Green's Theorem

Theorem
Let $$\Gamma \ $$ be a positively oriented piecewise smooth simple closed curve in $$\R^2 \ $$, and let $$U = \text{ int}(\Gamma) \ $$. If $$A \ $$ and $$B \ $$ are functions of $$(x,y) \ $$ defined on an open region containing $$U \ $$ and have continuous partial derivatives in such a set, then


 * $$\oint_{\Gamma} (A\, dx + B\, dy) = \iint_{U} \left(\frac{\partial B}{\partial x} - \frac{\partial A}{\partial y}\right)\ dA \ $$