This tag is for questions about Green's theorem. Green's theorem gives the relationship between a line integral around a simple closed curve $C$ and a double integral over the plane region $D$ bounded by $C$.
Green's theorem is named after George Green and is the two-dimensional special case of the more general Kelvin–Stokes theorem.
Theorem:
Let $C$ be a positively oriented, piecewise smooth, simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $L$ and $M$ are functions of $(x, y)$ defined on an open region containing $D$ and have continuous partial derivatives there, then
$$\oint_{C} (L \\ \mathrm{dx} + M \\ \mathrm{dy}) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) \\ \mathrm{dx} \\ \mathrm{dy}$$ where the path of integration along $C$ is counterclockwise.
