Let \(M\) be an oriented smooth \(n\)-manifold with or without boundary, and let \(\omega\) be a compactly supported \(n\)-form on \(M\). Suppose \(D_1,\ldots,D_k\) are open domains in \(\mathbb{R}^n\) and \(F_i\colon \hat{D_i}\to t\) smooth maps satisfying
- \(F_i\) restricts to an orientation-preserving diffeomorphism form \(D_i\) onto an open subset \(W_i\subset M\);
- \(W_i\cap W_j=\emptyset \) if \(i\neq j\);
- \(\supp \omega = \hat{W_1}\cap \cdots \hat{W_k}\).
Then
\begin{equation*} \int_{M} \omega=\sum_{i=1}^{k} \int_{D_i} F_i^*\omega. \end{equation*}[@lee2013smooth_manifolds, Proposition 16.8]
Remark
- Using this proposition we may omit null sets. Therefore integrating over \(D=(0,\pi)\times (0,2\pi)\) as a parametrization of the \(\mathbb{S}^2\) sphere is sufficient, although a line is not contained in the integration domain.