The microscopic vacuum Maxwell equations: smooth and weak branches

Adopt as a physical postulate of this framework the following four equations for the fields of The electric and magnetic fields as fixed-time spatial vector fields and the sources of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current on the laboratory region $U\cong I\times \mathrm{\Omega }$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with ${\mu }_0>0$, $c>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ and the SI units of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(a) [Smooth branch] Let $E,B$ be jointly $C^1$ fields with the additional derivatives required by each equation existing and continuous, and let $\rho ,J$ be continuous. Then, with the divergence and curl of Divergence and curl of a $C^1$ vector field and the partial derivative ${\mathrm{\partial }}_t$ at fixed position of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$,

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

(b) [Weak branch] Let instead $E,B$ be locally integrable ${\mathbb{R}}^3$-valued fields and $\rho ,J$ locally integrable scalar and vector fields on $I\times \mathrm{\Omega }$, identified with their regular distributions (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions). Let $\mathrm{div}$, $\mathrm{curl}$ and ${\mathrm{\partial }}_t$ act as distributional derivatives, componentwise, and let $\psi$ range over $C_c^{\mathrm{\infty }}(I\times \mathrm{\Omega })$ (Test function space d of an open set). Then the same four identities hold in the sense of Distribution: the 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 for every such $\psi$, all derivatives being distributional derivatives (Distributional derivative).

(c) [Scope] The postulate contains no magnetic charge: $\mathrm{div}B=0$ and no magnetic-source term appears in $\mathrm{curl}E$. Sources that are not locally integrable functions (point or sheet sources) are covered by neither branch and require their own definition. The equations are adopted classical vacuum physics in the named inertial chart; they are not deduced from charge conservation, from empirical agreement or from the special-relativity postulates, and this item by itself asserts no covariance under changes of inertial frame.