Maxwell in inhomogeneous medium

E_{i}

= electric field

B_{i}

= magnetic field

P_{i}

= polarization field

M_{i}

= magnetization field

ϵ_{0}

= permittivity of free space

{ϵ_{r, i}}^{j}

= relative permittivity of material

{ϵ_{i}}^{j}

= permittivity of material

{χ_{e, i}}^{j}

= electric susceptibility

μ_{0}

= permeability of free space

{μ_{r, i}}^{j}

= relative permeability of material

{μ_{i}}^{j}

= permeability of material

{χ_{m, i}}^{j}

= magnetic susceptibility

D_{i} = ϵ_{0} E_{i} + P_{i}

P_{i} = ϵ_{0} {χ_{e, i}}^{j} E_{j}

D_{i} = ϵ_{0} E_{i} + ϵ_{0} {χ_{e, i}}^{j} E_{j}

D_{i} = ϵ_{0} (δ_{i}^{j} + {χ_{e, i}}^{j}) E_{j}

D_{i} = ϵ_{0} {ϵ_{r, i}}^{j} E_{j}

D_{i} = {ϵ_{i}}^{j} E_{j}

B_{i} = μ_{0} H_{i} + M_{i}

M_{i} = μ_{0} {χ_{m, i}}^{j} H_{j}

B_{i} = μ_{0} H_{i} + μ_{0} {χ_{m, i}}^{j} H_{j}

B_{i} = μ_{0} (δ_{i}^{j} + {χ_{m, i}}^{j}) H_{j}

B_{i} = μ_{0} {μ_{r, i}}^{j} H_{j}

B_{i} = {μ_{i}}^{j} H_{j}

E_{i} = {(ϵ^{- 1})_{i}}^{j} D_{j}

H_{i} = {(μ^{- 1})_{i}}^{j} B_{j}

{D^{i}}_{, i} = ρ_{f r e e}

... where

ρ_{f r e e}

= free charge (densitized?)

{B^{i}}_{, i} = 0

D_{i, t} = {ϵ_{i}}^{j k} H_{k, j} - J_{i, f r e e}

B_{i, t} = - {ϵ_{i}}^{j k} E_{k, j}

D_{i, t} = {ϵ_{i}}^{j k} (μ^{- 1} B)_{k, j} - J_{i, f r e e}

B_{i, t} = - {ϵ_{i}}^{j k} (ϵ^{- 1} D)_{k, j}

D_{i, t} = {ϵ_{i}}^{j k} ({(μ^{- 1})_{k}}^{l} B_{l})_{, j} - J_{i, f r e e}

B_{i, t} = - {ϵ_{i}}^{j k} ({(ϵ^{- 1})_{k}}^{l} D_{l})_{, j}

D_{i, t} - γ_{i m} ϵ^{m j k} {(μ^{- 1})_{k}}^{l} {B_{l}}_{, j} = {ϵ_{i}}^{j k} {(μ^{- 1})_{k}}^{l}_{, j} B_{l} - J_{i, f r e e}

B_{i, t} + γ_{i m} ϵ^{m j k} {(ϵ^{- 1})_{k}}^{l} D_{l, j} = {ϵ_{i}}^{j k} {(ϵ^{- 1})_{k}}^{l}_{, j} D_{l}

D_{i, t} - \frac{1}{\sqrt{γ}} γ_{i m} {\bar{ϵ}}^{m j k} {(μ^{- 1})_{k}}^{l} {B_{l}}_{, j} = {ϵ_{i}}^{j k} {(μ^{- 1})_{k}}^{l}_{, j} B_{l} - J_{i, f r e e}

B_{i, t} + \frac{1}{\sqrt{γ}} γ_{i m} {\bar{ϵ}}^{m j k} {(ϵ^{- 1})_{k}}^{l} D_{l, j} = {ϵ_{i}}^{j k} {(ϵ^{- 1})_{k}}^{l}_{, j} D_{l}

(\begin{matrix} D_{i, t} \\ B_{i, t} \end{matrix}) + \frac{1}{\sqrt{γ}} (\begin{matrix} γ_{i m} & 0 \\ 0 & γ_{i m} \end{matrix}) (\begin{matrix} 0 & - {\bar{ϵ}}^{m j k} \\ {\bar{ϵ}}^{m j k} & 0 \end{matrix}) (\begin{matrix} {(μ^{- 1})_{k}}^{l} & 0 \\ 0 & {(ϵ^{- 1})_{k}}^{l} \end{matrix}) (\begin{matrix} {B_{l}}_{, j} \\ D_{l, j} \end{matrix}) = (\begin{matrix} {ϵ_{i}}^{j k} {(μ^{- 1})_{k}}^{l}_{, j} B_{l} - J_{i, f r e e} \\ {ϵ_{i}}^{j k} {(ϵ^{- 1})_{k}}^{l}_{, j} D_{l} \end{matrix})