Green's Theorem

Definition

Let $F(x,y)=(M(x,y),N(x,y))$ ${\oint }_{C}Fdr={\oint }_C(Mdx+Ndy)={\iint }_R\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy$ where $C$ is a simple closed curve in a plane, and $R$ is the region bounded by $C$