0687fc06e432a5c6c1c40c00622e)
于是曲线γ被分成n小段.在第j小段上,力场对质点所做的功可以近似地表示为
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0051_0344.jpg?sign=1734425262-9QqiZ1BMAOQUy4GbDbpyRpgAJlunYj79-0-e939bd4d7011156a6a771c9f2a124a5e)
这里
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0051_0345.jpg?sign=1734425262-MhHh5s9UvXRPGByCJzp1v0ZJ0emeGxET-0-068fe30b191407be8d998c2736e60795)
于是,力场对这质点所做的功可以近似地表示为:
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0051_0346.jpg?sign=1734425262-1EdCwMELrblt16mIwjOcPWcnHXd1mlAD-0-3034b16daf210286a5e9f0f3e9acaf1b)
当|π|→0时,上式的极限就应是所求的功W:
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0051_0347.jpg?sign=1734425262-81wEuxefAnw3lQwHPEswTh0o3GOKLbFM-0-f651118a7f2c78625a2af00cffa96d06)
设P(x, y,z),Q(x, y,z)和R(x, y,z)是F(x, y,z)在三个坐标轴方向的分量,则(1.1)式又可以写成以下形式:
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0051_0348.jpg?sign=1734425262-m3bq9Ww3jPudsYorQqNTwJMqEFoUdQ7E-0-d60696f9230913d06d40db9b8af28f2c)
从以上讨论得到启发,引出了第二型曲线积分的定义.
设γ是一条连续参数曲线
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0051_0349.jpg?sign=1734425262-T7gRzVYG26kckOzSsImRtemJOhf3cgJe-0-654844e089b75e575350f97cd6dc70a2)
为确定起见,我们假定参数增加方向为曲线的正方向.
定义 设γ是如上所述的一条有向连续曲线,P(M)=P(x, y,z)是在γ上连续的一个数值函数.给曲线γ的参数区间[α,β]任意一个分割
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0051_0350.jpg?sign=1734425262-xbo8GDpurQ69B1bjaJRZ5Tdv3ReLvbVm-0-8f5fa544a224db31bd6bc2b90af8a2cb)
于是γ被剖分为曲线段
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0051_0351.jpg?sign=1734425262-PPx4dP9wL3NDcOYAwKdxDdTUfTpjOYhi-0-c3bcd557ebc7abe86c94f61cf50a25b8)
这里
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0051_0352.jpg?sign=1734425262-Tv7fvVHAPGdlx0PjG1jA8LtujkICnunB-0-e53649c20f550caf317b4aca4e1a0d50)
在每一曲线段γj上任意选取一点
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0051_0353.jpg?sign=1734425262-5AidWWZEQlW4HCIA8OxbaFzGjRvVfkAJ-0-868e31ba49e47c5c612bc799f205afd3)
然后作和数
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0052_0354.jpg?sign=1734425262-xXneVwW2a2nEtpPmSxKNB8uS3qYNtSxV-0-daa3727f5453e8b4dc9ab0bf47f8f812)
当|π|→0时,和数(1.2)的极限(如果存在)就定义函数P沿有向曲线γ对x坐标的曲线积分,记为
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0052_0355.jpg?sign=1734425262-7DURHaXbTDSy8XDKcL7F7A7Tq4L08Cyk-0-2d61d02b4cfd9342ed24d508f4762ced)
用类似的方式,可以定义函数Q对y坐标的曲线积分和函数R对z坐标的曲线积分:
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0052_0356.jpg?sign=1734425262-57QPTeX1B9GiuUVl2HJLVDKMdgnYJgw9-0-f79d87f2fd1a9873db2f604ce65721ed)
以上这些对坐标的曲线积分,统统被称为第二型曲线积分.我们还约定记
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0052_0357.jpg?sign=1734425262-19EfDgBUwXoza3AqqvsvU8Sgif8rhqeV-0-11f978f33bec94d4044a88bf6cc471df)
这积分的向量式写法是
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0052_0358.jpg?sign=1734425262-mWlHyTHIfsuALgxsEKWQUdlPffgBNXJE-0-3c65494242dae9216b4a77b610278202)
其中
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0052_0359.jpg?sign=1734425262-xn8Sa6mxtYXzChkJvuAWNMJdvjrzuYJB-0-1062214eae0ca0386e4cb288fb6b8f48)
如果有向曲线γ的始端与终端相衔接,那么我们就说γ是一条闭有向曲线.对于沿闭有向曲线的积分,常常把积分号写作例如
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0052_0361.jpg?sign=1734425262-aQBczaEk0WB7AMVjfLIwqhudHhdS9cPq-0-c7d9102a10106834f6f30e3455bcd65f)
等等.
从定义容易看出,第二型曲线积分具有以下重要性质(假定各等式右端的积分存在):
1.线性
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0053_0362.jpg?sign=1734425262-fgEuhrRkIaQ9bLHKeqq7aGu0qdLdSJf0-0-eb5e313b058621b17021c948c1808e66)
——这里α和β是常数;
2.可加性
设γ1和γ2是两有向曲线,γ1的终端就是γ2的始端,我们用记号γ=γ1+γ2表示由γ1和γ2连接起来作成的有向曲线,则有
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0053_0363.jpg?sign=1734425262-BES1YSEoitvOLg0GEvh3fNL0Zv4hR8MW-0-0b27532a66882ef59a955f638ed00aa3)
3.有向性
如果用记号——γ表示由有向曲线γ反转定向而得到的有向曲线,那么就有
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0053_0364.jpg?sign=1734425262-Szlulmpw3R4MHOzREMIWir4vcCdmmJeg-0-9035eee65e98ce4a278aa3626f4affd6)
注记 平面曲线
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0053_0365.jpg?sign=1734425262-I904iWc4IimsKoG3kZ4O6YmFW4qKx6sB-0-af3526f681fa5985e43cd28e7111c2da)
可以看做空间曲线的特殊情形.沿这样的曲线显然有
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0053_0366.jpg?sign=1734425262-vPvS7QntgZahppUYLQOmUG7HkVpxmLO8-0-a40c3bf42b257496ea6225b57df1b115)
——因为沿这曲线因而,对于平面曲线γ,只须考虑以下形式的积分:
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0053_0367.jpg?sign=1734425262-Zo8gzR27Wl8bgH4z8T4DeDFgvk7uokgE-0-a3e9eeaf936b63299ef3f2fe9ac5cf98)
l. b第二型曲线积分的计算
设γ是一条连续可微的参数曲线,它的向量方程为
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0053_0368.jpg?sign=1734425262-al6qvi4xwp6CGBSnI4rA1zZdYKmh9P5C-0-27985c9ea5391466df83b59e49471e4a)
用分量表示,曲线γ的方程可以写成
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0054_0369.jpg?sign=1734425262-MqxTwOPppIOzspJu1fenDR0MObpjqDsI-0-c7635292e776f924961cd4e903b9837d)
为确定起见,我们假定γ以参数增加的方向为正方向.
定理 设γ是如上所述的一条有向曲线,P, Q和R是在γ上连续的函数.则有
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0054_0370.jpg?sign=1734425262-AiQGWdmSN4CdkJI6rADUY6YXblhtbvsS-0-3866bf875911f3b1248a281c84098a88)
证明 因为x'(t)在闭区间[α,β]上有界,可设
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0054_0371.jpg?sign=1734425262-GkwlhTOPOpR7meBS07is4OdqvkBftzQD-0-5f7c5f49e9289e21ae6d948846b6b23d)
又因为复合函数P(x(t),y(t),z(t))在闭区间[α,β]一致连续,所以对任何ε>0,存在δ>0,使得只要
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0054_0372.jpg?sign=1734425262-LJ3NJ7kJMtbf8hYFRVLYS4FrmWZOmWKl-0-7c1f10f84d75a6795267973c53fe2a44)
就有
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0054_0373.jpg?sign=1734425262-crXQbjNZd7lkA96biYikbTuiGeEjruCq-0-ad08bbdd3454b5b55f1aeb8807ed3f0e)
对于[α,β]的分割
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0054_0374.jpg?sign=1734425262-WMoMUTSTqUfT5k7j7Y5x7iW9hT9paPMh-0-e03b2385e4291231856e848a48157dc9)
和任意选取的
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0054_0375.jpg?sign=1734425262-dosHqKmPLUNUcPAxJOgBJr0nbtpp8mhH-0-207ff858fa297970208cbc486f23efbe)
只要
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0054_0376.jpg?sign=1734425262-4VTdqBUZqNTFj5UX4y57sh1i1PpOcRYi-0-d6b395a2754d6c26f0a2502b90b51c61)
就有
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0054_0377.jpg?sign=1734425262-aKpzEHUPM1FW8nkAfmPVqJBT8pbeJJ2h-0-ea45c95759e28190822f9aa651398db8)
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0055_0378.jpg?sign=1734425262-GSGvwHCrghhfqST5UGBpxCw6mmHwWqz6-0-e5a30eb2e2cbfb1c8037af382c4ad85a)
这证明了
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0055_0379.jpg?sign=1734425262-b9xKNzdlNTi6IOkBe2ohxtf9HrHS1lot-0-0c1e6c8c4f3e5e4a9d81a35e5896af89)
至于对y坐标的和对Z坐标的另外两个积分,可以用相同的办法处理.□
例1 设质量为m的质点沿任意连续曲线γ从空间位置A移动到位置B.试计算重力对这质点做的功W.
解 设在OXYZ直角坐标系中,OZ轴是竖直向上的.则功W可以表示为
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0055_0380.jpg?sign=1734425262-TOcJUusbp1o0vCFX9q4td54hMLerMJFp-0-5ebdfe9a1964cafe3d1c25f295f21464)
根据定义容易得到
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0055_0381.jpg?sign=1734425262-2RRrysT9Mq1WoORNPqImQzPXUST5SaUJ-0-bf80b8ee078b39cbdd49869d1874ff9c)
因而
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0055_0382.jpg?sign=1734425262-yg8eIl2PMmNdlC2ElI6lnvUKvuzp42B0-0-bb355960588ddced1859090e2545b1e2)
我们看到:重力场对质点所做的功,只与起点与终点的位置有关,与经过的路径无关.
例2 试计算
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0056_0383.jpg?sign=1734425262-VTMb2xE8is0LML1uBDDsiN4xwXdWDGJc-0-ef03ba0de2b2151e1a6a4829f9ade4bf)
这里C是OXY平面上中心在原点半径为a的圆周,E是以OX轴和OY轴为对称轴并且两半轴长度分别为a和b的椭圆周.
解 我们写出C的参数方程
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0056_0384.jpg?sign=1734425262-UuvSHOxpMByXeTiWPkXuHil10Hg2dDqu-0-67eaba3ed736cc0ffcec0621d4c52d63)
用上面定理中的公式进行计算得
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0056_0385.jpg?sign=1734425262-Jvh0x1QY2NEUXB1zEwE1Meg6bVj6qlFV-0-c20005e454888486dfa9a0002cebbdd1)
同样可得
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0056_0386.jpg?sign=1734425262-lvZE9r4na0ZMM6e1CU8YnntRyV2Q0pQN-0-dfbb6b48956ec5f355f2bee6c441d655)
在例2中,我们看到,对于γ=C或者γ=E的情形,积分
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0056_0387.jpg?sign=1734425262-QSinSqp7usZNpv7lBnaToCSaOR7dcGPq-0-f180fdb81275351ca9b74bc761bc3b75)
正好等于γ所围图形的面积.这一结论可以推广于很一般的情形,我们将在以后作进一步的讨论.
例3 试计算
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0056_0388.jpg?sign=1734425262-RD3P7flpeDuZm31SbpMQb9sUSvZYp2Di-0-68f5bf4346b6b7431b523b0d9885ced6)
这里C和E如例2中所述.
解 用参数表示进行计算得
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0056_0389.jpg?sign=1734425262-ryfndjt8UWbZTAWfthoZHYyqmbMAISZO-0-e2d34f04d102529328d3c3bc67d64de0)
同样可得
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0057_0390.jpg?sign=1734425262-OdBF9W3HzAe1J2zLQrtR6EI3ITkn01xg-0-059e484a2c1a9831e55ab2c9ad2807b4)
例4 试计算
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0057_0391.jpg?sign=1734425262-RFxlaJKjPzCdsfiX8WBxb9hhY3BYPRX2-0-a7269e94715ad49441d31b310e9fa4c1)
这里C同上两例中所述.
解 用参数表示进行计算可得
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0057_0392.jpg?sign=1734425262-Bpr7mzAHH7KpuzTz4WUifG4ZAL3dOF0H-0-9d808e5b60407597d498a581ebbc3fe2)
例5 试计算
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0057_0393.jpg?sign=1734425262-XTemUQ8AQhYVV7CKIoK00LEdVTJxrwGK-0-8d7d363517b821a5658085f5dedd58f2)
这里H是k圈螺旋线:
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0057_0394.jpg?sign=1734425262-uLW522RRatJxTUo2i3V3SkKh0k1rGmdn-0-ad91d96bee665dcad6bbbfb384b02b08)
解 我们有
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0057_0395.jpg?sign=1734425262-fPyaowTLFf6Az2ldMHULy7QNvxpkUhgU-0-43b278a9bc3ab6fd3820fd67cdbc5e8e)
l. c与第一型曲线积分的联系
考查连续可微曲线C:
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0057_0396.jpg?sign=1734425262-20OQ4kYM6ruJfdu5NklU3lptAwCRCKvA-0-19189622652b4bbb40fbd34ab445c089)
这里假设
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0057_0397.jpg?sign=1734425262-pFFMA0mgW99YaQvXs7VjCxN8Am7l51vG-0-e2d3aaf56801efdb2047fe6eeb6efae6)
我们约定以参数增加的方向为曲线C的正方向.于是,沿C正方向的切线单位向量为
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0057_0398.jpg?sign=1734425262-QfJ5V9PHYZ4mnIgsKMR7ZIq9bdKZoRac-0-78ab32f0f62ef1b93225f67c147b1403)
我们把这向量的分量cosα,cosβ,cosγ叫做有向曲线C的方向数:
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0058_0399.jpg?sign=1734425262-p87hSkOLLpTihDvkckRSHnWQrZDrbfEA-0-af665f1bc47c11effdf6861346e5b323)
设函数P(x, y,z),Q(x, y,z)和R(x, y,z)在曲线C上连续,则有
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0058_0400.jpg?sign=1734425262-c5AhDxgjsVOR4LIYuaPmOHPaLgbdSZRU-0-1f133b691c1397b3e3d574212c15e223)
这样,借助于方向数cosα,cosβ和cosγ,我们把第二型曲线积分形式上表示为第一型曲线积分
![](https://epubservercos.yuewen.com/F2E018/15279417505131406/epubprivate/OEBPS/Images/figure1_0058_0401.jpg?sign=1734425262-wua7TSpf3EReZJ0Lzf78uOw1Xh44IzPr-0-4d0a8714c541b70810c657f60bf0786b)
请注意,第二型曲线积分与第一型曲线积分相比较,有一个根本不同之处:第二型曲线积分是有向的,而第一型曲线积分是无向的.在上面的公式中,之所以能用第一型曲线积分表示第二型曲线积分,是因为在被积函数中引入了方向数——当曲线反转定向时,各方向数都改变符号.