Green circulation and flux on a disk

General Stokes agrees with both planar Green formulas: For a compact smooth planar region $D$ oriented by $dx\wedge dy$ and smooth $P,Q$ on a neighborhood, general Stokes gives ${\int }_{\mathrm{\partial }D}(Pdx+Qdy)={\int }_D(Q_x-P_y)dxdy,$ ${\int }_{\mathrm{\partial }D}(Pdy-Qdx)={\int }_D(P_x+Q_y)dxdy\mathrm{.}$ When $D$ also has the supplied finite elementary Green decomposition required by the classical results, these are exactly their circulation and outward-flux formulas. Outer boundary curves run counterclockwise and holes clockwise.

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.