$E_i$ = electric field
$B_i$ = magnetic field
$P_i$ = polarization field
$M_i$ = magnetization field
$\epsilon_0$ = permittivity of free space
${\epsilon_{r,i}}^j$ = relative permittivity of material
${\epsilon_i}^j$ = permittivity of material
${\chi_{e,i}}^j$ = electric susceptibility
$\mu_0$ = permeability of free space
${\mu_{r,i}}^j$ = relative permeability of material
${\mu_i}^j$ = permeability of material
${\chi_{m,i}}^j$ = magnetic susceptibility

$D_i = \epsilon_0 E_i + P_i$
$P_i = \epsilon_0 {\chi_{e,i}}^j E_j$
$D_i = \epsilon_0 E_i + \epsilon_0 {\chi_{e,i}}^j E_j$
$D_i = \epsilon_0 (\delta_i^j + {\chi_{e,i}}^j) E_j$
$D_i = \epsilon_0 {\epsilon_{r,i}}^j E_j$
$D_i = {\epsilon_i}^j E_j$

$B_i = \mu_0 H_i + M_i$
$M_i = \mu_0 {\chi_{m,i}}^j H_j$
$B_i = \mu_0 H_i + \mu_0 {\chi_{m,i}}^j H_j$
$B_i = \mu_0 (\delta_i^j + {\chi_{m,i}}^j) H_j$
$B_i = \mu_0 {\mu_{r,i}}^j H_j$
$B_i = {\mu_i}^j H_j$


$E_i = {(\epsilon^{-1})_i}^j D_j$
$H_i = {(\mu^{-1})_i}^j B_j$

${D^i}_{,i} = \rho_{free}$ ... where $\rho_{free}$ = free charge (densitized?)
${B^i}_{,i} = 0$

$D_{i,t} = {\epsilon_i}^{jk} H_{k,j} - J_{i,free}$
$B_{i,t} = -{\epsilon_i}^{jk} E_{k,j}$

$D_{i,t} = {\epsilon_i}^{jk} (\mu^{-1} B)_{k,j} - J_{i,free}$
$B_{i,t} = -{\epsilon_i}^{jk} (\epsilon^{-1} D)_{k,j}$

$D_{i,t} = {\epsilon_i}^{jk} ({(\mu^{-1})_k}^l B_l)_{,j} - J_{i,free}$
$B_{i,t} = -{\epsilon_i}^{jk} ({(\epsilon^{-1})_k}^l D_l)_{,j}$

$D_{i,t} - \gamma_{im} \epsilon^{mjk} {(\mu^{-1})_k}^l {B_l}_{,j} = {\epsilon_i}^{jk} {{(\mu^{-1})_k}^l}_{,j} B_l - J_{i,free}$
$B_{i,t} + \gamma_{im} \epsilon^{mjk} {(\epsilon^{-1})_k}^l D_{l,j} = {\epsilon_i}^{jk} {{(\epsilon^{-1})_k}^l}_{,j} D_l$

$D_{i,t} - \frac{1}{\sqrt\gamma} \gamma_{im} \bar\epsilon^{mjk} {(\mu^{-1})_k}^l {B_l}_{,j} = {\epsilon_i}^{jk} {{(\mu^{-1})_k}^l}_{,j} B_l - J_{i,free}$
$B_{i,t} + \frac{1}{\sqrt\gamma} \gamma_{im} \bar\epsilon^{mjk} {(\epsilon^{-1})_k}^l D_{l,j} = {\epsilon_i}^{jk} {{(\epsilon^{-1})_k}^l}_{,j} D_l$

$\pmatrix{ D_{i,t} \\ B_{i,t} } + \frac{1}{\sqrt\gamma} \pmatrix{ \gamma_{im} & 0 \\ 0 & \gamma_{im} } \pmatrix{ 0 & -\bar\epsilon^{mjk} \\ \bar\epsilon^{mjk} & 0 } \pmatrix{ {(\mu^{-1})_k}^l & 0 \\ 0 & {(\epsilon^{-1})_k}^l } \pmatrix{ {B_l}_{,j} \\ D_{l,j} } = \pmatrix{ {\epsilon_i}^{jk} {{(\mu^{-1})_k}^l}_{,j} B_l - J_{i,free} \\ {\epsilon_i}^{jk} {{(\epsilon^{-1})_k}^l}_{,j} D_l }$