Compact-support Stokes on the upper half-space

Compact-support Stokes on Euclidean space: For $n\ge 1$ and $\eta \in {\mathrm{\Omega }}_c^{n-1}({\mathbb{R}}^n)$, with the standard orientation, ${\int }_{{\mathbb{R}}^n}d\eta =0$.

Induced boundary orientation: For an oriented manifold with boundary, orient $T_p\mathrm{\partial }M$ by the outward-normal-first rule: an outward vector first, followed by a positive boundary determinant, is a positive determinant of $T_{p}M$.

Integral of a compactly supported top form: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. For an oriented smooth manifold $M^n$, possibly with boundary, and $\omega \in {\mathrm{\Omega }}_c^n(M)$, choose a smooth partition $({\rho }_i)$ subordinate to connected interior or boundary charts $(U_i,{\varphi }_i)$. For $n\ge 1$ set ${\int }_M\omega ={\sum }_i{I}_{{\varphi }_i}({\rho }_i\omega )\mathrm{.}$ Each product has compact support in its chart and only finitely many are nonzero, by lem-a-locally-finite-sum-is-finite-near-the-compact-support-of-a-form. For $n=0$ set ${\int }_M\omega ={\sum }_{p\in \mathrm{supp}\omega }\epsilon (p)\omega (p)\mathrm{.}$ A zero-manifold is discrete; the singleton open cover of a compact subset has a finite subcover. Thus this sum too is finite. Empty support or empty $M$ gives zero. Independence of the choices is discharged by thm-global-form-integration-is-independent-of-the-atlas-partition-and-refinement.

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.