= Generalized Stokes theorem
{c}
{wiki=Generalized_Stokes_theorem}
If $X$ is an oriented $n$-manifold with boundary, $F:\partial X\hookrightarrow X$ is the inclusion, and $\omega$ is a compactly supported $(n-1)$-form, then
$$
\int_Xd\omega=\int_{\partial X}F^*\omega,
$$
where the boundary has the <outward-normal-first boundary orientation>. A subordinate <partition of unity> reduces the theorem to the fundamental theorem of calculus in oriented coordinate half-spaces; local finiteness and compact support ensure that only finitely many terms contribute.
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