Boundary presentations adapted to a simple solid region in a coordinate direction

Let $(k,D,{\gamma }_1,{\gamma }_2)$ be a simple description of a solid $E\subseteq {\mathbb{R}}^3$ in the direction $k$ (Simple solid regions in a coordinate direction and their cyclic coordinate projection), and write

${\mathrm{\Gamma }}_2:=\{p:{\pi }_k(p)\in D,p_k={\gamma }_2({\pi }_k(p))\},{\mathrm{\Gamma }}_1:=\{p:{\pi }_k(p)\in D,p_k={\gamma }_1({\pi }_k(p))\}$

for the upper and lower graph of the description. Let $\mathrm{\Sigma }=((D_1,{\phi }_1),\dots ,(D_P,{\phi }_P))$ be a compatible finite patch presentation in the sense of Finitely patched regular surfaces, their area, scalar integrals, and flux, each $(D_j,{\phi }_j)$ a regular parametrized surface patch of Regular parametrized surface patches on compact Jordan parameter regions, whose patch images cover $\mathrm{\partial }E$ and are contained in $\mathrm{\partial }E$. Write ${\phi }_{j,u},{\phi }_{j,v}$ for the two parameter derivatives, and ${\phi }_{j,u}\times {\phi }_{j,v}$ for the oriented area vector of Unit normal fields, orientations, and flux through a regular surface patch, whose coordinates are those of The cross product in ${\mathbb{R}}^3$.

The presentation $\mathrm{\Sigma }$ is adapted to the description $(k,D,{\gamma }_1,{\gamma }_2)$ when the index set $\{1,\dots ,P\}$ is partitioned into three sublists ${\mathrm{\Sigma }}^{+}$, ${\mathrm{\Sigma }}^{-}$ and ${\mathrm{\Sigma }}^0$, supplied with the presentation, such that all of the following hold.

1. Upper faces. For $j\in {\mathrm{\Sigma }}^{+}$, the image of ${\phi }_j$ lies in the graph of ${\gamma }_2$ and the $k$th coordinate of ${\phi }_{j,u}\times {\phi }_{j,v}$ is positive on the interior of $D_j$.
2. Lower faces. For $j\in {\mathrm{\Sigma }}^{-}$, the image of ${\phi }_j$ lies in ${\mathrm{\Gamma }}_1$ and the $k$th coordinate of ${\phi }_{j,u}\times {\phi }_{j,v}$ is negative on the interior of $D_j$.
3. Lateral faces. For $j\in {\mathrm{\Sigma }}^0$, the $k$th coordinate of ${\phi }_{j,u}\times {\phi }_{j,v}$ vanishes on the interior of $D_j$.
4. The graph faces cover the base. Writing $V_j:={\pi }_k[{\phi }_j[D_j^{\circ }]]$ for the projected image of the $j$th patch, the projected images of the upper sublist are pairwise disjoint and fill $D$ up to content zero, and the same holds for the lower sublist: for each of ${\mathrm{\Sigma }}^{+}$ and ${\mathrm{\Sigma }}^{-}$ the sets $V_j$ with $j$ in that sublist are pairwise disjoint and $D\setminus \bigcup V_j$ has content zero in the sense of Measure zero and content zero in ${\mathbb{R}}^m$ by countable and finite cube covers.
5. Both graph sublists are nonempty. ${\mathrm{\Sigma }}^{+}\mathrm{\ne }\mathrm{\varnothing }$ and ${\mathrm{\Sigma }}^{-}\mathrm{\ne }\mathrm{\varnothing }$; the lateral sublist ${\mathrm{\Sigma }}^0$ may be empty.

The partition into the three sublists is part of the supplied data, exactly as the description $(k,D,{\gamma }_1,{\gamma }_2)$ is; nothing here is inferred from the set $E$ or from the unordered collection of patch images. Interiors are those of Interior, closure, boundary, limit point, isolated point and dense subset of a metric space and integrals over the projected images are those of The Riemann integral of a bounded function over a bounded Jordan measurable set.

• The conditions are on the sign of one coordinate, not on outwardness. Clauses 1 to 3 are analytic: by Each coordinate of the oriented area vector is the Jacobian determinant of the matching cyclic projection the $k$th coordinate of the oriented area vector is the Jacobian determinant of ${\pi }_k\circ {\phi }_j$, so clause 1 says that the projection of an upper patch is orientation-preserving on the parameter interior and clause 3 says that a lateral patch projects with vanishing Jacobian determinant. That the induced normals of the graph faces then point out of $E$ is a theorem, At interior base points, the graph faces of an adapted presentation induce the outward unit normal, rather than part of this definition.

• A graph face is not required to be a graph patch. Clause 1 asks only that the patch image lie in ${\mathrm{\Gamma }}_2$; it does not ask that ${\phi }_j$ be the map $w\mapsto (w,{\gamma }_2(w))$ read in the projected coordinates. That is what admits the eight spherical octants and the four quarter-cylinders: their graph functions have unbounded gradient at the equator or at the silhouette, so they are not $C^1$ on a neighbourhood of the closed base and could not parametrize a patch, while the octants and quarters themselves are patches in every direction at once.

• Why the lateral condition is imposed on the interior. The parameter region of a patch is the closure of its interior, and the $k$th coordinate of the oriented area vector is continuous on the whole region, so a vanishing condition on the interior already forces vanishing everywhere on the region. Stating it on the interior keeps the three clauses in the same form and matches where clauses 1 and 2 can be stated at all, since the oriented area vector may vanish on a parameter boundary.

• What clause 4 is for. It is the only clause that ties the presentation to the base quantitatively: without it, one tiny upper patch in ${\mathrm{\Gamma }}_2$ together with one tiny lower patch in ${\mathrm{\Gamma }}_1$ could satisfy clauses 1, 2, 3 and 5 while covering almost none of either graph. Pairwise disjointness and the content-zero residue are what make the sum of the graph-face fluxes an integral over the whole of $D$.