Green's theorem

Statement

Let $C$ be a positively oriented, piecewise smooth, simple closed curve in $\mathbb{R}^2$, and let $D$ be the bounded region enclosed by $C$. Let $L$ and $M$ be real functions with continuous partial derivatives on an open set that contains $D \cup C$. Then $$\oint_C (L \, dx + M \, dy) = \iint_D \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dA.$$

Remarks

Positive orientation means that $C$ goes counterclockwise, so $D$ is on the left. Proof sketch: first let $D$ be a region between the graphs of two functions of $x$. Write the double integral of $\partial L / \partial y$ as an iterated integral, and use the fundamental theorem of calculus. This gives $\oint_C L \, dx = -\iint_D \partial L / \partial y \, dA$. The $M$ term is similar, and general regions are cut into such pieces. Example: with $L = -y$ and $M = x$, the right side is $2$ times the area of D. Green's theorem is the planar case of Stokes' theorem. George Green stated a similar result in 1828, and Riemann gave a proof in 1851.

Explanations

L1 human

The theorem says that the total circulation of a plane field around a closed curve equals the sum of the small circulations inside. The small circulations of neighbouring squares cancel along shared edges, so only the outer boundary is left.