Integral Maxwell and constraints

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, the ambient convention under which the Lebesgue measure and integration interface used below is stated (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open, and let $g:U\to \mathbb{R}$ be continuous.

For a closed axis-parallel cube $Q={\prod }_{i<n}[a_i,a_i+\mathrm{\ell }]\subseteq U$ with $\mathrm{\ell }>0$, the restriction $g{\mathrm{\mid }}_Q$ is continuous on the compact nondegenerate rectangle $Q$, hence bounded and Borel and Riemann integrable; denote by ${\int }_{Q}gd{\lambda }_n$ the Lebesgue integral of $g$ over $Q$ with respect to the restriction of Lebesgue measure, which by the comparison of the Riemann and Lebesgue integrals equals the Riemann integral of $g$ over $Q$. Suppose that

${\int }_{Q}gd{\lambda }_n=0$

for every closed axis-parallel cube $Q\subseteq U$ with side $\mathrm{\ell }>0$. Then $g=0$ on $U$.

In particular, for $n=2$, a continuous function on an open subset of the plane whose integral over every closed axis-parallel square contained in its domain vanishes is identically zero.

The hypothesis requires every closed axis-parallel cube, with every centre and every positive side length. A single cube does not suffice, and neither do the cubes centred at one fixed point: on $U={\mathbb{R}}^n$ the nonzero continuous function $g(x)=x_0-p_0$ has ${\int }_{Q}gd{\lambda }_n=0$ for every closed cube $Q$ centred at $p$, by cancellation in its first coordinate. This family also differs from the family of closed cubes that merely contain $p$.

The localization statement is standard: a continuous function that is orthogonal to every axis-parallel cube integral vanishes. The proof above is written in the library's own measure and Darboux interfaces, using the multidimensional Darboux integral of The lower and upper Darboux integrals over a nondegenerate rectangle in ${\mathbb{R}}^m$ and the comparison with the Lebesgue integral of Riemann–Lebesgue comparison for distribution test integrands; the sharpness witness is the coordinate-cancellation computation, proved directly by mirror-symmetric grid refinements rather than by a separate symmetry or change-of-variables theorem.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, the convention of the surface-measure interface cited below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be open, let $S\subset \mathrm{\Omega }$ be a compact embedded $C^2$ hypersurface whose surface measure is that of the compact embedded $C^1$ class of Surface integration on compact C1 hypersurfaces (the $C^2$ data are special cases of the $C^1$ data), and let $a:S\to \mathbb{R}$ be continuous. Suppose that

${\int }_{S}a\psi dS=0$

for every $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ (Test function space d of an open set), the pairing being the signed surface integral of Surface integration on compact C1 hypersurfaces of the continuous function $a(\psi {\mathrm{\mid }}_S)$. For complex tests the integral is defined componentwise in real and imaginary parts; the proof below uses real tests. Then $a=0$ on $S$. Equivalently, if two continuous densities on $S$ pair equally against every test function, then they are equal pointwise on $S$. The conclusion is pointwise and uses the continuity of $a$; a merely locally integrable density would satisfy only an almost-everywhere conclusion, which this statement does not claim.

The pointwise conclusion is the standard reduction of a vanishing pairing against every ambient smooth test function to a local positivity argument on the hypersurface. The proof above keeps the library's own chart-measure interface: the graph density of the published treatments appears here only through the strict positivity and continuity of the regular chart density of Chart and partition independence of surface measure, so no product-extension assertion for arbitrary planar tests is used.

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.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, the convention of the surface-measure interface inherited from Surface distributions and interfacial jumps (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 the units and constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let ${\mathcal{D}}^{\mathrm{\prime }}(U)$ be the space of distributions on $U$ (Distribution, Test function space d of an open set); recall that the pairing $⟨T,\psi ⟩$ is bilinear with no conjugation and that the distributional partial derivatives are $⟨{\mathrm{\partial }}_{\mu }T,\psi ⟩=-⟨T,{\mathrm{\partial }}_{\mu }\psi ⟩$ (Distributional derivative).

(a) [Fields as distributions] Locally integrable fields and source densities on $U$ are identified with their regular distributions (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions).

(b) [Point sources] Let $I_0\subseteq \mathbb{R}$ be a nonempty open interval, let $z:I_0\to {\mathbb{R}}^3$ be a $C^1$ trajectory whose graph map $\gamma (t)=(z(t),t)$ is proper into the open set $U$, and let $q\in \mathbb{R}$ (coulombs). By Point-support spacetime densities: distributions, continuity equation and endpoint fluxes,

$⟨{\rho }_{\mathrm{p}\mathrm{t}},\psi ⟩:=q{\int }_{I_0}\psi (z(t),t)dt,⟨J_{\mathrm{p}\mathrm{t}},\psi ⟩:=q{\int }_{I_0}{z}^{\mathrm{\prime }}(t)\psi (z(t),t)dt$

define distributions on $U$ with ${\mathrm{\partial }}_t{\rho }_{\mathrm{p}\mathrm{t}}+\mathrm{div}{J}_{\mathrm{p}\mathrm{t}}=0$. If instead $I_0=(a,b)$ has finite endpoints, $z$ extends continuously to $[a,b]$, the absolute improper integral ${\int }_a^b\mathrm{\mid }{z}^{\mathrm{\prime }}(t)\mathrm{\mid }dt$ is finite and both endpoint events lie in $U$, the same formulas define distributions with

${\mathrm{\partial }}_t{\rho }_{\mathrm{p}\mathrm{t}}+\mathrm{div}{J}_{\mathrm{p}\mathrm{t}}=q{\delta }_{\gamma (a)}-q{\delta }_{\gamma (b)}$

(Dirac delta and its derivatives); a finite-history point source therefore contributes an endpoint term unless that term is included explicitly, and the pairings above exclude it only by the stated properness or endpoint hypotheses.

(c) [Stationary-sheet sources] Let $S\subset \mathrm{\Omega }$ be a fixed compact embedded $C^2$ hypersurface with a chosen unit normal $n$ pointing from the side ${\mathrm{\Omega }}_{-}$ to the side ${\mathrm{\Omega }}_{+}$ (Surface distributions and interfacial jumps, Surface integration on compact C1 hypersurfaces), and let $\sigma :S\times I\to \mathbb{R}$ and $K:S\times I\to {\mathbb{R}}^3$ be continuous with $K(x,t)\cdot n(x)=0$ for all $(x,t)$; $\sigma$ carries C m${}^{-2}$ and $K$ carries A m${}^{-1}$ (a tangential surface current density; the sheet is stationary). Then

$⟨{\rho }_S,\psi ⟩:={\int }_I{\int }_S\sigma (x,t)\psi (x,t)dS(x)dt,⟨J_S,\psi ⟩:={\int }_I{\int }_{S}K(x,t)\psi (x,t)dS(x)dt$

define distributions on $U$: for $\psi$ with compact support the inner integrals are continuous in $t$ on a compact parameter interval and the functionals are bounded by $(\sup \mathrm{\mid }\sigma \psi \mathrm{\mid })\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}(S)\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}$ of the time projection of the support, and likewise in the vector case. A normal sheet current (one with $K\cdot n\mathrm{\ne }0$), a line current and a moving sheet are not source types of this definition.

(d) [Unrestricted weak fields and optional interface branches] The general field class for the solution predicate (e) consists of all locally integrable pairs $(E,B)$ on $U$, also when the source has a stationary-sheet part. No classical one-sided trace or off-sheet $C^1$ regularity is required to define those four distributional pairings. Additional trace hypotheses are optional hypotheses of interface and jump results, not admission requirements of the general weak model.

For a pointwise classical jump branch, one may additionally assume that $E,B$ are $C^1$ off $S\times I$ and, around every $(x_0,t_0)\in S\times I$, each side is the restriction of a $C^1$ function on an open spacetime neighbourhood of the corresponding closed side of a smaller stationary graph cylinder. These joint one-sided extensions make the field traces and first spatial and temporal derivatives continuous up to that side, locally uniformly on compact cylinders.

For the broader weak-trace branch, retain spacetime $C^1$ regularity on $(\mathrm{\Omega }\setminus S)\times I$ and assume the original one-sided spatial $C^1$ extensions at each time, with the explicit additional conditions that the piecewise first spatial and temporal derivatives are in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(U)$ and the field traces are measurable and locally integrable on $S\times I$ for $dSdt$. This branch supports sheet-distribution and almost-everywhere jump identities; spatial continuity at each time gives pointwise spatial identities for almost every time, and joint continuity of the jump densities gives the further all-time upgrade. Separate spatial $C^1$ traces at each time alone do not imply that joint regularity or upgrade. Whenever such traces exist, $[F]:=F_{+}-F_{-}$ denotes their jump in the chosen normal convention.

(e) [Weak Maxwell equations with a distributional source pair] Let $E,B$ be a field pair of class (d), let $\rho ={\rho }_v+{\rho }_s$ and $J=J_v+J_s$ be source distributions on $U$ whose volume parts ${\rho }_v,J_v$ are locally integrable volume densities with the charge and current quantity types of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current (the weak predicate requires this stated $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ regularity, not the classical continuous-source regularity) and whose singular parts $({\rho }_s,J_s)$ are point sources as in (b) or stationary-sheet sources as in (c), and let ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Say that $E,B$ satisfy the weak Maxwell equations with the source pair $(\rho ,J)$ when, for every $\psi \in C_c^{\mathrm{\infty }}(U)$, the four pairings

$⟨\mathrm{div}E-\rho \mathrm{/}{\epsilon }_0,\psi ⟩,⟨\mathrm{div}B,\psi ⟩,⟨\mathrm{curl}E+{\mathrm{\partial }}_{t}B,\psi ⟩,⟨\mathrm{curl}B-{\mu }_{0}J-c^{-2}{\mathrm{\partial }}_{t}E,\psi ⟩$

all vanish, all derivatives being distributional derivatives of the regular distributions of the fields and the source pairings being those of (b) and (c). This clause defines a solution predicate. When the singular parts vanish, its four identities are the same as the locally integrable weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches. That earlier postulate explicitly excludes point and sheet distributions and does not imply their physical adoption here; adoption for the declared singular-source models is made separately, after this definition. No physical field equation is deduced merely from defining this predicate.

(f) [Scope] No product of two distributions is formed anywhere: the fields occur only through their distributional derivatives paired with a test function, and the source densities $\sigma ,K$ occur only inside the ordinary surface and time integrals of (b) and (c); products such as ${\rho }_{s}E$ or $J_s\times B$ are not defined by this item. The definition asserts no existence theorem for fields with singular sources, no conservation law (continuity for these sources is proved separately where used) and no regularisation. Sources that are not locally integrable and are not point or stationary-sheet sources, moving sheets, line currents, and arbitrary products of distributions are outside the definition.

The distributional treatment of point charges and surface sources follows the standard classical presentations cited in the frontmatter; the pairings are defined with the library's own distribution and surface-measure conventions, and the continuity statements quoted in (b) are the conclusions of Point-support spacetime densities: distributions, continuity equation and endpoint fluxes, consumed here as the exact interface of the point-source pairings.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, the convention of the imported piecewise $C^1$ boundary and surface-measure definitions (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 the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values; let $E,B$ be fields of The electric and magnetic fields as fixed-time spatial vector fields and $\rho ,J$ source fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current on $\mathrm{\Omega }\times I$; and let $(V,{\nu }_V)$ and $(D,r)$ be the volume and patch test geometries of Fixed volume and patch test geometries, outward normals and induced orientations. Assume the fields are jointly $C^1$ on $\mathrm{\Omega }\times I$ and that the additional derivatives appearing below exist and are continuous there; for instance $E,B$ jointly of class $C^2$ and $\rho ,J$ of class $C^1$ is sufficient.

(a) [Differential implies integral] If the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches holds pointwise on $\mathrm{\Omega }\times I$, then for every volume test geometry $(V,{\nu }_V)$ and every $t\in I$,

${\oint }_{\mathrm{\partial }V}E(\cdot ,t)\cdot {\nu }_{V}dS=\frac{1}{{\epsilon }_0}{\int }_V\rho (\cdot ,t)d{\lambda }_3,{\oint }_{\mathrm{\partial }V}B(\cdot ,t)\cdot {\nu }_{V}dS=0,$

and for every patch test geometry $(D,r)$ the flux functions $t\mapsto {\int }_{S}B(\cdot ,t)\cdot n_{S}dS$ and $t\mapsto {\int }_{S}E(\cdot ,t)\cdot n_{S}dS$ are differentiable on $I$ and, for every $t\in I$,

${\oint }_{\mathrm{\partial }S}E(\cdot ,t)\cdot d\mathbf{r}=-\frac{d}{dt}{\int }_{S}B(\cdot ,t)\cdot n_{S}dS,{\oint }_{\mathrm{\partial }S}B(\cdot ,t)\cdot d\mathbf{r}={\mu }_0{\int }_{S}J(\cdot ,t)\cdot n_{S}dS+c^{-2}\frac{d}{dt}{\int }_{S}E(\cdot ,t)\cdot n_{S}dS\mathrm{.}$

(b) [Integral implies differential] Conversely, suppose the two volume identities of (a) hold for every volume test geometry and every $t\in I$, and that for every patch test geometry the two flux functions are differentiable on $I$ and the two patch identities of (a) hold for every $t\in I$. Then the four smooth-branch equations of The microscopic vacuum Maxwell equations: smooth and weak branches hold pointwise on $\mathrm{\Omega }\times I$:

$\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0,\mathrm{div}B=0,\mathrm{curl}E=-{\mathrm{\partial }}_{t}B,\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E\mathrm{.}$

The equivalence is asserted for the exact class of test geometries of Fixed volume and patch test geometries, outward normals and induced orientations, which contains the open interiors of all closed axis-parallel boxes compactly contained in $\mathrm{\Omega }$ and the affine square patches; it is not a recovery from one selected volume or one selected patch, and the divergence and curl of Divergence and curl of a $C^1$ vector field and the time derivative of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ are the operators used.