# 格林公式

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

## 定理

$\iint\limits_{D}(\frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y})dxdy=\oint_{L^{+}}(Pdx+Qdy)$

## D 为一个简单区域时的证明

$\int_{C} L\, dx = \iint_{D} \left(- \frac{\partial L}{\partial y}\right)\, dA\qquad\mathrm{(1)}$

$\int_{C} M\, dy = \iint_{D} \left(\frac{\partial M}{\partial x}\right)\, dA\qquad\mathrm{(2)}$

$D = \{(x,y)|a\le x\le b, g_1(x) \le y \le g_2(x)\}$

 $\iint_{D} \left(\frac{\partial L}{\partial y}\right)\, dA$ $=\int_a^b\!\!\int_{g_1(x)}^{g_2(x)} \left[\frac{\partial L (x,y)}{\partial y}\, dy\, dx \right]$ $= \int_a^b \Big\{L(x,g_2(x)) - L(x,g_1(x)) \Big\} \, dx\qquad\mathrm{(3)}$

$\int_{C_1} L(x,y)\, dx = \int_a^b \Big\{L(x,g_1(x))\Big\}\, dx$

$\int_{C_3} L(x,y)\, dx = -\int_{-C_3} L(x,y)\, dx = - \int_a^b [L(x,g_2(x))]\, dx$

$\int_{C_4} L(x,y)\, dx = \int_{C_2} L(x,y)\, dx = 0$

 $\int_{C} L\, dx$ $= \int_{C_1} L(x,y)\, dx + \int_{C_2} L(x,y)\, dx + \int_{C_3} L(x,y)\, dx + \int_{C_4} L(x,y)\, dx$ $= -\int_a^b [L(x,g_2(x))]\, dx + \int_a^b [L(x,g_1(x))]\, dx\qquad\mathrm{(4)}$

(3)和(4)相加，便得到(1)。类似地，也可以得到(2)。

## 参考文献

1. ^ George Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Nottingham, England: T. Wheelhouse, 1828). Green did not actually derive the form of "Green's theorem" which appears in this article; rather, he derived a form of the "divergence theorem", which appears on pages 10-12 of his Essay.
In 1846, the form of "Green's theorem" which appears in this article was first published, without proof, in an article by Augustin Cauchy: A. Cauchy (1846) "Sur les intégrales qui s'étendent à tous les points d'une courbe fermée" (On integrals that extend over all of the points of a closed curve), Comptes rendus, 23: 251-255. (The equation appears at the bottom of page 254, where (S) denotes the line integral of a function k along the curve s that encloses the area S.)
A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his inaugural dissertation: Bernhard Riemann (1851) Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (Basis for a general theory of functions of a variable complex quantity), (Göttingen, (Germany): Adalbert Rente, 1867); see pages 8 - 9.
2. ^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3
3. ^ Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7