Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$} with $n\ge 2$, and let $M$ be an \reftext{def:oriented-smooth-manifold-boundary-2026a}{oriented smooth manifold with boundary} of dimension $n$ that is compact as defined in \ref{def:smooth-manifold-with-boundary-2026a}.

Let $\omega$ be a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential $(n-1)$-form} on $M$ and let $d\omega$ denote the \reftext{def:exterior-derivative-smooth-form-manifold-boundary-2026a}{exterior derivative} of $\omega$.

Let $\partial M$ be the \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} of $M$, regarded as a compact oriented smooth manifold (without boundary) of dimension $n-1$ with the \reftext{def:induced-orientation-boundary-manifold-2026a}{induced boundary orientation}, and let $\iota^{*}\omega$ denote the \reftext{def:restriction-form-boundary-manifold-2026a}{restriction} of $\omega$ to $\partial M$ where $\iota:\partial M\to M$ is the inclusion map.

Then

where both sides are \reftext{def:integral-form-oriented-manifold-boundary-2026a}{integrals of smooth top-degree forms over compact oriented smooth manifolds}. If $\partial M=\varnothing$, then the right-hand side is understood to be $0$.