Fixed-frequency reciprocal tensor medium

Fix a real angular frequency $\omega \in \mathbb{R}$ and a bounded open region $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with $C^1$ boundary in a material-rest frame. Here a material-rest frame is the inertial frame in which every modelled material element has fixed spatial coordinates, as stipulated by the framework's material-closure postulate. A map in $C^1(\bar{\mathrm{\Omega }})$ means the restriction of a $C^1$ map on an open neighborhood of $\bar{\mathrm{\Omega }}$. Let $\epsilon ,\mu :\bar{\mathrm{\Omega }}\to {\mathbb{C}}^{3\times 3}$ be continuous coefficient fields with units $\mathrm{F}{\mathrm{m}}^{-1}$ and $\mathrm{H}{\mathrm{m}}^{-1}$. They may be spatially varying and complex. Let $E_j,H_j\in C^1(\bar{\mathrm{\Omega }};{\mathbb{C}}^3)$ and $J_j\in C^0(\bar{\mathrm{\Omega }};{\mathbb{C}}^3)$, for $j=1,2$, be phasor amplitudes with SI units $\mathrm{V}{\mathrm{m}}^{-1}$, $\mathrm{A}{\mathrm{m}}^{-1}$, and $\mathrm{A}{\mathrm{m}}^{-2}$. The corresponding real fields use $e^{-i\omega t}$, for example $E_j^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}(x,t)=\mathrm{Re}(E_j(x)e^{-i\omega t})$. Their displacement and magnetic-induction amplitudes are $D_j=\epsilon E_j$ in $\mathrm{C}{\mathrm{m}}^{-2}$ and $B_j=\mu H_j$ in $\mathrm{T}$.

The fixed-frequency constitutive and source equations are

$D_j=\epsilon E_j,B_j=\mu H_j,\mathrm{curl}{E}_j=i\omega B_j,\mathrm{curl}{H}_j=J_j-i\omega D_j\mathrm{.}$

All operators act componentwise on the spatial variable. For $u,v\in {\mathbb{C}}^3$, define the bilinear dot product $u\cdot v={\sum }_{r=1}^3{u}_r{v}_r$ and the complex-bilinear cross product by the usual right-handed component formula. Neither product conjugates either input. The transpose $T$ is the ordinary matrix transpose, without complex conjugation. This definition calls the coefficient model reciprocal when $\epsilon (x{)}^T=\epsilon (x)$ and $\mu (x{)}^T=\mu (x)$ at every $x\in \bar{\mathrm{\Omega }}$. It places no Hermitian, positive-definite, or passivity condition on these complex matrices.

This is a fixed-frequency phasor data class. Its possibly complex coefficients are not inferred from the framework's real, instantaneous material branch; a separate response model is required to justify any such interpretation.

Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch

The macroscopic Maxwell equations in terms of D, H and the free sources

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