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 $\omega$ be a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{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 \ref{thm:smooth-partition-unity-compact-manifold-boundary-2026a}. The \textbf{integral of $\omega$ over $M$} is defined by

where $\chi_i\omega$ and its chart representative $(\chi_i\omega)_{\alpha_i}$ are as in \ref{thm:integral-manifold-independence-choices-2026a}, and each summand is an integral of a compactly supported continuous $n$-form in the sense of \ref{def:integral-compactly-supported-n-form-euclidean-2026a}. By \ref{thm:integral-manifold-independence-choices-2026a}, 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 \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} of a compact oriented $M$, then $\partial M$, with the smooth structure from \ref{thm:boundary-smooth-manifold-structure-2026a} and the \reftext{def:induced-orientation-boundary-manifold-2026a}{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 $\partial M$.