GREEN'S THEOREM IN THE PLANE

Let $P,\ Q,\ \partial P/\partial y,\ \partial Q/\partial x$ be single-valued and continuous in a simply-connected region $\cal R$ bounded by a simple closed curve $C$ . Then
$\displaystyle \oint_C [Pdx + Qdy] = \underset{\cal R}{\iint}\left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)dxdy\cdots(22)$
where $\oint_C$ is used to emphasize that $C$ is closed and that it is described in the positive direction.

This theorem is also true for regions bounded by two or more closed curves (i.e. multiply-connected regions).