Agreement of general and classical surface Stokes

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.

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.

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 classical Stokes theorem for a $C^2$ patch over a finite elementary Green region: Let $(D,\phi )$ be a $C^2$ patch over a finite elementary Green region (def-the-induced-boundary-chain-of-a-c2-surface-patch), with positive boundary chain $\mathrm{\partial }D=({\sigma }_1,\dots ,{\sigma }_m)$ and induced boundary chain $\phi (\mathrm{\partial }D)$, and let $F$ be a $C^1$ vector field on an open set $U\subseteq {\mathbb{R}}^3$ containing $\phi [D]$. Then the circulation around the induced boundary chain equals the flux of the curl in the induced orientation: ${\oint }_{\phi (\mathrm{\partial }D)}F\cdot d\mathbf{r}={\int }_D⟨(\mathrm{curl}F)\circ \phi ,{\phi }_u\times {\phi }_v⟩\mathrm{.}$ The right-hand side is the flux of $\mathrm{curl}F$ through the patch in the orientation induced by $\phi$, in the sense of def-oriented-unit-normal-and-flux-of-a-surface-patch.

Divergence and curl of a $C^1$ vector field: Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open and let $F=(F_0,\dots ,F_{n-1}):U\to {\mathbb{R}}^n$ be $C^1$ in the componentwise Euclidean sense of def-ck-euclidean-maps-and-diffeomorphisms. Then the divergence of $F$ is $\mathrm{div}F:={\sum }_{i<n}{\mathrm{\partial }}_i{F}_i$, the function $U\to \mathbb{R}$ whose value at $p$ is ${\sum }_{i<n}{\mathrm{\partial }}_i{F}_i(p)$. The partial derivatives are those of def-directional-and-partial-derivatives, and the sum is the finite sum used throughout def-euclidean-inner-product. Since each ${\mathrm{\partial }}_i{F}_i$ is continuous on $U$, so is $\mathrm{div}F$. Now let $n=3$ and let $F:U\to {\mathbb{R}}^3$ be $C^1$ on an open $U\subseteq {\mathbb{R}}^3$. Following def-cross-product-in-r3, write the three coordinates of a point and of a vector as $x,y,z$ rather than $0,1,2$, so that $F=(F_x,F_y,F_z)$ means $F=(F_0,F_1,F_2)$ and ${\mathrm{\partial }}_x,{\mathrm{\partial }}_y,{\mathrm{\partial }}_z$ are ${\mathrm{\partial }}_0,{\mathrm{\partial }}_1,{\mathrm{\partial }}_2$. With that naming, the curl of $F$ is $\mathrm{curl}F:=({\mathrm{\partial }}_y{F}_z-{\mathrm{\partial }}_z{F}_y,{\mathrm{\partial }}_z{F}_x-{\mathrm{\partial }}_x{F}_z,{\mathrm{\partial }}_x{F}_y-{\mathrm{\partial }}_y{F}_x)$, a map $U\to {\mathbb{R}}^3$ each of whose coordinates is continuous on $U$. In this naming the divergence reads $\mathrm{div}F={\mathrm{\partial }}_x{F}_x+{\mathrm{\partial }}_y{F}_y+{\mathrm{\partial }}_z{F}_z$. Both operators are defined pointwise from the first partial derivatives of the components, so no differentiability of $F$ beyond $C^1$ is used and no orientation or metric structure enters beyond the standard coordinates of def-jacobian-matrix-and-gradient. For a $C^1$ scalar function $f$ on $U$, the gradient $\mathrm{\nabla }f=({\mathrm{\partial }}_{0}f,\dots ,{\mathrm{\partial }}_{n-1}f)$ is that of def-jacobian-matrix-and-gradient; in the three-coordinate naming, $\mathrm{\nabla }f=({\mathrm{\partial }}_{x}f,{\mathrm{\partial }}_{y}f,{\mathrm{\partial }}_{z}f)$.