${ A} _u$ = Electromagnetic 4-potential

Faraday tensor:
${{{ F} _u} _v} = {{{{ A} _v} _{,u}}{-{{{ A} _u} _{,v}}}}$

Maxwell equations:
${{{{ F} ^u} ^v} _{;v}} = {{{\mu}} \cdot {{{ J} ^u}}}$
${{{{ \star F} ^u} ^v} _{;v}} = {0}$
${{{\frac{1}{2}}} {{{{{{ \epsilon} ^u} ^v} ^a} ^b}} {{{{{ F} _a} _b} _{;v}}}} = {0}$

hyperbolic balance law:

${\int{{\left({{\frac{\partial { U} ^I}{\partial t}} + {\frac{\partial { F} ^I}{\partial x}}}\right)}}d { x} ^4} = {\int{{{ S} ^I}}d { x} ^4}$

separating state variables:

${{{{ D} ^i} _{,t}}{-{{\left( {{\frac{1}{\mu}}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{ B} _k}}\right)} _{,j}}}} = {{ J} ^i}$
${{{{ B} ^i} _{,t}} + {{\left( {{\frac{1}{\epsilon}}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{ D} _k}}\right)} _{,j}}} = {0}$

${{{{ D} ^i} _{,t}}{-{{\left( {{\frac{1}{\mu}}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{{ g} _k} _l}} {{{ B} ^l}}\right)} _{,j}}}} = {{ J} ^i}$
${{{{ B} ^i} _{,t}} + {{\left( {{\frac{1}{\epsilon}}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{{ g} _k} _l}} {{{ D} ^l}}\right)} _{,j}}} = {0}$

${{{{ D} ^i} _{,t}}{-{{{{ B} ^l}} {{{{ g} _k} _l}} {{{{{{ \epsilon} ^i} ^j} ^k} _{,j}}} {{\frac{1}{\mu}}}}}{-{{{{ B} ^l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{{{ g} _k} _l} _{,j}}} {{\frac{1}{\mu}}}}}{-{{{{{ B} ^l} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{\mu}}}}} + {{{{ B} ^l}} {{{ \mu} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{{\mu}^{2}}}}}} = {{ J} ^i}$
${{{{ B} ^i} _{,t}} + {{{{ D} ^l}} {{{{ g} _k} _l}} {{{{{{ \epsilon} ^i} ^j} ^k} _{,j}}} {{\frac{1}{\epsilon}}}} + {{{{ D} ^l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{{{ g} _k} _l} _{,j}}} {{\frac{1}{\epsilon}}}} + {{{{{ D} ^l} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{\epsilon}}}}{-{{{{ D} ^l}} {{{ \epsilon} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{{\epsilon}^{2}}}}}}} = {0}$

${{{{ D} ^i} _{,t}}{-{{{{ B} ^l}} {{{{ g} _k} _l}} {{\frac{1}{\mu}}} {{{\left( \frac{1}{\sqrt{g}}\right)} _{,j}}} {{{{{ \bar{\epsilon}} ^i} ^j} ^k}}}}{-{{{{ B} ^l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{{{ g} _k} _l} _{,j}}} {{\frac{1}{\mu}}}}}{-{{{{{ B} ^l} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{\mu}}}}} + {{{{ B} ^l}} {{{ \mu} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{{\mu}^{2}}}}}} = {{ J} ^i}$
${{{{ B} ^i} _{,t}} + {{{{ D} ^l}} {{{{ g} _k} _l}} {{\frac{1}{\epsilon}}} {{{\left( \frac{1}{\sqrt{g}}\right)} _{,j}}} {{{{{ \bar{\epsilon}} ^i} ^j} ^k}}} + {{{{ D} ^l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{{{ g} _k} _l} _{,j}}} {{\frac{1}{\epsilon}}}} + {{{{{ D} ^l} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{\epsilon}}}}{-{{{{ D} ^l}} {{{ \epsilon} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{{\epsilon}^{2}}}}}}} = {0}$

${{{{ D} ^i} _{,t}} + {{{\frac{1}{2}}} {{{ B} ^l}} {{{ g} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{\mu}}} {{\frac{1}{g}}}}{-{{{{ B} ^l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{{{ g} _k} _l} _{,j}}} {{\frac{1}{\mu}}}}}{-{{{{{ B} ^l} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{\mu}}}}} + {{{{ B} ^l}} {{{ \mu} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{{\mu}^{2}}}}}} = {{ J} ^i}$
${{{{ B} ^i} _{,t}}{-{{{\frac{1}{2}}} {{{ D} ^l}} {{{ g} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{\epsilon}}} {{\frac{1}{g}}}}} + {{{{ D} ^l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{{{ g} _k} _l} _{,j}}} {{\frac{1}{\epsilon}}}} + {{{{{ D} ^l} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{\epsilon}}}}{-{{{{ D} ^l}} {{{ \epsilon} _{,j}}} {{{{ g} _k} _l}} {{{{{ \epsilon} ^i} ^j} ^k}} {{\frac{1}{{\epsilon}^{2}}}}}}} = {0}$

Conserved quantities:
${{ U} ^I} = {\left[\begin{array}{c} { D} ^i\\ { B} ^i\end{array}\right]}$

Flux:
${{ F} ^I} = {\left[\begin{array}{c} -{{{\frac{1}{\mu}}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{ n} _j}} {{{ B} _k}}}\\ {{\frac{1}{\epsilon}}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{ n} _j}} {{{ D} _k}}\end{array}\right]}$

${{{ \left[\begin{array}{c} { D} ^i\\ { B} ^i\end{array}\right]} _{,t}} + {{{\left[\begin{array}{cc} 0& {-{\frac{1}{\mu}}} {{{{{ \epsilon} ^i} ^l} ^k}}\\ {{\frac{1}{\epsilon}}} {{{{{ \epsilon} ^i} ^l} ^k}}& 0\end{array}\right]}} {{{ \left[\begin{array}{c} { D} _k\\ { B} _k\end{array}\right]} _{,j}}} {{{ n} _l}} {{{ n} ^j}}}} = {\left[\begin{array}{c} { J} ^i\\ 0\end{array}\right]}$

neglecting normal gradient (which should emerge as an extra extrinsic curvature source term):

${{{ \left[\begin{array}{c} { D} ^i\\ { B} ^i\end{array}\right]} _{,t}} + {{{\left[\begin{array}{cc} 0& {-{\frac{1}{\mu}}} {{{{ g} ^i} ^m}} {{{{{ \epsilon} _m} _l} _k}} {{{ n} ^l}}\\ {{\frac{1}{\epsilon}}} {{{{ g} ^i} ^m}} {{{{{ \epsilon} _m} _l} _k}} {{{ n} ^l}}& 0\end{array}\right]}} {{{ \left[\begin{array}{c} { D} ^k\\ { B} ^k\end{array}\right]} _{,j}}} {{{ n} ^j}}}} = {\left[\begin{array}{c} { J} ^i\\ 0\end{array}\right]}$

or in lowered form:
${{{ \left[\begin{array}{c} { D} _i\\ { B} _i\end{array}\right]} _{,t}} + {{{\left[\begin{array}{cc} 0& {-{\frac{1}{\mu}}} {{{{ g} _i} _m}} {{{{{ \epsilon} ^m} ^l} ^j}} {{{ n} _l}}\\ {{\frac{1}{\epsilon}}} {{{{ g} _i} _m}} {{{{{ \epsilon} ^m} ^l} ^j}} {{{ n} _l}}& 0\end{array}\right]}} {{{ \left[\begin{array}{c} { D} _j\\ { B} _j\end{array}\right]} _{,k}}} {{{ n} ^k}}}} = {\left[\begin{array}{c} { J} ^i\\ 0\end{array}\right]}$