Green's theorem

Level AdvancedDifficulty ★★★★★Theorem⌖ Open in the map

What is it?

In the plane, the circulation around a closed curve equals the integral of the curl inside it. A computational gem: the area of a polygon from its vertices (the shoelace formula) is Green's theorem.

Statement

\oint_{\partial D}(P\,\dd x + Q\,\dd y) = \iint_D\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\dd A.

Formulas

$A = \frac12\oint (x\,\dd y - y\,\dd x) = \frac12\sum_{i}\big(x_i y_{i+1} - x_{i+1}y_i\big)$
shoelace formula for polygon area

Where it shows up in computing

Mesh processing (discrete differential geometry)★★★★★frequentComputer graphics
Polygon areas, centroids and orientation tests in GIS and graphics use the shoelace formula.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚛ Physics and simulation

Stokes' theorem→Electromagnetism (Maxwell's equations)★★★★★

3D Computer graphics

Mesh processing (discrete differential geometry)★★★★★

What depends on it

Stokes' theorem