Stokes theorem

Overview

Stokes theorem is one of the most important theorems in differential geometry. It relates integral of differential forms that are exact on a manifold to what the integral does on the boundary. It can be seen as a generalization of the fundamental theorem of calculus.

Statement

Let be a smooth -manifold with boundary, and a compactly supported differential form on . Then

Where denotes the exterior derivative of .

Note if we are being careful on the right-hand side, the differential form should technically be where is the inclusion to the boundary.