Independence of the Manifold Integral from Chart and Partition Choices

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 $\omega=(\omega_\alpha)_{\alpha\in A}$ be a smooth differential $n$-form on $M$.

Let $N\in\mathbb{N}$, $\alpha_1,\dots,\alpha_N\in A$, and let $\chi_1,\dots,\chi_N$ be a smooth partition of unity subordinate to $U_{\alpha_1},\dots,U_{\alpha_N}$ as in Smooth Partitions of Unity on a Compact Smooth Manifold with Boundary. For each $i\in\{1,\dots,N\}$, let $\chi_i\omega$ denote the smooth differential $n$-form on $M$ whose chart representative in each chart $(U_\alpha,\varphi_\alpha)$ assigns to $x\in\Omega_\alpha$ the pointwise scalar multiple of the alternating form $\omega_{\alpha,x}$ by the real number $\chi_{i,\alpha}(x)$, where $\chi_{i,\alpha}$ is the chart representative of $\chi_i$.

Then the following hold.

1. For each $i\in\{1,\dots,N\}$, the representative $(\chi_i\omega)_{\alpha_i}$ of $\chi_i\omega$ in the chart $(U_{\alpha_i},\varphi_{\alpha_i})$ is a continuous differential $n$-form on the admissible domain $\Omega_{\alpha_i}$ that is compactly supported in $\Omega_{\alpha_i}$, in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain.

2. The real number

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

with each summand an integral in the sense of Integral of a Compactly Supported Continuous n-Form on a Euclidean or Half-Space Domain, is well defined.

3. The value $S$ does not depend on the choices made: if $N'\in\mathbb{N}$, $\beta_1,\dots,\beta_{N'}\in A$, and $\chi_1',\dots,\chi_{N'}'$ form another smooth partition of unity subordinate to $U_{\beta_1},\dots,U_{\beta_{N'}}$ as in Smooth Partitions of Unity on a Compact Smooth Manifold with Boundary, then

$$\sum_{i=1}^{N}\int_{\Omega_{\alpha_i}}(\chi_i\omega)_{\alpha_i}=\sum_{j=1}^{N'}\int_{\Omega_{\beta_j}}(\chi_j'\omega)_{\beta_j}.$$

The proof of claim 3 rests on the pullback invariance of the integral under orientation-preserving smooth diffeomorphisms, Pullback Invariance of the Integral under Orientation-Preserving Smooth Diffeomorphisms, applied to the transition maps of the oriented atlas.