# 格林公式

## 定理

$\iint \limits _{D}\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)\mathrm {d} x\mathrm {d} y=\oint _{L^{+}}(P\mathrm {d} x+Q\mathrm {d} y)$ ## D 为一个简单区域时的证明

$\int _{C}L\,dx=\iint _{D}\left(-{\frac {\partial L}{\partial y}}\right)\,dA\qquad \mathrm {(1)}$ $\int _{C}M\,dy=\iint _{D}\left({\frac {\partial M}{\partial x}}\right)\,dA\qquad \mathrm {(2)}$ $D=\{(x,y)|a\leq x\leq b,g_{1}(x)\leq y\leq g_{2}(x)\}$ $\iint _{D}\left({\frac {\partial L}{\partial y}}\right)\,dA$ $=\int _{a}^{b}\!\!\int _{g_{1}(x)}^{g_{2}(x)}\left[{\frac {\partial L(x,y)}{\partial y}}\,dy\,dx\right]$ $=\int _{a}^{b}{\Big \{}L(x,g_{2}(x))-L(x,g_{1}(x)){\Big \}}\,dx\qquad \mathrm {(3)}$ $\int _{C_{1}}L(x,y)\,dx=\int _{a}^{b}{\Big \{}L(x,g_{1}(x)){\Big \}}\,dx$ $\int _{C_{3}}L(x,y)\,dx=-\int _{-C_{3}}L(x,y)\,dx=-\int _{a}^{b}[L(x,g_{2}(x))]\,dx$ $\int _{C_{4}}L(x,y)\,dx=\int _{C_{2}}L(x,y)\,dx=0$ $\int _{C}L\,dx$ $=\int _{C_{1}}L(x,y)\,dx+\int _{C_{2}}L(x,y)\,dx+\int _{C_{3}}L(x,y)\,dx+\int _{C_{4}}L(x,y)\,dx$ $=-\int _{a}^{b}[L(x,g_{2}(x))]\,dx+\int _{a}^{b}[L(x,g_{1}(x))]\,dx\qquad \mathrm {(4)}$ (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页面存档备份，存于互联网档案馆） 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页面存档备份，存于互联网档案馆） (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
4. Stewart, James. Calculus 6th. Thomson, Brooks/Cole.