Agreement with classical Gauss flux in Euclidean space

Divergence theorem for a volume form: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M^n$ be oriented with boundary, $n\ge 1$, let $\mu$ be a positive smooth volume form, and let $X$ be a compactly supported smooth vector field. Then ${\int }_M({\mathrm{div}}_{\mu }X)\mu ={\int }_{\mathrm{\partial }M}{j}^{\ast }({\iota }_X\mu ),$ with outward-normal-first orientation. For compact $M$ every smooth $X$ is allowed.

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.

The divergence theorem on an elementary solid region: Let $E$ be an elementary solid region with presentation $\mathrm{\Sigma }=((D_1,{\phi }_1),\dots ,(D_P,{\phi }_P))$ (def-elementary-solid-region) and let $F$ be a $C^1$ vector field on an open set containing $E$. Then ${\iiint }_E\mathrm{div}F={\iint }_{\mathrm{\partial }E}⟨F,n⟩,$ where the left side is the integral of $\mathrm{div}F$ over $E$ and the right side is the flux of $F$ over the presentation $\mathrm{\Sigma }$, that is ${\sum }_{j=1}^P{\int }_{D_j}⟨F({\phi }_j),{\phi }_{j,u}\times {\phi }_{j,v}⟩$. At every interior parameter point whose projection lies in the interior of the relevant base, the orientation in which that flux is taken is the outward one, by cor-every-face-of-an-elementary-solid-region-is-outward-oriented.

Unit normal fields, orientations, and flux through a regular surface patch: For a regular patch $(D,\phi )$, the parametrization induces on its interior the unit normal $N_{\phi }=\frac{{\phi }_u\times {\phi }_v}{\mathrm{\parallel }{\phi }_u\times {\phi }_v{\mathrm{\parallel }}_2}\mathrm{.}$ The denominator is positive there by regularity and thm-surface-area-density-is-cross-product-norm, and the vector is orthogonal to the tangent plane (def-tangent-plane-of-a-regular-surface-patch). Choosing $N_{\phi }$ rather than $-N_{\phi }$ is an orientation. For a continuous vector field $F$, the flux in the orientation induced by $\phi$ is ${\int }_D(F\circ \phi )\cdot ({\phi }_u\times {\phi }_v)$. This is the scalar Riemann integral of a continuous function on $D$ (def-surface-area-and-scalar-surface-integral-of-a-patch, def-euclidean-inner-product); replacing the orientation by its negative negates the integrand.

Divergence as an exterior derivative: For a positive volume form $\mu$ and smooth vector field $X$ on an oriented smooth $n$-manifold, $n\ge 1$, with boundary allowed, $d({\iota }_X\mu )=({\mathrm{div}}_{\mu }X)\mu$. No tangency assumption on $X$ at the boundary is needed.

Simple solid regions in a coordinate direction and their cyclic coordinate projection: A simple description gives a compact Jordan measurable solid $E$. In particular, the supplied simple descriptions in an elementary-solid presentation make $E$ compact and Jordan measurable in ${\mathbb{R}}^3$.

A bounded set in ${\mathbb{R}}^m$ is Jordan measurable iff its boundary is null, equivalently of content zero: A metric-bounded set $E\subseteq {\mathbb{R}}^m$ is Jordan measurable if and only if its boundary $\mathrm{\partial }E$ is null, equivalently has content zero.