Volume-form divergence on the Euclidean ball

Agreement with classical Gauss flux in Euclidean space: For $\mu =dx\wedge dy\wedge dz$ and a smooth Euclidean field $F$, the volume-form divergence is ${\mathrm{\partial }}_x{F}_x+{\mathrm{\partial }}_y{F}_y+{\mathrm{\partial }}_z{F}_z$. For a surface parametrization $r(u,v)$, $r^{\ast }({\iota }_F\mu )=(F(r)\cdot (r_u\times r_v))du\wedge dv\mathrm{.}$ Consequently the volume-form divergence theorem agrees with the classical Gauss flux theorem on compact smooth regions that also admit the supplied elementary-solid presentation required by that classical theorem.

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.