Em Elementary Mathematical Prerequisites

Fix a vector $\mathbf{k}\in {\mathbb{R}}^3$, a parameter $\omega >0$ and amplitudes $\mathbf{e},\mathbf{b}\in {\mathbb{C}}^3$. Here ${\mathbb{C}}^3$ is the set of functions $\{0,1,2\}\to \mathbb{C}$ under componentwise operations, the complex analogue of the convention for ${\mathbb{R}}^n$ in The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$; we relabel the function values at $0,1,2$ by $1,2,3$ and write the coordinates of $\mathbf{e}$ as $(e_1,e_2,e_3)$ and those of $\mathbf{b}$ as $(b_1,b_2,b_3)$. For $\mathbf{x}\in {\mathbb{R}}^3$, $⟨\mathbf{k},\mathbf{x}⟩$ is the Euclidean inner product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, the real number ${\sum }_{j=1}^3{k}_j{x}_j$; for $t\in \mathbb{R}$ put

$\theta (\mathbf{x},t):=⟨\mathbf{k},\mathbf{x}⟩-\omega t\in \mathbb{R}\mathrm{.}$

Let $\exp$ be the complex exponential of The complex exponential by its power series (the power series ${\sum }_{n\ge 0}{z}^n\mathrm{/}n!$, convergent for every $z\in \mathbb{C}$), let $\mathrm{Re}$ be the real part of Real and imaginary parts, complex conjugation, and modulus, and let $i$ be the imaginary unit of The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$.

For each $t\in \mathbb{R}$ define maps $\mathbf{E}(\cdot ,t),\mathbf{B}(\cdot ,t):{\mathbb{R}}^3\to {\mathbb{R}}^3$ componentwise by

$E_j(\mathbf{x},t):=\mathrm{Re}(e_j\exp (i\theta (\mathbf{x},t)\left)\right),B_j(\mathbf{x},t):=\mathrm{Re}(b_j\exp (i\theta (\mathbf{x},t)\left)\right),j\in \{1,2,3\}\mathrm{.}$

Equivalently, $\mathbf{E}(\cdot ,t)$ is the real part, taken componentwise, of the complex vector $\mathbf{e}\exp (i\theta (\mathbf{x},t))$, and likewise for $\mathbf{B}(\cdot ,t)$ with $\mathbf{b}$ in place of $\mathbf{e}$.

Well-definedness and conventions

For every $(\mathbf{x},t)\in {\mathbb{R}}^3\times \mathbb{R}$ the number $\theta (\mathbf{x},t)$ is real, so $i\theta (\mathbf{x},t)$ is a complex number and $\exp (i\theta (\mathbf{x},t))$ is defined by the everywhere-convergent exponential series; multiplying by the complex number $e_j$ or $b_j$ and taking real parts is likewise defined. Both amplitudes may be zero, in which case the corresponding field vanishes identically. The joint map $(\mathbf{x},t)\mapsto (\mathbf{E}(\mathbf{x},t),\mathbf{B}(\mathbf{x},t))$ has ${\mathbb{R}}^3\times \mathbb{R}$, an open subset of ${\mathbb{R}}^4$, as its domain, so the $C^k$ conventions of $C^k$ Euclidean maps and diffeomorphisms apply to any regularity statement made about it; this definition asserts no regularity of its own beyond well-definedness. The exponential is entire with ${\exp }^{\mathrm{\prime }}=\exp$ (The complex exponential is entire and its complex derivative is itself), which is what makes the encoded expressions differentiable in the plane-wave form; derivations that differentiate them use Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ and the chain rule for holomorphic maps where they are applied.

Complex notation here is an encoding of real fields and of real linear equations, not a complex physical field. The symbols $\mathbf{e}$ and $\mathbf{b}$ are constant complex amplitudes, not fields; the complex quantity $\mathbf{e}\exp (i\theta (\mathbf{x},t))$ is an auxiliary object whose real part is the field. Real parts, imaginary parts and complex conjugation are the explicit operations of Real and imaginary parts, complex conjugation, and modulus. Since $\mathrm{Re}$ is additive and commutes with real scalar multiples, any identity obtained from the encoded expressions by sums, real scalar multiples, partial differentiation or integration may be projected by $\mathrm{Re}$ term by term; identities that multiply two encoded real fields may not be read off from a product of complex expressions without conjugation, and the correct second-order average rule is proved separately. No electromagnetic interpretation, equation of motion or physical identification is part of this definition.

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.

Throughout, assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); the surface integral of Surface integration on compact C1 hypersurfaces and the domains of Bounded C1 domains and their outward normals are developed under that convention.

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be open. Let $S\subset \mathrm{\Omega }$ be a compact embedded $C^2$ hypersurface (Codimension and hypersurfaces, Embedded submanifolds and slice charts): every $p\in S$ has an open neighbourhood $U\subseteq {\mathbb{R}}^3$ and a $C^2$ diffeomorphism $\phi :U\to \phi (U)$ onto an open subset of ${\mathbb{R}}^3$ with $\phi (S\cap U)=\phi (U)\cap ({\mathbb{R}}^2\times \{0\})$, in the sense of $C^k$ Euclidean maps and diffeomorphisms. Such an $S$ is a compact embedded $C^1$ hypersurface, so the surface integral and the surface measure $S(A)={\int }_S{\mathbf{1}}_{A}dS$ of Surface integration on compact C1 hypersurfaces and Chart and partition independence of surface measure apply; that measure is finite. The slice-chart definition has no manifold boundary; $S$ has no edge. If $S$ is empty, all surface functionals below are zero and the estimates are read separately without taking a supremum over the empty set.

A two-sided presentation of $S$ consists of a continuous unit normal field $n:S\to {\mathbb{R}}^3$ together with sides: $n(x)$ is orthogonal to $S$ at $x$ in the sense that $⟨n(x),{\gamma }^{\mathrm{\prime }}(0)⟩=0$ for every $C^1$ path $\gamma :(-1,1)\to S$ with $\gamma (0)=x$ and $\parallel n(x){\parallel }_2=1$; and ${\mathrm{\Omega }}_{-},{\mathrm{\Omega }}_{+}\subseteq \mathrm{\Omega }$ are open sets with

$\mathrm{\Omega }\setminus S={\mathrm{\Omega }}_{-}\cup {\mathrm{\Omega }}_{+},{\mathrm{\Omega }}_{-}\cap {\mathrm{\Omega }}_{+}=\mathrm{\varnothing },$

such that for every $x\in S$ there is $\epsilon =\epsilon (x)>0$ with $x+sn(x)\in {\mathrm{\Omega }}_{+}$ for all $0<s<\epsilon$ and $x+sn(x)\in {\mathrm{\Omega }}_{-}$ for all $-\epsilon <s<0$. The normal $n$ points from the side ${\mathrm{\Omega }}_{-}$ to the side ${\mathrm{\Omega }}_{+}$; the orientation convention for normals is that of Unit normal fields, orientations, and flux through a regular surface patch.

The surface distribution ${\delta }_S$. For $\psi \in \mathcal{D}(\mathrm{\Omega })$ (Test function space d of an open set) define

${\delta }_S(\psi ):={\int }_S\psi dS,$

where the right-hand side is the surface integral of the restriction of $\psi$ to $S$, taken in the sense of Surface integration on compact C1 hypersurfaces; complex-valued $\psi$ are integrated componentwise in real and imaginary parts. For a continuous density $a:S\to \mathbb{R}$ define

$a{\delta }_S(\psi ):={\int }_{S}a\psi dS\mathrm{.}$

Both are complex-linear functionals on $\mathcal{D}(\mathrm{\Omega })$, and both are distributions: ${\delta }_S,a{\delta }_S\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$ (Distribution).

Jumps across $S$. Let $F:\mathrm{\Omega }\setminus S\to {\mathbb{R}}^3$ be a field that is of class $C^1$ on ${\mathrm{\Omega }}_{-}$ and on ${\mathrm{\Omega }}_{+}$ separately and has one-sided continuous traces: there are continuous maps $F_{+},F_{-}:S\to {\mathbb{R}}^3$ with

$F_{+}(x)={\lim }_{\begin{array}{c}y\to x\\ y\in {\mathrm{\Omega }}_{+}\end{array}}F(y),F_{-}(x)={\lim }_{\begin{array}{c}y\to x\\ y\in {\mathrm{\Omega }}_{-}\end{array}}F(y)$

for every $x\in S$, the limits taken in ${\mathbb{R}}^3$. The jump of $F$ across $S$ is the continuous map

$[F]:=F_{+}-F_{-}:S\to {\mathbb{R}}^3\mathrm{.}$

No continuity of $F$ across $S$ is assumed or implied; $F$ need not even be defined on $S$, and the two traces may differ.

Well-definedness and finite order

The surface functionals are distributions. Fix a compact $K\subseteq \mathrm{\Omega }$ and $\psi \in {\mathcal{D}}_K=\{\psi \in \mathcal{D}(\mathrm{\Omega }):\mathrm{supp}\psi \subseteq K\}$ with the seminorm $p_0(\psi )={\sup }_K\mathrm{\mid }\psi \mathrm{\mid }$ of Fixed support test function frechet space. Since $\psi$ vanishes outside $K$ and is continuous, ${\sup }_S\mathrm{\mid }\psi \mathrm{\mid }\le p_0(\psi )$; and ${\int }_S\mathrm{\mid }a\psi \mathrm{\mid }dS\le \parallel a{\parallel }_{\mathrm{\infty },S}{\sup }_S\mathrm{\mid }\psi \mathrm{\mid }S(S)$ with $\parallel a{\parallel }_{\mathrm{\infty },S}<\mathrm{\infty }$ by compactness of $S$ and continuity of $a$. Hence

$\mathrm{\mid }{\delta }_S(\psi )\mathrm{\mid }\le S(S)p_0(\psi ),\mathrm{\mid }a{\delta }_S(\psi )\mathrm{\mid }\le \parallel a{\parallel }_{\mathrm{\infty },S}S(S)p_0(\psi ),$

a finite-order estimate of order $0$ with constants independent of $K$; by the local finite-order characterization (Local finite order characterization of distributions) both functionals are distributions. This is the sense in which boundedness on the compact interface and finiteness of the surface measure give continuity of finite order. Linearity is immediate from linearity of the integral; the pairing is complex bilinear with no conjugation, as in Distribution.

The field has locally integrable components. The hypersurface $S$ is a Lebesgue-null set of ${\mathbb{R}}^3$: by the slice-chart property it is covered by finitely many compact pieces each of which is, in suitable $C^2$ coordinates, a graph over a compact subset of a coordinate plane, hence has content zero (The graph of a continuous function on a compact Euclidean set has content zero), and each piece has content zero in the ambient coordinates because it is the image under the $C^1$ chart inverse of a compact content-zero set (A $C^1$ map sends a compact set of content zero to a set of content zero); a finite union of compact content-zero sets is compact and null (Subsets and countable unions of null subsets of ${\mathbb{R}}^m$ are null), hence has content zero (For compact subsets of ${\mathbb{R}}^m$, measure zero and content zero coincide), so $S$ has content zero in the sense of Measure zero and content zero in ${\mathbb{R}}^m$ by countable and finite cube covers. Now $F$ is continuous on $\mathrm{\Omega }\setminus S={\mathrm{\Omega }}_{-}\cup {\mathrm{\Omega }}_{+}$. For each $x\in S$, the two one-sided limits are finite; hence there is a neighbourhood $U_x$ of $x$ on which $\mathrm{\mid }F\mathrm{\mid }$ is bounded on each side (use the defining limit with tolerance $1$ for each of $F_{+}(x)$ and $F_{-}(x)$). Away from $S$, continuity also gives a neighbourhood with a finite bound at every point. For a compact $K\subseteq \mathrm{\Omega }$, these neighbourhoods cover $K$; take a finite subcover and the maximum of its finitely many bounds. This gives a finite $M_K$ such that $\mathrm{\mid }F\mathrm{\mid }\le M_K$ on $K\setminus S$. Extend $F$ by zero on the null closed set $S$; the extension is measurable because the two side restrictions are continuous, and

${\int }_K\parallel F{\parallel }_2\le M_K{\lambda }_3(K)<\mathrm{\infty },$

the contributions from $K\cap S$ being null. Thus the components of $F$ are locally integrable on $\mathrm{\Omega }$ in the sense of Regular distribution from a locally integrable function, and define a regular distribution; the trace condition is exactly what makes the boundary values $F_{\pm }$ well-defined continuous functions on $S$, so the jump $[F]$ is continuous and bounded on $S$.

Conventions and exclusions

The sign convention is fixed by the normal: $n$ points from ${\mathrm{\Omega }}_{-}$ to ${\mathrm{\Omega }}_{+}$, so $[F]=F_{+}-F_{-}$ is measured in the direction of $n$. This convention is used by the interface-derivative lemma on this page, where $\mathrm{div}F$ and $\mathrm{curl}F$ acquire the surface terms $n\cdot [F]{\delta }_S$ and $n\times [F]{\delta }_S$; reversing $n$ reverses the signs of both sides consistently. The definition assumes a fixed stationary interface $S$: no time dependence of $S$ is part of the data, and the stationary jump formulas are not asserted for moving interfaces. The density $a$ is continuous, so $a{\delta }_S$ is defined by the displayed integral; no product of two arbitrary distributions, and no product of ${\delta }_S$ with a merely locally integrable field, is defined here. The definition is stated under ${\mathrm{A}\mathrm{C}}_{\omega }$ because the surface measure it invokes is; the positivity of the surface measure of nonempty open pieces of $S$, and the finite-order estimates above, are its only uses here.

Assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Integral form. Let $n,m\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open, and let $g:U\to {\mathbb{R}}^m$ be continuous. Write $g_i(x):=g(x)(i)$ for $0\le i<m$, using the canonical function coordinates of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. For a measurable $V\subseteq U$ write, componentwise,

${\int }_{V}gd{\lambda }_n:=(\int g_0{\chi }_{V}d{\lambda }_n,\dots ,\int g_{m-1}{\chi }_{V}d{\lambda }_n),$

each component $0\le i<m$, when integrable, being the Lebesgue integral of $g_i$ on $V$, extended by zero outside $V$ (Integrable real and complex functions, and their integrals, Integral over a measurable subset). A ball $V\subseteq U$ is compactly contained in $U$ when its closure is a compact subset of $U$; for a Euclidean ball this means $\bar{V}\subseteq U$. If ${\int }_{V}gd{\lambda }_n=0$ for every ball $V$ compactly contained in $U$, then $g=0$ on $U$.

Disc form. Let $W\subseteq {\mathbb{R}}^3$ be open and let $F:W\to {\mathbb{R}}^3$ be continuous. Write $F_i(x):=F(x)(i)$ for $0\le i<3$ and $e_0=(1,0,0)$, $e_1=(0,1,0)$, $e_2=(0,0,1)$, the canonical standard vectors of The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$. Extend the basis labels cyclically by $e_j:=e_{j<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>3}$ for $j\ge 3$. Thus $e_{i+1}\times e_{i+2}=e_i$ for each $i\in \{0,1,2\}$. For such $i$, $q\in W$ and $r>0$ put

${\phi }_{i,q}(s,t):=q+s{e}_{i+1}+t{e}_{i+2},D_r:=\{(s,t)\in {\mathbb{R}}^2:s^2+t^2<r^2\},$

with $\overline{D_r}$ the closed disc. If for every $i$, every $q\in W$ and every $r>0$ with ${\phi }_{i,q}(\overline{D_r})\subseteq W$ one has

${\int }_{\overline{D_r}}(F_i\circ {\phi }_{i,q})d{\lambda }_2=0,$

then $F=0$ on $W$.

Conventions. The displayed disc condition is the vanishing of the flux of $F$ through the flat disc ${\phi }_{i,q}(\overline{D_r})$ in the orientation with unit normal $e_i$: the affine parametrization is a regular patch whose tangent vectors $e_{i+1},e_{i+2}$ have cross product $e_i$ of norm one (Regular parametrized surface patches on compact Jordan parameter regions, The cross product in ${\mathbb{R}}^3$, The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$), so the flux of Unit normal fields, orientations, and flux through a regular surface patch is the Riemann integral ${\int }_{\overline{D_r}}(F_i\circ {\phi }_{i,q})d(s,t)$; that integral equals the displayed Lebesgue integral because the integrand is continuous on the compact Jordan region $\overline{D_r}$ (A closed disc of radius $r\ge 0$ has Jordan content $\pi r^2$, The Riemann integral of a bounded function over a bounded Jordan measurable set, A continuous real function on a compact Jordan measurable set is Riemann integrable over that set, Borel Darboux integrands in finite dimension). Both clauses are pointwise conclusions on their open domains. The first clause is the converse from universal integral relations to a pointwise equation; a single vanishing integral over one ball or one disc supplies no conclusion, and the hypothesis ranges over the full family described.

• The proof uses, for each clause, the hypothesis only on balls centred at the point where the function is assumed nonzero and with radius half of a continuity threshold; the full family in the Statement is what makes the lemma a localization statement, but no choice from the family is made, and the argument is choice-free apart from the declared ${\mathrm{A}\mathrm{C}}_{\omega }$ used by the surface and measurability suppliers.
• The disc clause is not a statement about a single loop or box: one special disc carries no information, and the components must be tested in all three coordinate directions at all admissible centres and radii.

Assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $a>0$ and let $\rho \in L^1({\lambda }_3)$ be a real or complex valued Lebesgue integrable function on ${\mathbb{R}}^3$ that vanishes ${\lambda }_3$-almost everywhere outside the closed ball $B_a:=\{y\in {\mathbb{R}}^3:\parallel y{\parallel }_2\le a\}$ (The class $L^1(\mu )$ of integrable functions, Open ball, closed ball and sphere in a metric space). Put

$Q:=\int \rho d{\lambda }_3,p:=\int y\rho (y)d{\lambda }_3\in {\mathbb{C}}^3,\parallel \rho {\parallel }_1:=\int \mid \rho \mid d{\lambda }_3,$

the vector $p$ being defined by $p_k:=\int y_k\rho (y)d{\lambda }_3(y)$ for the canonical coordinates $k<3$, namely $k\in \{0,1,2\}$, and $⟨p,x⟩:={\sum }_{k<3}{p}_k{x}_k$ being the complex-bilinear coordinate contraction, with no complex conjugation, which agrees with the real Euclidean pairing when $p$ is real; if $\rho$ is real-valued then $Q$ is real and $p\in {\mathbb{R}}^3$, while $R(x)$ below is real in that case and complex in general. Then for every $x\in {\mathbb{R}}^3$ with $r:=\parallel x{\parallel }_2\ge 2a>0$,

$\int \frac{\rho (y)}{\parallel x-y{\parallel }_2}d{\lambda }_3(y)=\frac{Q}{r}+\frac{⟨p,x⟩}{r^3}+R(x),R(x):={\int }_{B_a}\rho (y)(\frac{1}{\parallel x-y{\parallel }_2}-\frac{1}{r}-\frac{⟨x,y⟩}{r^3})d{\lambda }_3(y),$

and the remainder satisfies the explicit bound

$\mid R(x)\mid \le \frac{16a^2\parallel \rho {\parallel }_1}{r^3}\mathrm{.}$

In the all-space kernel integral define $K_x(y)=\parallel x-y{\parallel }_2^{-1}$ for $y\mathrm{\ne }x$ and $K_x(x)=0$, so its integrand means $\rho K_x$ and is an everywhere-defined measurable function. This convention cannot affect the integral: $x\mathrm{\notin }{B}_a$ and the density vanishes almost everywhere outside $B_a$. All integrals are Lebesgue integrals; the closed ball $B_a$ is Borel and ${\chi }_{B_a}\rho$ is integrable. The constant $16$ is permissible but not sharp, and $Q$, $p$ and $\parallel \rho {\parallel }_1$ are finite. No derivative of the remainder in $x$ is asserted.