Excitation and free-current forms

Let the laboratory region be $U=I\times \mathrm{\Omega }$, with nonempty open $I\subseteq \mathbb{R}$ and nonempty open $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$, in the fixed positively oriented chart $x^0=ct,x^1,x^2,x^3$. Write $dt=d{x}^0\mathrm{/}c$ and $vo{l}_3=d{x}^1\wedge d{x}^2\wedge d{x}^3$. Use the framework fields $E,B,D,H$ and the free sources ${\rho }_f,J_f$, with their SI types, units, and regularity as defined by the existing laboratory, electromagnetic-field, and polarization/magnetization definitions. Whenever a first exterior derivative is used, the field coefficients are at least $C^1$.

Define the spatial forms

$\mathcal{E}=E_{1}d{x}^1+E_{2}d{x}^2+E_{3}d{x}^3,$

$\mathcal{B}=B_{1}d{x}^2\wedge d{x}^3+B_{2}d{x}^3\wedge d{x}^1+B_{3}d{x}^1\wedge d{x}^2,$

$\mathcal{D}=D_{1}d{x}^2\wedge d{x}^3+D_{2}d{x}^3\wedge d{x}^1+D_{3}d{x}^1\wedge d{x}^2,$

$\mathcal{H}=H_{1}d{x}^1+H_{2}d{x}^2+H_{3}d{x}^3,$

and

${\mathcal{J}}_{\mathrm{s}\mathrm{p}}=(J_f{)}_{1}d{x}^2\wedge d{x}^3+(J_f{)}_{2}d{x}^3\wedge d{x}^1+(J_f{)}_{3}d{x}^1\wedge d{x}^2\mathrm{.}$

Define the spacetime field-strength, excitation, and free-current forms by

$F=\mathcal{B}-dt\wedge \mathcal{E},G=\mathcal{D}+dt\wedge \mathcal{H},\mathcal{J}={\rho }_{f}vo{l}_3-dt\wedge {\mathcal{J}}_{\mathrm{s}\mathrm{p}}\mathrm{.}$

Thus $F_{0i}=-E_i\mathrm{/}c$ in $x^0=ct$ coordinates, consistently with the framework electromagnetic tensor. The integral of $F$ over an oriented spacetime 2-surface has units Wb; the integral of $G$ over such a surface and of $\mathcal{J}$ over an oriented spacetime 3-chain has units C. No relation between $G$ and the Lorentzian Hodge dual of $F$ is part of this definition; the vacuum closure is proved only in the separate vacuum branch.

For a $C^1$ $k$-form $\omega =\frac{1}{k!}{\omega }_{{\mu }_1\cdots {\mu }_k}d{x}^{{\mu }_1}\wedge \cdots \wedge d{x}^{{\mu }_k}$, for $1\le k\le 3$, in this fixed chart, with Greek indices in $\{0,1,2,3\}$ and repeated indices summed, define its laboratory-coordinate exterior derivative by

$d\omega =\frac{1}{k!}{\mathrm{\partial }}_{\lambda }{\omega }_{{\mu }_1\cdots {\mu }_k}d{x}^{\lambda }\wedge d{x}^{{\mu }_1}\wedge \cdots \wedge d{x}^{{\mu }_k}\mathrm{.}$

When the coefficients are smooth this is the usual exterior derivative. For a continuous spatial 2-form ${\beta }_t$ and a finite real smooth singular 2-cycle $z={\sum }_{\sigma }{c}_{\sigma }\sigma$ in $\mathrm{\Omega }$, define its cycle flux at time $t$ by ${\mathrm{\Phi }}_z(t)={\sum }_{\sigma }{c}_{\sigma }{\int }_{{\mathrm{\Delta }}^2}{\sigma }^{\ast }{\beta }_t$, using the standard affine orientation and continuous-chain integral convention of Continuous differential-form integrals over smooth singular chains in Euclidean domains.

Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields

The electromagnetic two-form, its observer decomposition and the four-current

Polarization, magnetization, bound and free sources, and the fields D and H

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Continuous differential-form integrals over smooth singular chains in Euclidean domains

Lorentzian Hodge star and periods

Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$

Continuously differentiable maps, local inverses, and local diffeomorphisms

Standard orientation of the affine simplex

Smooth singular chain and cochain complexes

Integral of a form over a smooth singular simplex

A continuous real function on a compact Jordan measurable set is Riemann integrable over that set

The local coordinate formula for the exterior derivative