The general Stokes theorem

Compact-support Stokes on the upper half-space: Give $H^n=\{x_n\ge 0\}$ the standard orientation, $n\ge 1$, and its face the outward-normal-first orientation. If $\eta \in {\mathrm{\Omega }}_c^{n-1}(H^n)$ and $j:\mathrm{\partial }{H}^n\hookrightarrow H^n$, then ${\int }_{H^n}d\eta ={\int }_{\mathrm{\partial }{H}^n}{j}^{\ast }\eta \mathrm{.}$ With $\eta ={\sum }_i{a}_{i}d{x}^1\wedge \cdots \wedge \widehat{d{x}^i}\wedge \cdots \wedge d{x}^n$, both sides are $(-1{)}^n{\int }_{{\mathbb{R}}^{n-1}}{a}_n(x^{\mathrm{\prime }},0)d{x}^{\mathrm{\prime }}$ for $n>1$, and $-a_1(0)$ for $n=1$.

Localization of Stokes by a partition of unity: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M^n$ be oriented with boundary, $n\ge 1$, $\eta \in {\mathrm{\Omega }}_c^{n-1}(M)$, and $({\rho }_i)$ a smooth chart partition. Then $\eta ={\sum }_i{\rho }_i\eta ,d\eta ={\sum }_{i}d({\rho }_i\eta ),{\sum }_{i}d{\rho }_i\wedge \eta =0,$ with only finitely many nonzero form summands. Boundary restrictions have the corresponding finite localization and compact support, so these identities can be integrated termwise.

Change of variables on oriented manifolds: Let $F:M\to N$ be a diffeomorphism of oriented smooth $n$-manifolds and $\omega \in {\mathrm{\Omega }}_c^n(N)$. If $F$ preserves orientation everywhere, ${\int }_M{F}^{\ast }\omega ={\int }_N\omega$; if it reverses orientation everywhere, ${\int }_M{F}^{\ast }\omega =-{\int }_N\omega$. If the sign varies between components, apply the appropriate signed equality on each component and add.

Integration on an oriented embedded submanifold: Let $j:S\hookrightarrow M$ be an oriented embedded smooth $k$-submanifold, with boundary allowed. For a smooth $k$-form $\omega$ on $M$ such that $j^{\ast }\omega$ has compact support on $S$, define ${\int }_S\omega :={\int }_S{j}^{\ast }\omega$. If $F:T\to S$ is an orientation-preserving diffeomorphism, this equals ${\int }_T(jF{)}^{\ast }\omega$. Compact support is required on $S$ itself.

The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold: If $M$ has dimension $n\ge 1$, the restrictions of boundary charts to their faces give $\mathrm{\partial }M$ the structure of a closed embedded smooth boundaryless $(n-1)$-manifold. For $n=0$, $\mathrm{\partial }M=\mathrm{\varnothing }$.

Compact-support Stokes on Euclidean space: The integral of the exterior derivative of a compactly supported $(n-1)$-form on standard oriented ${\mathbb{R}}^n$ is zero.