A nonzero period obstructs exactness and bounding

A compactly supported primitive has zero total derivative integral: If $M^n$ is oriented and boundaryless, $n\ge 1$, and $\eta \in {\mathrm{\Omega }}_c^{n-1}(M)$, then ${\int }_{M}d\eta =0$. In particular, on a compact such manifold every exact smooth top form has zero integral. The compact-support assumption is on the primitive $\eta$, not merely on $d\eta$.

Closed forms have zero boundary integral: For oriented $M^n$ with boundary, $n\ge 1$, if $\eta \in {\mathrm{\Omega }}_c^{n-1}(M)$ is closed, then ${\int }_{\mathrm{\partial }M}{j}^{\ast }\eta =0$. When $M$ is compact, no separate support assumption on the smooth closed form is needed.

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.

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.