Maxwell hyperbolic conservation law

${ A} _u$ = Electromagnetic 4-potential

Faraday tensor:
${{{ F} _u} _v} = {{{{ A} _v} _{;u}}{-{{{ A} _u} _{;v}}}}$
due to symmetry of connections...
${{{ F} _u} _v} = {{{{ A} _v} _{,u}}{-{{{ A} _u} _{,v}}}}$

dense:

${\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} { {{ F} _t} _t}& { {{ F} _t} _x}& { {{ F} _t} _y}& { {{ F} _t} _z}\\ { {{ F} _x} _t}& { {{ F} _x} _x}& { {{ F} _x} _y}& { {{ F} _x} _z}\\ { {{ F} _y} _t}& { {{ F} _y} _x}& { {{ F} _y} _y}& { {{ F} _y} _z}\\ { {{ F} _z} _t}& { {{ F} _z} _x}& { {{ F} _z} _y}& { {{ F} _z} _z}\end{array}\right]}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} 0& {-{{ {{ A} _t} _{,x}}}} + {{ {{ A} _x} _{,t}}}& {-{{ {{ A} _t} _{,y}}}} + {{ {{ A} _y} _{,t}}}& {-{{ {{ A} _t} _{,z}}}} + {{ {{ A} _z} _{,t}}}\\ {{ {{ A} _t} _{,x}}}{-{{ {{ A} _x} _{,t}}}}& 0& {-{{ {{ A} _x} _{,y}}}} + {{ {{ A} _y} _{,x}}}& {-{{ {{ A} _x} _{,z}}}} + {{ {{ A} _z} _{,x}}}\\ {{ {{ A} _t} _{,y}}}{-{{ {{ A} _y} _{,t}}}}& {{ {{ A} _x} _{,y}}}{-{{ {{ A} _y} _{,x}}}}& 0& {-{{ {{ A} _y} _{,z}}}} + {{ {{ A} _z} _{,y}}}\\ {{ {{ A} _t} _{,z}}}{-{{ {{ A} _z} _{,t}}}}& {{ {{ A} _x} _{,z}}}{-{{ {{ A} _z} _{,x}}}}& {{ {{ A} _y} _{,z}}}{-{{ {{ A} _z} _{,y}}}}& 0\end{array}\right]}}$
${\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} { {{ F} _t} _t}& { {{ F} _t} _x}& { {{ F} _t} _y}& { {{ F} _t} _z}\\ { {{ F} _x} _t}& { {{ F} _x} _x}& { {{ F} _x} _y}& { {{ F} _x} _z}\\ { {{ F} _y} _t}& { {{ F} _y} _x}& { {{ F} _y} _y}& { {{ F} _y} _z}\\ { {{ F} _z} _t}& { {{ F} _z} _x}& { {{ F} _z} _y}& { {{ F} _z} _z}\end{array}\right]}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} 0& {-{{ {{ A} _t} _{,x}}}} + {{ {{ A} _x} _{,t}}}& {-{{ {{ A} _t} _{,y}}}} + {{ {{ A} _y} _{,t}}}& {-{{ {{ A} _t} _{,z}}}} + {{ {{ A} _z} _{,t}}}\\ {{ {{ A} _t} _{,x}}}{-{{ {{ A} _x} _{,t}}}}& 0& {-{{ {{ A} _x} _{,y}}}} + {{ {{ A} _y} _{,x}}}& {-{{ {{ A} _x} _{,z}}}} + {{ {{ A} _z} _{,x}}}\\ {{ {{ A} _t} _{,y}}}{-{{ {{ A} _y} _{,t}}}}& {{ {{ A} _x} _{,y}}}{-{{ {{ A} _y} _{,x}}}}& 0& {-{{ {{ A} _y} _{,z}}}} + {{ {{ A} _z} _{,y}}}\\ {{ {{ A} _t} _{,z}}}{-{{ {{ A} _z} _{,t}}}}& {{ {{ A} _x} _{,z}}}{-{{ {{ A} _z} _{,x}}}}& {{ {{ A} _y} _{,z}}}{-{{ {{ A} _z} _{,y}}}}& 0\end{array}\right]}}$

${{ E} ^u} = {{{{{ F} ^u} ^v}} {{{ n} _v}}}$

${{ E} _i} = {{{{ A} _t} _{,i}}{-{{{ A} _i} _{,t}}}}$

${{ B} ^u} = {{{{{{{ \epsilon} ^u} ^v} ^a} ^b}} {{{{ F} _a} _b}} {{{ n} _v}}}$

${{ B} _i} = {{{{{{ \epsilon} _i} ^j} ^k}} {{{{ A} _k} _{,j}}}}$

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

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

${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{cccccc} 0& 0& 0& \frac{{{{{ {(n_1)}} _2}} {{{{ g} _1} _3}}}{-{{{{ {(n_1)}} _3}} {{{{ g} _1} _2}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& \frac{{-{{{{ {(n_1)}} _1}} {{{{ g} _1} _3}}}} + {{{{ {(n_1)}} _3}} {{{{ g} _1} _1}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_1)}} _1}} {{{{ g} _1} _2}}}{-{{{{ {(n_1)}} _2}} {{{{ g} _1} _1}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}\\ 0& 0& 0& \frac{{{{{ {(n_1)}} _2}} {{{{ g} _2} _3}}}{-{{{{ {(n_1)}} _3}} {{{{ g} _2} _2}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& \frac{{-{{{{ {(n_1)}} _1}} {{{{ g} _2} _3}}}} + {{{{ {(n_1)}} _3}} {{{{ g} _1} _2}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_1)}} _1}} {{{{ g} _2} _2}}}{-{{{{ {(n_1)}} _2}} {{{{ g} _1} _2}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}\\ 0& 0& 0& \frac{{{{{ {(n_1)}} _2}} {{{{ g} _3} _3}}}{-{{{{ {(n_1)}} _3}} {{{{ g} _2} _3}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& \frac{{-{{{{ {(n_1)}} _1}} {{{{ g} _3} _3}}}} + {{{{ {(n_1)}} _3}} {{{{ g} _1} _3}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_1)}} _1}} {{{{ g} _2} _3}}}{-{{{{ {(n_1)}} _2}} {{{{ g} _1} _3}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}\\ \frac{{-{{{{ {(n_1)}} _2}} {{{{ g} _1} _3}}}} + {{{{ {(n_1)}} _3}} {{{{ g} _1} _2}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_1)}} _1}} {{{{ g} _1} _3}}}{-{{{{ {(n_1)}} _3}} {{{{ g} _1} _1}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& \frac{{-{{{{ {(n_1)}} _1}} {{{{ g} _1} _2}}}} + {{{{ {(n_1)}} _2}} {{{{ g} _1} _1}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& 0& 0\\ \frac{{-{{{{ {(n_1)}} _2}} {{{{ g} _2} _3}}}} + {{{{ {(n_1)}} _3}} {{{{ g} _2} _2}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_1)}} _1}} {{{{ g} _2} _3}}}{-{{{{ {(n_1)}} _3}} {{{{ g} _1} _2}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& \frac{{-{{{{ {(n_1)}} _1}} {{{{ g} _2} _2}}}} + {{{{ {(n_1)}} _2}} {{{{ g} _1} _2}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& 0& 0\\ \frac{{-{{{{ {(n_1)}} _2}} {{{{ g} _3} _3}}}} + {{{{ {(n_1)}} _3}} {{{{ g} _2} _3}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_1)}} _1}} {{{{ g} _3} _3}}}{-{{{{ {(n_1)}} _3}} {{{{ g} _1} _3}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& \frac{{-{{{{ {(n_1)}} _1}} {{{{ g} _2} _3}}}} + {{{{ {(n_1)}} _2}} {{{{ g} _1} _3}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& 0& 0\end{array}\right]}$

with a Cartesian metric:
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{cccccc} 0& 0& 0& 0& {\frac{1}{\mu}} {{ {(n_1)}} _3}& -{{\frac{1}{\mu}} {{ {(n_1)}} _2}}\\ 0& 0& 0& -{{\frac{1}{\mu}} {{ {(n_1)}} _3}}& 0& {\frac{1}{\mu}} {{ {(n_1)}} _1}\\ 0& 0& 0& {\frac{1}{\mu}} {{ {(n_1)}} _2}& -{{\frac{1}{\mu}} {{ {(n_1)}} _1}}& 0\\ 0& -{{\frac{1}{\epsilon}} {{ {(n_1)}} _3}}& {\frac{1}{\epsilon}} {{ {(n_1)}} _2}& 0& 0& 0\\ {\frac{1}{\epsilon}} {{ {(n_1)}} _3}& 0& -{{\frac{1}{\epsilon}} {{ {(n_1)}} _1}}& 0& 0& 0\\ -{{\frac{1}{\epsilon}} {{ {(n_1)}} _2}}& {\frac{1}{\epsilon}} {{ {(n_1)}} _1}& 0& 0& 0& 0\end{array}\right]}$

${{{\frac{\partial { F} ^I}{\partial { U} ^J}}} {{{ U} ^J}}} = {\left[\begin{array}{c} {\frac{1}{\mu}}{\left({{-{{{{ {(n_1)}} _2}} {{{ B} _3}}}} + {{{{ {(n_1)}} _3}} {{{ B} _2}}}}\right)}\\ {\frac{1}{\mu}}{\left({{{{{ {(n_1)}} _1}} {{{ B} _3}}}{-{{{{ {(n_1)}} _3}} {{{ B} _1}}}}}\right)}\\ {\frac{1}{\mu}}{\left({{-{{{{ {(n_1)}} _1}} {{{ B} _2}}}} + {{{{ {(n_1)}} _2}} {{{ B} _1}}}}\right)}\\ {\frac{1}{\epsilon}}{\left({{{{{ {(n_1)}} _2}} {{{ D} _3}}}{-{{{{ {(n_1)}} _3}} {{{ D} _2}}}}}\right)}\\ {\frac{1}{\epsilon}}{\left({{-{{{{ {(n_1)}} _1}} {{{ D} _3}}}} + {{{{ {(n_1)}} _3}} {{{ D} _1}}}}\right)}\\ {\frac{1}{\epsilon}}{\left({{{{{ {(n_1)}} _1}} {{{ D} _2}}}{-{{{{ {(n_1)}} _2}} {{{ D} _1}}}}}\right)}\end{array}\right]}$

So the flux jacobian is linear wrt the state vector, therefore ${{{\frac{\partial { F} ^I}{\partial { U} ^J}}} {{{ U} ^J}}} = {{ U} ^I}$ . Remember, this is not always the case, esp with nonlinear terms.

with the normal aligned to the x-axis:
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{cccccc} 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& \frac{1}{\mu}\\ 0& 0& 0& 0& -{\frac{1}{\mu}}& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& -{\frac{1}{\epsilon}}& 0& 0& 0\\ 0& \frac{1}{\epsilon}& 0& 0& 0& 0\end{array}\right]}$

I think my formulation is wrong, and that's why I am getting $|n|^2 = \Sigma_i (n_i)^2$ instead of $(n_i n^i)$.
${{|n|}^{2}} = {{{{ {(n_1)}} _1}^{2}} + {{{ {(n_1)}} _2}^{2}} + {{{ {(n_1)}} _3}^{2}}}$

${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\left[\begin{array}{cccccc} 0& 0& 1& 0& 0& 0\\ 0& -{\frac{\sqrt{\epsilon}}{\sqrt{\mu}}}& 0& 0& 0& \frac{\sqrt{\epsilon}}{\sqrt{\mu}}\\ \frac{\sqrt{\epsilon}}{\sqrt{\mu}}& 0& 0& 0& -{\frac{\sqrt{\epsilon}}{\sqrt{\mu}}}& 0\\ 0& 0& 0& 1& 0& 0\\ 1& 0& 0& 0& 1& 0\\ 0& 1& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccccc} -{{{\frac{1}{\sqrt{\epsilon}}}} {{\frac{1}{\sqrt{\mu}}}}}& 0& 0& 0& 0& 0\\ 0& -{{{\frac{1}{\sqrt{\epsilon}}}} {{\frac{1}{\sqrt{\mu}}}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {{\frac{1}{\sqrt{\epsilon}}}} {{\frac{1}{\sqrt{\mu}}}}& 0\\ 0& 0& 0& 0& 0& {{\frac{1}{\sqrt{\epsilon}}}} {{\frac{1}{\sqrt{\mu}}}}\end{array}\right]}} {{\left[\begin{array}{cccccc} 0& 0& \frac{\sqrt{\mu}}{{{2}} {{\sqrt{\epsilon}}}}& 0& \frac{1}{2}& 0\\ 0& -{\frac{\sqrt{\mu}}{{{2}} {{\sqrt{\epsilon}}}}}& 0& 0& 0& \frac{1}{2}\\ 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& -{\frac{\sqrt{\mu}}{{{2}} {{\sqrt{\epsilon}}}}}& 0& \frac{1}{2}& 0\\ 0& \frac{\sqrt{\mu}}{{{2}} {{\sqrt{\epsilon}}}}& 0& 0& 0& \frac{1}{2}\end{array}\right]}}}$

${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{cccccc} 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& \frac{1}{\mu}\\ 0& 0& 0& 0& -{\frac{1}{\mu}}& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& -{\frac{1}{\epsilon}}& 0& 0& 0\\ 0& \frac{1}{\epsilon}& 0& 0& 0& 0\end{array}\right]}$

now in the frame of the normal:
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\left[\begin{array}{cccccc} { {(n_1)}} _1& { {(n_2)}} _1& { {(n_3)}} _1& 0& 0& 0\\ { {(n_1)}} _2& { {(n_2)}} _2& { {(n_3)}} _2& 0& 0& 0\\ { {(n_1)}} _3& { {(n_2)}} _3& { {(n_3)}} _3& 0& 0& 0\\ 0& 0& 0& { {(n_1)}} _1& { {(n_2)}} _1& { {(n_3)}} _1\\ 0& 0& 0& { {(n_1)}} _2& { {(n_2)}} _2& { {(n_3)}} _2\\ 0& 0& 0& { {(n_1)}} _3& { {(n_2)}} _3& { {(n_3)}} _3\end{array}\right]}} {{\left[\begin{array}{cccccc} 0& 0& 1& 0& 0& 0\\ 0& -{\frac{\sqrt{\epsilon}}{\sqrt{\mu}}}& 0& 0& 0& \frac{\sqrt{\epsilon}}{\sqrt{\mu}}\\ \frac{\sqrt{\epsilon}}{\sqrt{\mu}}& 0& 0& 0& -{\frac{\sqrt{\epsilon}}{\sqrt{\mu}}}& 0\\ 0& 0& 0& 1& 0& 0\\ 1& 0& 0& 0& 1& 0\\ 0& 1& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccccc} -{\frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}}& 0& 0& 0& 0& 0\\ 0& -{\frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}& 0\\ 0& 0& 0& 0& 0& \frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}\end{array}\right]}} {{\left[\begin{array}{cccccc} 0& 0& \frac{\sqrt{\mu}}{{{2}} {{\sqrt{\epsilon}}}}& 0& \frac{1}{2}& 0\\ 0& -{\frac{\sqrt{\mu}}{{{2}} {{\sqrt{\epsilon}}}}}& 0& 0& 0& \frac{1}{2}\\ 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& -{\frac{\sqrt{\mu}}{{{2}} {{\sqrt{\epsilon}}}}}& 0& \frac{1}{2}& 0\\ 0& \frac{\sqrt{\mu}}{{{2}} {{\sqrt{\epsilon}}}}& 0& 0& 0& \frac{1}{2}\end{array}\right]}} {{\left[\begin{array}{cccccc} { {(n_1)}} _1& { {(n_1)}} _2& { {(n_1)}} _3& 0& 0& 0\\ { {(n_2)}} _1& { {(n_2)}} _2& { {(n_2)}} _3& 0& 0& 0\\ { {(n_3)}} _1& { {(n_3)}} _2& { {(n_3)}} _3& 0& 0& 0\\ 0& 0& 0& { {(n_1)}} _1& { {(n_1)}} _2& { {(n_1)}} _3\\ 0& 0& 0& { {(n_2)}} _1& { {(n_2)}} _2& { {(n_2)}} _3\\ 0& 0& 0& { {(n_3)}} _1& { {(n_3)}} _2& { {(n_3)}} _3\end{array}\right]}}}$

${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\left[\begin{array}{cccccc} \frac{{{{ {(n_3)}} _1}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}& -{\frac{{{{ {(n_2)}} _1}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}}& { {(n_1)}} _1& 0& -{\frac{{{{ {(n_3)}} _1}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}}& \frac{{{{ {(n_2)}} _1}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}\\ \frac{{{{ {(n_3)}} _2}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}& -{\frac{{{{ {(n_2)}} _2}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}}& { {(n_1)}} _2& 0& -{\frac{{{{ {(n_3)}} _2}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}}& \frac{{{{ {(n_2)}} _2}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}\\ \frac{{{{ {(n_3)}} _3}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}& -{\frac{{{{ {(n_2)}} _3}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}}& { {(n_1)}} _3& 0& -{\frac{{{{ {(n_3)}} _3}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}}& \frac{{{{ {(n_2)}} _3}} {{\sqrt{\epsilon}}}}{\sqrt{\mu}}\\ { {(n_2)}} _1& { {(n_3)}} _1& 0& { {(n_1)}} _1& { {(n_2)}} _1& { {(n_3)}} _1\\ { {(n_2)}} _2& { {(n_3)}} _2& 0& { {(n_1)}} _2& { {(n_2)}} _2& { {(n_3)}} _2\\ { {(n_2)}} _3& { {(n_3)}} _3& 0& { {(n_1)}} _3& { {(n_2)}} _3& { {(n_3)}} _3\end{array}\right]}} {{\left[\begin{array}{cccccc} -{\frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}}& 0& 0& 0& 0& 0\\ 0& -{\frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}& 0\\ 0& 0& 0& 0& 0& \frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}\end{array}\right]}} {{\left[\begin{array}{cccccc} \frac{{{{ {(n_3)}} _1}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}& \frac{{{{ {(n_3)}} _2}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}& \frac{{{{ {(n_3)}} _3}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}& {\frac{1}{2}} {{ {(n_2)}} _1}& {\frac{1}{2}} {{ {(n_2)}} _2}& {\frac{1}{2}} {{ {(n_2)}} _3}\\ -{\frac{{{{ {(n_2)}} _1}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}}& -{\frac{{{{ {(n_2)}} _2}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}}& -{\frac{{{{ {(n_2)}} _3}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}}& {\frac{1}{2}} {{ {(n_3)}} _1}& {\frac{1}{2}} {{ {(n_3)}} _2}& {\frac{1}{2}} {{ {(n_3)}} _3}\\ { {(n_1)}} _1& { {(n_1)}} _2& { {(n_1)}} _3& 0& 0& 0\\ 0& 0& 0& { {(n_1)}} _1& { {(n_1)}} _2& { {(n_1)}} _3\\ -{\frac{{{{ {(n_3)}} _1}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}}& -{\frac{{{{ {(n_3)}} _2}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}}& -{\frac{{{{ {(n_3)}} _3}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}}& {\frac{1}{2}} {{ {(n_2)}} _1}& {\frac{1}{2}} {{ {(n_2)}} _2}& {\frac{1}{2}} {{ {(n_2)}} _3}\\ \frac{{{{ {(n_2)}} _1}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}& \frac{{{{ {(n_2)}} _2}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}& \frac{{{{ {(n_2)}} _3}} {{\sqrt{\mu}}}}{{{2}} {{\sqrt{\epsilon}}}}& {\frac{1}{2}} {{ {(n_3)}} _1}& {\frac{1}{2}} {{ {(n_3)}} _2}& {\frac{1}{2}} {{ {(n_3)}} _3}\end{array}\right]}}}$

${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{cccccc} 0& 0& 0& 0& \frac{{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _2}}}{-{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _1}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _3}}}{-{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _1}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}\\ 0& 0& 0& \frac{{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _1}}}{-{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _2}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& 0& \frac{{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _3}}}{-{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _2}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}\\ 0& 0& 0& \frac{{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _1}}}{-{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _3}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _2}}}{-{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _3}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& 0\\ 0& \frac{{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _1}}}{-{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _2}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _1}}}{-{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _3}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& 0& 0\\ \frac{{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _2}}}{-{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _1}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& \frac{{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _2}}}{-{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _3}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& 0& 0\\ \frac{{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _3}}}{-{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _1}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _3}}}{-{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _2}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& 0& 0& 0\end{array}\right]}$

R(v) = $\left[\begin{array}{c} {{{{v_1}}} \cdot {{{ {(n_3)}} _1}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}{-{{{{v_2}}} \cdot {{{ {(n_2)}} _1}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}} + {{{{v_3}}} \cdot {{{ {(n_1)}} _1}}}{-{{{{v_5}}} \cdot {{{ {(n_3)}} _1}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}} + {{{{v_6}}} \cdot {{{ {(n_2)}} _1}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}\\ {{{{v_1}}} \cdot {{{ {(n_3)}} _2}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}{-{{{{v_2}}} \cdot {{{ {(n_2)}} _2}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}} + {{{{v_3}}} \cdot {{{ {(n_1)}} _2}}}{-{{{{v_5}}} \cdot {{{ {(n_3)}} _2}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}} + {{{{v_6}}} \cdot {{{ {(n_2)}} _2}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}\\ {{{{v_1}}} \cdot {{{ {(n_3)}} _3}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}{-{{{{v_2}}} \cdot {{{ {(n_2)}} _3}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}} + {{{{v_3}}} \cdot {{{ {(n_1)}} _3}}}{-{{{{v_5}}} \cdot {{{ {(n_3)}} _3}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}} + {{{{v_6}}} \cdot {{{ {(n_2)}} _3}} {{\sqrt{\epsilon}}} {{\frac{1}{\sqrt{\mu}}}}}\\ {{{{v_1}}} \cdot {{{ {(n_2)}} _1}}} + {{{{v_2}}} \cdot {{{ {(n_3)}} _1}}} + {{{{v_4}}} \cdot {{{ {(n_1)}} _1}}} + {{{{v_5}}} \cdot {{{ {(n_2)}} _1}}} + {{{{v_6}}} \cdot {{{ {(n_3)}} _1}}}\\ {{{{v_1}}} \cdot {{{ {(n_2)}} _2}}} + {{{{v_2}}} \cdot {{{ {(n_3)}} _2}}} + {{{{v_4}}} \cdot {{{ {(n_1)}} _2}}} + {{{{v_5}}} \cdot {{{ {(n_2)}} _2}}} + {{{{v_6}}} \cdot {{{ {(n_3)}} _2}}}\\ {{{{v_1}}} \cdot {{{ {(n_2)}} _3}}} + {{{{v_2}}} \cdot {{{ {(n_3)}} _3}}} + {{{{v_4}}} \cdot {{{ {(n_1)}} _3}}} + {{{{v_5}}} \cdot {{{ {(n_2)}} _3}}} + {{{{v_6}}} \cdot {{{ {(n_3)}} _3}}}\end{array}\right]$

L(v) = $\left[\begin{array}{c} {{{\frac{1}{2}}} {{{v_1}}} \cdot {{{ {(n_3)}} _1}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{\frac{1}{2}}} {{{v_2}}} \cdot {{{ {(n_3)}} _2}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{\frac{1}{2}}} {{{v_3}}} \cdot {{{ {(n_3)}} _3}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{\frac{1}{2}}} {{{v_4}}} \cdot {{{ {(n_2)}} _1}}} + {{{\frac{1}{2}}} {{{v_5}}} \cdot {{{ {(n_2)}} _2}}} + {{{\frac{1}{2}}} {{{v_6}}} \cdot {{{ {(n_2)}} _3}}}\\ {-{{{\frac{1}{2}}} {{{v_1}}} \cdot {{{ {(n_2)}} _1}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}}}{-{{{\frac{1}{2}}} {{{v_2}}} \cdot {{{ {(n_2)}} _2}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}}}{-{{{\frac{1}{2}}} {{{v_3}}} \cdot {{{ {(n_2)}} _3}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}}} + {{{\frac{1}{2}}} {{{v_4}}} \cdot {{{ {(n_3)}} _1}}} + {{{\frac{1}{2}}} {{{v_5}}} \cdot {{{ {(n_3)}} _2}}} + {{{\frac{1}{2}}} {{{v_6}}} \cdot {{{ {(n_3)}} _3}}}\\ {{{{v_1}}} \cdot {{{ {(n_1)}} _1}}} + {{{{v_2}}} \cdot {{{ {(n_1)}} _2}}} + {{{{v_3}}} \cdot {{{ {(n_1)}} _3}}}\\ {{{{v_4}}} \cdot {{{ {(n_1)}} _1}}} + {{{{v_5}}} \cdot {{{ {(n_1)}} _2}}} + {{{{v_6}}} \cdot {{{ {(n_1)}} _3}}}\\ {-{{{\frac{1}{2}}} {{{v_1}}} \cdot {{{ {(n_3)}} _1}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}}}{-{{{\frac{1}{2}}} {{{v_2}}} \cdot {{{ {(n_3)}} _2}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}}}{-{{{\frac{1}{2}}} {{{v_3}}} \cdot {{{ {(n_3)}} _3}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}}} + {{{\frac{1}{2}}} {{{v_4}}} \cdot {{{ {(n_2)}} _1}}} + {{{\frac{1}{2}}} {{{v_5}}} \cdot {{{ {(n_2)}} _2}}} + {{{\frac{1}{2}}} {{{v_6}}} \cdot {{{ {(n_2)}} _3}}}\\ {{{\frac{1}{2}}} {{{v_1}}} \cdot {{{ {(n_2)}} _1}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{\frac{1}{2}}} {{{v_2}}} \cdot {{{ {(n_2)}} _2}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{\frac{1}{2}}} {{{v_3}}} \cdot {{{ {(n_2)}} _3}} {{\sqrt{\mu}}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{\frac{1}{2}}} {{{v_4}}} \cdot {{{ {(n_3)}} _1}}} + {{{\frac{1}{2}}} {{{v_5}}} \cdot {{{ {(n_3)}} _2}}} + {{{\frac{1}{2}}} {{{v_6}}} \cdot {{{ {(n_3)}} _3}}}\end{array}\right]$

left transform code:

real const tmp1 = 1. / (eig)->sqrt_1_mu;
real const tmp2 = n3_l.$x$ * tmp1;
real const tmp3 = (eig)->sqrt_1_eps * tmp2;
real const tmp4 = 1. / 2.;
real const tmp5 = (X)->ptr[0] * tmp3;
real const tmp6 = n3_l.$y$ * tmp1;
real const tmp7 = (eig)->sqrt_1_eps * tmp6;
real const tmp8 = (X)->ptr[1] * tmp7;
real const tmp9 = n3_l.$z$ * tmp1;
real const tmp10 = (eig)->sqrt_1_eps * tmp9;
real const tmp11 = (X)->ptr[2] * tmp10;
real const tmp12 = (X)->ptr[3] * n2_l.$x$;
real const tmp13 = (X)->ptr[4] * n2_l.$y$;
real const tmp14 = (X)->ptr[5] * n2_l.$z$;
real const tmp15 = tmp4 * tmp13;
real const tmp16 = tmp4 * tmp14;
real const tmp17 = tmp4 * tmp12;
real const tmp18 = tmp15 + tmp16;
real const tmp19 = tmp4 * tmp11;
real const tmp20 = tmp17 + tmp18;
real const tmp21 = tmp4 * tmp8;
real const tmp22 = tmp4 * tmp5;
real const tmp23 = n2_l.$x$ * tmp1;
real const tmp24 = (eig)->sqrt_1_eps * tmp23;
real const tmp25 = (X)->ptr[0] * tmp24;
real const tmp26 = tmp4 * tmp25;
real const tmp27 = n2_l.$y$ * tmp1;
real const tmp28 = (eig)->sqrt_1_eps * tmp27;
real const tmp29 = (X)->ptr[1] * tmp28;
real const tmp30 = tmp4 * tmp29;
real const tmp31 = n2_l.$z$ * tmp1;
real const tmp32 = (eig)->sqrt_1_eps * tmp31;
real const tmp33 = (X)->ptr[2] * tmp32;
real const tmp34 = tmp4 * tmp33;
real const tmp35 = (X)->ptr[3] * n3_l.$x$;
real const tmp36 = (X)->ptr[4] * n3_l.$y$;
real const tmp37 = (X)->ptr[5] * n3_l.$z$;
real const tmp38 = tmp4 * tmp36;
real const tmp39 = tmp4 * tmp37;
real const tmp40 = tmp4 * tmp35;
real const tmp41 = tmp38 + tmp39;
real const tmp42 = tmp40 + tmp41;
real const (Y)->ptr[0] = tmp19 + tmp20 + tmp21 + tmp22;
real const (Y)->ptr[1] = -tmp26 + -tmp30 + -tmp34 + tmp42;
real const (Y)->ptr[2] = (X)->ptr[0] * n_l.$x$ + (X)->ptr[2] * n_l.$z$ + (X)->ptr[1] * n_l.$y$;
real const (Y)->ptr[3] = (X)->ptr[3] * n_l.$x$ + (X)->ptr[5] * n_l.$z$ + (X)->ptr[4] * n_l.$y$;
real const (Y)->ptr[4] = -tmp22 + -tmp21 + -tmp19 + tmp20;
real const (Y)->ptr[5] = tmp26 + tmp30 + tmp34 + tmp42;

right transform code:

real const tmp1 = 1. / (eig)->sqrt_1_eps;
real const tmp2 = n3_l.$x$ * tmp1;
real const tmp3 = (eig)->sqrt_1_mu * tmp2;
real const tmp4 = n2_l.$x$ * tmp1;
real const tmp5 = (eig)->sqrt_1_mu * tmp4;
real const tmp6 = n3_l.$y$ * tmp1;
real const tmp7 = (eig)->sqrt_1_mu * tmp6;
real const tmp8 = n2_l.$y$ * tmp1;
real const tmp9 = (eig)->sqrt_1_mu * tmp8;
real const tmp10 = n3_l.$z$ * tmp1;
real const tmp11 = (eig)->sqrt_1_mu * tmp10;
real const tmp12 = n2_l.$z$ * tmp1;
real const tmp13 = (eig)->sqrt_1_mu * tmp12;
real const (Y)->ptr[0] = (X)->ptr[0] * tmp3 + -(X)->ptr[1] * tmp5 + (X)->ptr[2] * n_l.$x$ + (X)->ptr[5] * tmp5 + -(X)->ptr[4] * tmp3;
real const (Y)->ptr[1] = (X)->ptr[0] * tmp7 + -(X)->ptr[1] * tmp9 + (X)->ptr[2] * n_l.$y$ + (X)->ptr[5] * tmp9 + -(X)->ptr[4] * tmp7;
real const (Y)->ptr[2] = (X)->ptr[0] * tmp11 + -(X)->ptr[1] * tmp13 + (X)->ptr[2] * n_l.$z$ + (X)->ptr[5] * tmp13 + -(X)->ptr[4] * tmp11;
real const (Y)->ptr[3] = (X)->ptr[0] * n2_l.$x$ + (X)->ptr[1] * n3_l.$x$ + (X)->ptr[3] * n_l.$x$ + (X)->ptr[5] * n3_l.$x$ + (X)->ptr[4] * n2_l.$x$;
real const (Y)->ptr[4] = (X)->ptr[0] * n2_l.$y$ + (X)->ptr[1] * n3_l.$y$ + (X)->ptr[3] * n_l.$y$ + (X)->ptr[5] * n3_l.$y$ + (X)->ptr[4] * n2_l.$y$;
real const (Y)->ptr[5] = (X)->ptr[0] * n2_l.$z$ + (X)->ptr[1] * n3_l.$z$ + (X)->ptr[3] * n_l.$z$ + (X)->ptr[5] * n3_l.$z$ + (X)->ptr[4] * n2_l.$z$;

combining:

${{{\left( {{\epsilon}} \cdot {{{ E} ^i}}\right)} _{,t}}{-{{\left( {{\frac{1}{\mu}}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{{{ \epsilon} _k} ^l} ^m}} {{{{ A} _m} _{,l}}}\right)} _{,j}}}} = {{ J} ^i}$
${{{\left( {{\epsilon}} \cdot {{\left({{{{ A} _t} _{,i}}{-{{{ A} _i} _{,t}}}}\right)}}\right)} _{,t}}{-{{\left( {{\frac{1}{\mu}}} {{{{{ \epsilon} ^i} ^j} ^k}} {{{{{ \epsilon} _k} ^l} ^m}} {{{{ A} _m} _{,l}}}\right)} _{,j}}}} = {{ J} ^i}$
${{{{{\epsilon}} \cdot {{{{{ A} _t} _{,i}} _{,t}}}}{-{{{\epsilon}} \cdot {{{{{ A} _i} _{,t}} _{,t}}}}}}{-{{\left( {{\frac{1}{\mu}}} {{\left({{{{{{ δ} ^i} ^l}} {{{{ δ} ^j} ^m}}}{-{{{{{ δ} ^i} ^m}} {{{{ δ} ^j} ^l}}}}}\right)}} {{{{ A} _m} _{,l}}}\right)} _{,j}}}} = {{ J} ^i}$
${{{{\epsilon}} \cdot {{{{{ A} _t} _{,t}} _{,i}}}}{-{{{\epsilon}} \cdot {{{{{ A} _i} _{,t}} _{,t}}}}}{-{{{\frac{1}{\mu}}} {{{{{ A} _j} _{,j}} _{,i}}}}} + {{{\frac{1}{\mu}}} {{{{{ A} _i} _{,j}} _{,j}}}}} = {{ J} _i}$
using Lorentz gauge: ${{-{{{ A} _t} _{,t}}} + {{{ A} _i} _{,i}}} = {0}$
${{{{{ A} _i} _{,t}} _{,t}}{-{{{{c}^{2}}} {{{{{ A} _i} _{,j}} ^{,j}}}}}} = {{{{\frac{1}{\epsilon}}} {{{ J} _i}}}{-{{{{c}^{2}}} {{{{ R} _i} _u}} {{{ A} _u}}}}}$
...
${{{{{ A} _t} _{,t}} _{,t}}{-{{{{c}^{2}}} {{{{{ A} _t} _{,j}} ^{,j}}}}}} = {{{{\frac{1}{\epsilon}}} {{{ J} _t}}}{-{{{{c}^{2}}} {{{{ R} _t} _u}} {{{ A} _u}}}}}$