Moving parametrized surface patches and their flux

Let $D\subseteq {\mathbb{R}}^2$ be a nonempty finite elementary Green region in the sense of Type I, Type II, and elementary regions for Green's theorem, written as $D=D_1\cup \cdots \cup D_N$ with its supplied decomposition into elementary Green regions of pairwise disjoint interiors and with the shared-arc structure of that definition, and let $\mathrm{\partial }D$ be its positive boundary chain (Positive orientation of elementary-region boundaries). We require in addition that $D$ be Jordan measurable; this is explained below and holds for every finite elementary Green region. Let $I\subseteq \mathbb{R}$ be an interval and let

$r:D\times I\to {\mathbb{R}}^3$

be a map that extends to a $C^2$ map ($C^k$ Euclidean maps and diffeomorphisms) on an open neighbourhood of $D\times I\subseteq {\mathbb{R}}^3$. Write $p=(u,v)$ for the parameter point, write $r_u(p,t),r_v(p,t),r_t(p,t)\in {\mathbb{R}}^3$ for the partial derivatives of that $C^2$ extension at $(p,t)$, and require

$r_u(p,t)\times r_v(p,t)\mathrm{\ne }0\text{for every }(p,t)\in D^{\circ }\times I^{\circ },$

the cross product being that of The cross product in ${\mathbb{R}}^3$. For every $t\in I^{\circ }$ this gives rank two on $D^{\circ }$. It does not imply the connectedness or interior injectivity required by Regular parametrized surface patches on compact Jordan parameter regions; neither is a standing hypothesis here. At endpoint or singleton times the rank condition is not imposed.

The swept set of the family is

$S:={\bigcup }_{t\in I}r(D\times \{t\})\subseteq {\mathbb{R}}^3,\mathrm{\Sigma }:=\{(r(p,t),t):p\in D,t\in I\}\subseteq {\mathbb{R}}^4\mathrm{.}$

A velocity field for $r$ is a pair $(W,v)$ consisting of an open set $W\subseteq {\mathbb{R}}^4$ with $\mathrm{\Sigma }\subseteq W$ and a map $v:W\to {\mathbb{R}}^3$ of class $C^1$ such that

$v(r(p,t),t)=r_t(p,t)\text{for every }(p,t)\in D\times I\mathrm{.}$

Such a pair is required to exist: the standing hypothesis of this definition is that a velocity field for $r$ is given, and $v$ denotes it. The equality above is what single-valuedness means here: whenever two parameter points at the same time map to the same spatial point, their time derivatives agree. Values at different times may differ.

Finally let $B:V\to {\mathbb{R}}^3$ be a $C^1$ field on an open set $V\subseteq {\mathbb{R}}^4$ containing $\mathrm{\Sigma }$. The flux of $B$ through the patch at time $t\in I$ is

$\mathrm{\Phi }(t):={\int }_D⟨B(r(p,t),t),r_u(p,t)\times r_v(p,t)⟩dudv,$

the integral being the Riemann integral over the compact Jordan measurable set $D$ of the displayed continuous integrand, and the pairing being the Euclidean inner product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. This is a parametrized flux, counted with the supplied parametrization. When $(D,r(\cdot ,t))$ satisfies the regular-patch definition, it agrees with Unit normal fields, orientations, and flux through a regular surface patch. In general different parameter points may represent the same spatial point with different normals. At a rank-two slice the unit normal means $N(p,t)=(r_u\times r_v)\mathrm{/}\parallel r_u\times r_v{\parallel }_2$ on $D^{\circ }$, not necessarily a function of the spatial image alone.

For an injectively parametrized material-sheet interpretation, require in addition that $r(\cdot ,t):D\to {\mathbb{R}}^3$ be injective for every $t\in I$. Injectivity alone does not imply $r_u\times r_v\mathrm{\ne }0$: a regular-slice interpretation also requires this condition on $D^{\circ }\times I$, including endpoint and singleton times. These additional requirements are not standing hypotheses and are not used by the flux formula above.

Well-definedness and conventions

Jordan measurability of $D$. Each piece is compact and Jordan measurable by Type I, Type II, and elementary regions for Green's theorem. The topological boundary of their finite union is contained in the finite union of their content-zero boundaries: a point outside all those boundaries is either interior to one piece or exterior to every piece. Combining finite cube covers proves content zero without Countable Choice. Thus $D$ is compact and Jordan measurable by A bounded set in ${\mathbb{R}}^m$ is Jordan measurable iff its boundary is null, equivalently of content zero; the positive boundary chain $\mathrm{\partial }D$ is separate supplied orientation data.

Extension independence. Each elementary piece is the closure of its interior by its strict graph inequalities, so $D=\overline{D^{\circ }}$. Two $C^2$ extensions agree on $D\times I$, and their spatial difference quotients agree at interior parameter points. Their continuous spatial partials therefore agree on all of $D\times I$: a nonzero difference at a boundary point would remain nonzero nearby and contradict the dense-interior equality. For nondegenerate $I$, the same argument on the dense set $D^{\circ }\times I^{\circ }$ makes $r_t$ extension-independent. For singleton $I=\{t_0\}$ it does not: the same zero restriction admits extensions $0$ and $(t-t_0)\chi (p)$ with different time derivatives. In that case the chosen extension is part of the velocity data; the flux still depends only on the spatial partials.

Integrability. For fixed $t\in I$, the map $p\mapsto B(r(p,t),t)$ is continuous on $D$ as a composition of continuous maps, the maps $p\mapsto r_u(p,t)$ and $p\mapsto r_v(p,t)$ are continuous on $D$, and the cross product and inner product are continuous bilinear operations; hence the integrand is continuous on the compact set $D$. A continuous real function on a compact Jordan measurable set is Riemann integrable over it (A continuous real function on a compact Jordan measurable set is Riemann integrable over that set), so $\mathrm{\Phi }(t)$ is defined for every $t\in I$.

Degenerate time domains. If $I$ is empty then $D\times I$ is empty, the family is vacuous and no flux value is asserted. If $I=\{t_0\}$ is a singleton, the compatibility condition still reads $v(r(p,t_0),t_0)=r_t(p,t_0)$, but it can always be met: choosing the extension $\tilde{r}(p,t):=r(p,t_0)$, which is constant in the time variable, gives $r_t\equiv 0$ and admits the compatible $C^1$ velocity field $v\equiv 0$. A different extension may require a different velocity field or may fail compatibility; as explained above, $r_t(p,t_0)$ is not determined by $r$ alone in this case. The flux $\mathrm{\Phi }(t_0)$ involves only $r_u\times r_v$, so it is the parametrized flux of the fixed map and does not depend on that choice. The condition $r_u\times r_v\mathrm{\ne }0$ is imposed only on $D^{\circ }\times I^{\circ }$, so rank failures on the boundary of the parameter region (seams) are permitted exactly as in Regular parametrized surface patches on compact Jordan parameter regions, and at an endpoint time of a nondegenerate $I$ the definition asserts no regularity of the slice.

No physical assertion. This is a definition of a mathematical object: a time-dependent parametrized family, a velocity field compatible with it, and the flux integral. It asserts nothing about electromagnetic fields, about which vector fields occur in nature, or about the derivative of $\mathrm{\Phi }$; the transport identity for ${\mathrm{\Phi }}^{\mathrm{\prime }}$ is proved separately.