Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$} and let $M$ be an \reftext{def:oriented-smooth-manifold-boundary-2026a}{oriented smooth manifold with boundary} of dimension $n$ that is compact in the sense of \ref{def:smooth-manifold-with-boundary-2026a}, with chosen \reftext{def:oriented-smooth-atlas-manifold-boundary-2026a}{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 \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{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 \ref{thm:smooth-partition-unity-compact-manifold-boundary-2026a}. For each $i\in\{1,\dots,N\}$, let $\chi_i\omega$ denote the \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{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 \ref{def:continuous-n-form-support-euclidean-domain-2026a}.

2. The real number

with each summand an integral in the sense of \ref{def:integral-compactly-supported-n-form-euclidean-2026a}, 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 \ref{thm:smooth-partition-unity-compact-manifold-boundary-2026a}, then

The proof of claim 3 rests on the pullback invariance of the integral under orientation-preserving smooth diffeomorphisms, \ref{thm:pullback-invariance-integral-diffeomorphism-euclidean-2026a}, applied to the transition maps of the oriented atlas.