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

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

Applications

Area Calculation

Green's theorem is very useful when it comes to calculating the area of a region. If we take $f(x,y)=y$ and $g(x,y)=x$, the area of the region $\Omega \,$, with boundary $\partial \Omega$ can be calculated by

$A={\frac {1}{2}}\oint \limits _{\partial \Omega }xdy-ydx$

This formula gives a relationship between the area of a region and the line integral around its boundary.

If the curve is parametrisized as $\left(x(t),y(t)\right)$, the area formula becomes

$A={\frac {1}{2}}\oint \limits _{\partial \Omega }(xy'-x'y)dt$