---
tag: 0HR3N02317
kind: statement
role: theorem
name: Green's theorem
msc: [26B20]
uses: [0D032NSZKY, 0YS8RX87F4, 0NKBMFC18R, 05AXG3QRKR, 0HPHKCGEQ0, 0AKRKVHW94]
sources:
  - url: https://en.wikipedia.org/wiki/Green%27s_theorem
    locator: Theorem
  - url: https://en.wikipedia.org/wiki/Green%27s_theorem
    locator: Proof when D is a simple region
  - url: https://en.wikipedia.org/wiki/Green%27s_theorem
    locator: History
same_as:
  wikidata: Q321237
  wikipedia: Green's theorem
---

## Statement

Let $C$ be a positively oriented, piecewise smooth, simple [[closed curve|0D032NSZKY]] 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|05AXG3QRKR]] 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|0NKBMFC18R]] 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|09CBQB6V67]].
Green's theorem is the planar case of [[Stokes' theorem|0AKRKVHW94]]. 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.