The induced boundary chain and circulation of a $C^2$ patch over a finite elementary Green region

A $C^2$ patch over a finite elementary Green region is a regular parametrized surface patch $(D,\phi )$ in the sense of Regular parametrized surface patches on compact Jordan parameter regions whose parameter region $D$ is supplied, in addition, with a decomposition making it a finite elementary Green region in the sense of Type I, Type II, and elementary regions for Green's theorem, and whose parametrization $\phi$ is of class $C^2$ on an open neighbourhood of $D$ ($C^k$ Euclidean maps and diffeomorphisms). Both requirements on $D$ are part of the data: it is a compact Jordan parameter region, so it is the closure of its nonempty connected interior, and it carries a supplied elementary decomposition.

Let $\mathrm{\partial }D=({\sigma }_1,\dots ,{\sigma }_m)$ be the positive boundary chain of that decomposition, the finite list of oriented piecewise-$C^1$ arcs of Positive orientation of elementary-region boundaries. Then the induced boundary chain is the list of arcs obtained by composing the positive boundary chain of the parameter region with the parametrization, namely

$\phi (\mathrm{\partial }D):=(\phi \circ {\sigma }_1,\dots ,\phi \circ {\sigma }_m),$

each entry a piecewise-$C^1$ path in ${\mathbb{R}}^3$ in the sense of Reversal, concatenation, closed paths, and oriented piecewise-C1 reparametrizations. For a continuous vector field $F$ on a set containing $\phi [D]$, the circulation of $F$ around the induced boundary chain is the finite sum

${\oint }_{\phi (\mathrm{\partial }D)}F\cdot d\mathbf{r}:={\sum }_{l=1}^m{\int }_{\phi \circ {\sigma }_l}F\cdot d\mathbf{r},$

with the vector line integrals of Scalar line integrals with respect to arc length and vector-field line integrals. The value does not depend on the order of the list, a finite sum of reals being independent of its order.

If instead $F$ is defined on an open set $U$ containing $\phi [D]$, continuity of $\phi$ and compactness of $D$ give an open neighbourhood $V$ of $D$ in the domain of $\phi$ with $\phi [V]\subseteq U$. On $V$, the pulled-back functions are the inner products of the field along the parametrization with the two parameter derivatives:

$P^{\ast }:=⟨F\circ \phi ,{\phi }_u⟩,Q^{\ast }:=⟨F\circ \phi ,{\phi }_v⟩,$

with the inner product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. The oriented area vector ${\phi }_u\times {\phi }_v$ and the flux it computes are those of Unit normal fields, orientations, and flux through a regular surface patch. A merely continuous field on an arbitrary set containing $\phi [D]$ is enough for circulation, but not for these neighbourhood-defined pullbacks.

• The orientation of the boundary is defined mechanically, not by a hand rule. Which way the induced boundary chain runs is decided entirely by Positive orientation of elementary-region boundaries in the parameter plane and then transported by $\phi$. The informal descriptions in the literature — walking along the curve with the head pointing along the normal and the surface on the left, or the right-hand rule — agree with this, but none of them is used here as a definition, and none of them is quoted as one. What makes the sign agreement a fact rather than a convention is that Green's theorem is proved on the parameter region.

• A closed disc is not a legal parameter region here. An elementary Green region is bounded by continuous piecewise-$C^1$ graphs over a nondegenerate interval, and the two semicircular graphs of a disc are not piecewise $C^1$ at the endpoints. Every parameter region used with this definition on this page is a rectangle; a disc-shaped patch image is obtained instead by a polar parametrization over a rectangle, whose induced boundary chain then has two radial edges that cancel and one degenerate edge.

• Why $C^2$ and where the elementary decomposition is spent. When $F$ is $C^1$ on the open set $U$, the class $C^2$ makes the parameter derivatives ${\phi }_u,{\phi }_v$ of class $C^1$, so the pullback coefficients $P^{\ast },Q^{\ast }$ are differentiable on a neighbourhood of $D$; then The curl flux integrand of a $C^2$ patch is a two-dimensional curl of the pulled-back field uses $C^2$ once more to exchange the mixed second parameter derivatives of $\phi$. The elementary decomposition of $D$ is what lets Green's theorem be applied on the parameter region, and the positive boundary chain it carries is what the induced chain is the image of.