An exact top form with nonzero integral on a disk

The general Stokes theorem: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M$ be an oriented smooth $n$-manifold with boundary, $n\ge 1$, and let $\eta \in {\mathrm{\Omega }}_c^{n-1}(M)$. With $j:\mathrm{\partial }M\hookrightarrow M$ and the outward-normal-first orientation, ${\int }_{M}d\eta ={\int }_{\mathrm{\partial }M}{j}^{\ast }\eta \mathrm{.}$ An empty boundary contributes zero; in dimension one its integral is a finite signed sum of point values.

Computing form integrals by finite parametrizations: Let $n\ge 1$, let $M^n$ be oriented, and let $\omega \in {\mathrm{\Omega }}_c^n(M)$. For $1\le i\le m$ let $D_i\subset {\mathbb{R}}^n$ be bounded open Jordan domains and $F_i:{\bar{D}}_i\to M$ continuous and smooth up to the boundary in target coordinates: near each parameter point, a target coordinate representative extends smoothly to a Euclidean neighborhood. Suppose $F_i{\mathrm{\mid }}_{D_i}$ is an orientation-preserving diffeomorphism onto an open $W_i\subset M$, the $W_i$ are pairwise disjoint, and $\mathrm{supp}\omega \subset {\bigcup }_i{\bar{W}}_i$. Then ${\int }_M\omega ={\sum }_{i=1}^m{\int }_{D_i}{F}_i^{\ast }\omega \mathrm{.}$ An empty family is allowed when the support is empty. No nonsingularity of $D{F}_i$ on $\mathrm{\partial }{D}_i$, and no $M$-valued extension across a genuine target boundary, is assumed.