Let \(M\) be an oriented smooth \(n\)-manifold with boundary, and let \(\omega\) be a compactly supported smooth \((n-1)\)-form on \(M\). Then
\begin{equation*} \int_{M} d\omega = \int_{\partial M} \omega, \end{equation*}where \(d\omega\) denotes the exterior derivative of \(\omega\). [@lee2013smooth_manifolds, Theorem 16.11]
Remarks
- The right hand side vanishes for \(\partial M = \emptyset\).