= Generalized Stokes theorem
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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.
For the counterclockwise unit-square boundary and $\omega=x\,dy$, the right-hand side is $1$, while the left-hand side is $\int_{[0,1]^2}dx\wedge dy=1$. The equation says that the boundary integral measures the total local circulation in the region. Reversing the <orientation of a smooth manifold> reverses both sides together.
In dimension one, the boundary of an interval consists of its endpoint with a plus sign and its starting point with a minus sign; the identity becomes $\int_a^b df=f(b)-f(a)$. In the plane it gives <Green theorem>. Applied to appropriate <differential forms> in three dimensions it gives both <Stokes theorem> and the <divergence theorem>. The boundary and the differential being integrated must belong to the stated domain: removing a singular point can make a spanning disc unavailable.
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