Surface Stokes on a graph disk

Agreement of general and classical surface Stokes: Let $S\subset {\mathbb{R}}^3$ be a compact oriented smooth embedded surface with boundary, and let $F$ be smooth on an open neighborhood of $S$. Set $\alpha =F_{x}dx+F_{y}dy+F_{z}dz$ and $\mu =dx\wedge dy\wedge dz$. Then $d\alpha ={\iota }_{\mathrm{curl}F}\mu ,{\int }_{\mathrm{\partial }S}\alpha ={\int }_S{\iota }_{\mathrm{curl}F}\mu \mathrm{.}$ On an oriented parametrization $r(u,v)$ the latter integrand is $(\mathrm{curl}F)(r)\cdot (r_u\times r_v)du\wedge dv$; on a boundary curve it is $F(r)\cdot r^{\mathrm{\prime }}dt$. On the common smooth patch scope this is the published classical Stokes theorem, using the standard Euclidean metric identification.

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.