Integral of a Smooth n-Form over a Compact Oriented Smooth Manifold with Boundary

Let $n\in$ $\mathbb{N}$ and let $M$ be an oriented smooth manifold with boundary of dimension $n$ that is compact in the sense of Smooth Atlas and Smooth Manifold with Boundary, with chosen oriented smooth atlas $((U_\alpha,\varphi_\alpha))_{\alpha\in A}$, and write $\Omega_\alpha=\varphi_\alpha(U_\alpha)$. Let $\omega$ be a smooth differential $n$-form on $M$.

Choose $N\in\mathbb{N}$, indices $\alpha_1,\dots,\alpha_N\in A$, and a smooth partition of unity $\chi_1,\dots,\chi_N$ subordinate to $U_{\alpha_1},\dots,U_{\alpha_N}$; such data exist by Smooth Partitions of Unity on a Compact Smooth Manifold with Boundary. The integral of $\omega$ over $M$ is defined by

$$\int_{M}\omega=\sum_{i=1}^{N}\int_{\Omega_{\alpha_i}}(\chi_i\omega)_{\alpha_i},$$

where $\chi_i\omega$ and its chart representative $(\chi_i\omega)_{\alpha_i}$ are as in Independence of the Manifold Integral from Chart and Partition Choices, and each summand is an integral of a compactly supported continuous $n$-form in the sense of Integral of a Compactly Supported Continuous n-Form on a Euclidean or Half-Space Domain. By Independence of the Manifold Integral from Chart and Partition Choices, the value of the sum does not depend on the choice of the indices or of the partition of unity, so the integral is well defined.

This definition applies in particular to integrals over the boundary: if $n\ge 2$ and $\partial M$ is the nonempty boundary of a compact oriented $M$, then $\partial M$, with the smooth structure from Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1 and the induced orientation, is itself a compact oriented smooth manifold with boundary (of dimension $n-1$, with empty boundary), and the present definition yields the integral over $\partial M$ of any smooth $(n-1)$-form on $\partial M$.