Green's Theorem

Green's Theorem is a vector identity that is equivalent to the curl theorem in two dimensions. It relates the line integral around a simple closed curve $\partial \Omega$ with the doubble integral over the plane region $\Omega \,$.

The theorem is named after the british mathematician George Green. It can be applied to variuos fields in physics, among others flow integrals.

Mathematical Statement

Let $\Omega \,$ be a region in $\mathbb {R} ^{2}$ with a positively oriented, piecewise smooth, simple closed boundary $\partial \Omega$. $f(x,y)$ and $g(x,y)$ are functions defined on a open region containing $\Omega \,$ and have continuous partial derivatives in that region. Then Green's Theorem states that

$\oint \limits _{\partial \Omega }(fdx+gdy)=\iint \limits _{\Omega }\left({\frac {\partial g}{\partial x}}-{\frac {\partial f}{\partial y}}\right)dxdy$

The theorem is equivalent to the curl theorem in the plane and can be written in a more compact form as

$\oint \limits _{\partial \Omega }\mathbf {F} \cdot d\mathbf {S} =\iint \limits _{\Omega }(\nabla \times \mathbf {F} )d\mathbf {A}$