Elementary solid regions: one boundary presentation adapted in all three coordinate directions

An elementary solid region is a compact set $E\subseteq {\mathbb{R}}^3$ supplied with a simple description in each of the three coordinate directions (Simple solid regions in a coordinate direction and their cyclic coordinate projection) together with one compatible finite patch presentation of $\mathrm{\partial }E$ that is adapted to a simple description of $E$ in each of the three coordinate directions (Boundary presentations adapted to a simple solid region in a coordinate direction, Finitely patched regular surfaces, their area, scalar integrals, and flux).

Explicitly, the data are: three simple descriptions $(x,D_x,{\gamma }_1^x,{\gamma }_2^x)$, $(y,D_y,{\gamma }_1^y,{\gamma }_2^y)$ and $(z,D_z,{\gamma }_1^z,{\gamma }_2^z)$, each describing the same set $E$; one compatible finite patch presentation $\mathrm{\Sigma }=((D_1,{\phi }_1),\dots ,(D_P,{\phi }_P))$ whose patch images cover $\mathrm{\partial }E$ and are contained in $\mathrm{\partial }E$; and, for each of the three directions $k$, a partition of $\{1,\dots ,P\}$ into sublists ${\mathrm{\Sigma }}_k^{+},{\mathrm{\Sigma }}_k^{-},{\mathrm{\Sigma }}_k^0$ making $\mathrm{\Sigma }$ adapted to the $k$th description. The boundary is that of Interior, closure, boundary, limit point, isolated point and dense subset of a metric space.

One presentation, three partitions. The patch list is the same in all three directions; only the sorting of its indices into upper, lower and lateral changes with $k$. That is what makes the three single-direction flux identities statements about one and the same boundary integral, and it is the whole content of the word "elementary" here.

• The descriptions and the presentation are supplied data. Nothing above is inferred from the set $E$: a compact set may admit several simple descriptions in a given direction, several compatible presentations of its boundary, and several sortings of a presentation, and no claim is made that any of these exists for an arbitrary compact set or that it is unique when it does. The convention matches the one Type I, Type II, and elementary regions for Green's theorem uses in the plane, where the decomposition is likewise part of the data.

• A patch may be lateral in one direction and a face in another. The six faces of a box illustrate both: the top and bottom faces are the upper and lower sublists for $k=z$ and lateral for $k=x$ and $k=y$. What cannot happen is a patch lateral in all three directions, and that is Every patch of an elementary solid region's presentation is a graph face in some direction, and at interior base points its normal is outward.

• What is not assumed. The solid need not be convex, its boundary need not be connected, and no patch is required to be a graph over a coordinate plane. Conversely, nothing here asserts that every compact solid with a piecewise smooth boundary can be presented this way; What the classical divergence and Stokes theorems here do and do not cover states what is and is not covered.