Fixed volume and patch test geometries, outward normals and induced orientations

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, the convention already carried by the imported finite piecewise $C^1$ boundary presentations and by the compact $C^1$ surface measure cited below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $U\cong I\times \mathrm{\Omega }$ be the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ open and $I\subseteq \mathbb{R}$ a nonempty open interval. This definition fixes the time-independent spatial test geometries of the integral form of the Maxwell equations.

(a) [Volume test geometry] A volume test geometry in $\mathrm{\Omega }$ is a pair $(V,{\nu }_V)$ where $V\subseteq \mathrm{\Omega }$ is a nonempty bounded open set with $\bar{V}\subseteq \mathrm{\Omega }$ and a specified finite piecewise $C^1$ presentation of its boundary (Specified finite piecewise C1 boundary presentations), and ${\nu }_V$ is the outward unit normal of that presentation off its edge set (Bounded C1 domains and their outward normals). The closed-surface flux of a field $F$ continuous on a neighbourhood of $\bar{V}$ is the finite face sum

${\oint }_{\mathrm{\partial }V}F\cdot {\nu }_{V}dS:={\sum }_j{\int }_{S_j}F\cdot {\nu }_{V}dS,$

with the face surface integrals of Surface integration on compact C1 hypersurfaces. For every closed axis-parallel box $K={\prod }_{i<3}[a_i,b_i]\subseteq \mathrm{\Omega }$ with $a_i<b_i$, its open interior $V={\prod }_{i<3}(a_i,b_i)$ is a volume test geometry and $\bar{V}=K$; its natural six-face presentation is the boundary of $K$, with outward normals $\pm e_i$. The volume variable is $V$, while $K$ denotes its compact closure. Each face is a degenerate three-dimensional box and hence has ${\lambda }_3$-measure zero by A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included; their finite union $K\setminus V$ is null. Thus for continuous $h$ on a neighbourhood of $K$, ${\int }_{V}hd{\lambda }_3={\int }_{K}hd{\lambda }_3$: the positive and negative integrals over the null difference vanish by A nonnegative integral over a null set vanishes and Integrable real and complex functions, and their integrals.

(b) [Patch test geometry] A patch test geometry in $\mathrm{\Omega }$ is a regular $C^2$ surface patch $(D,r)$ over a finite elementary Green region in the sense of The induced boundary chain and circulation of a $C^2$ patch over a finite elementary Green region (Type I, Type II, and elementary regions for Green's theorem, Regular parametrized surface patches on compact Jordan parameter regions): $D\subseteq {\mathbb{R}}^2$ is a finite elementary Green region, $r$ is of class $C^2$ on an open neighbourhood of $D$ ($C^k$ Euclidean maps and diffeomorphisms), $r_u\times r_v\mathrm{\ne }0$ on the interior of $D$ (The cross product in ${\mathbb{R}}^3$) and no interior parameter point shares its image with a distinct point of $D$. Its induced boundary chain $\mathrm{\partial }S=r(\mathrm{\partial }D)$ and its circulation ${\oint }_{\mathrm{\partial }S}F\cdot d\mathbf{r}$ for a field continuous on the image are those of The induced boundary chain and circulation of a $C^2$ patch over a finite elementary Green region (Scalar line integrals with respect to arc length and vector-field line integrals); its oriented flux of a field $F$ continuous on the image is

${\int }_{S}F\cdot n_{S}dS:={\int }_D(F\circ r)\cdot (r_u\times r_v)dudv,$

the scalar Riemann integral of the continuous integrand on the compact Jordan parameter region, in the orientation induced by $r$ (Unit normal fields, orientations, and flux through a regular surface patch). The pair $(D,r)$ is part of the data. Every affine graph patch $r(u,v)=(u,v,\phi (u,v))$ over an axis-parallel square $D=[c_1,c_1+\mathrm{\ell }]\times [c_2,c_2+\mathrm{\ell }]$ with $\mathrm{\ell }>0$, whose graph lies in $\mathrm{\Omega }$ with $\phi$ affine (for instance constant), and more generally every affine regular patch over such a square, is a patch test geometry: the square is a finite elementary Green region, and $r$ is smooth and regular with ${\phi }_u,{\phi }_v$ constant.

(c) [Regularity convention] When an integral identity of this framework is asserted for a volume test geometry $V$, respectively for a patch test geometry $S$, with respect to a field $F$ of stated class $C^k$, it concerns fields defined and of class $C^k$ on an open neighbourhood of $\bar{V}$ in $\mathrm{\Omega }$, respectively of the image $r[D]$ in $\mathrm{\Omega }$; when the identity involves a time derivative of a flux, the field is defined with the stated regularity on a neighbourhood of that compact set times the compact time interval of the assertion. No regularity away from these neighbourhoods is asserted.

(d) [Conventions and scope] Test geometries are fixed spatial subsets of $\mathrm{\Omega }$ and carry no motion. A patch's orientation is the one induced by its parametrization, and its induced boundary chain has the compatible orientation of The induced boundary chain and circulation of a $C^2$ patch over a finite elementary Green region, so that the classical Stokes theorem applies with a plus sign. The definition retains the ${\mathrm{A}\mathrm{C}}_{\omega }$ convention of the imported boundary and surface-measure definitions; it asserts no differential equation, no physical law and no existence of fields.

The conventions recorded here are the classical textbook conventions for the divergence and Stokes test geometries: outward normals on closed volume boundaries, induced right-handed orientation on patch boundaries, and flux and circulation signs tied to those orientations. The mathematical content of every clause is carried by the imported items cited in place; the textbooks supply orientation and notation only.