Localization of Stokes by a partition of unity

Form calculus extends locally across a manifold boundary: On smooth manifolds with boundary, the coordinate exterior derivative, pullback naturality, graded Leibniz rule, support containment, and Cartan identity hold for smooth forms: $d(F^{\ast }\alpha )=F^{\ast }(d\alpha ),d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1{)}^{\mathrm{deg}\alpha }\alpha \wedge d\beta ,$ $\mathrm{supp}d\alpha \subseteq \mathrm{supp}\alpha ,{\mathcal{L}}_X\alpha =d({\iota }_X\alpha )+{\iota }_{X}d\alpha \mathrm{.}$ For arbitrary smooth vector fields at boundary points, ${\mathcal{L}}_X$ is defined by local Euclidean extensions; a two-sided flow inside the manifold is not required.

Local finiteness near compact support: If $(C_i{)}_{i\in I}$ is a locally finite family of closed subsets of a manifold and $K$ is compact, only finitely many $C_i$ meet $K$. There is an open neighborhood of $K$ disjoint from all the other $C_i$. In particular, for a smooth partition of unity $({\rho }_i)$ and $\omega \in {\mathrm{\Omega }}_c^k(M)$, only finitely many ${\rho }_i\omega$ are nonzero.

Linearity and additivity of the form integral: For compactly supported smooth top forms $\omega ,\eta$ on an oriented $M^n$ and $a,b\in \mathbb{R}$, ${\int }_M(a\omega +b\eta )=a{\int }_M\omega +b{\int }_M\eta \mathrm{.}$ Also ${\int }_M\omega ={\sum }_C{\int }_C\omega {\mathrm{\mid }}_C$, where $C$ ranges over connected components with their restricted orientations; only finitely many meet $\mathrm{supp}\omega$.

Smooth partitions of unity exist on manifolds with boundary: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Every open cover of a smooth manifold with boundary admits a smooth partition of unity subordinate to it.

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 $n=0$, $\mathrm{\partial }M=\mathrm{\varnothing }$.