$dx^0 = c dt, \partial_0 = \frac{1}{c} \partial_t$
using a metric and look at the dual:
${{{ \eta} _a} _b} = {\left[ \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{matrix} \right]}$
${{ n} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} -1 \\ 0 \\ 0 \\ 0\end{matrix} \right]}}$
four-potential as a one-form:
${{ A} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} {\frac{1}{c}} {\phi} \\ {A_{x}} \\ {A_{y}} \\ {A_{z}}\end{matrix} \right]}}$
$F = dA$
${{{ F} _a} _b} = {\overset{d\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{x}}_{,{{0}}}}}}} + { \phi_{,{{x}}}}}\right)} & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{y}}_{,{{0}}}}}}} + { \phi_{,{{y}}}}}\right)} & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{z}}_{,{{0}}}}}}} + { \phi_{,{{z}}}}}\right)} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{x}}_{,{{0}}}}}} - { \phi_{,{{x}}}}}\right)} & 0 & {-{ {A_{y}}_{,{{x}}}}} + { {A_{x}}_{,{{y}}}} & {-{ {A_{z}}_{,{{x}}}}} + { {A_{x}}_{,{{z}}}} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{y}}_{,{{0}}}}}} - { \phi_{,{{y}}}}}\right)} & {-{ {A_{x}}_{,{{y}}}}} + { {A_{y}}_{,{{x}}}} & 0 & {-{ {A_{z}}_{,{{y}}}}} + { {A_{y}}_{,{{z}}}} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{z}}_{,{{0}}}}}} - { \phi_{,{{z}}}}}\right)} & {-{ {A_{x}}_{,{{z}}}}} + { {A_{z}}_{,{{x}}}} & {-{ {A_{y}}_{,{{z}}}}} + { {A_{z}}_{,{{y}}}} & 0\end{matrix} \right]}}$
$\partial_a F_{bc}$:
${{{{ F} _b} _c} _{,a}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{x}}_{,{{0}}{{0}}}}}}} + { \phi_{,{{0}}{{x}}}}}\right)} & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{y}}_{,{{0}}{{0}}}}}}} + { \phi_{,{{0}}{{y}}}}}\right)} & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{z}}_{,{{0}}{{0}}}}}}} + { \phi_{,{{0}}{{z}}}}}\right)} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{x}}_{,{{0}}{{0}}}}}} - { \phi_{,{{0}}{{x}}}}}\right)} & 0 & {-{ {A_{y}}_{,{{0}}{{x}}}}} + { {A_{x}}_{,{{0}}{{y}}}} & {-{ {A_{z}}_{,{{0}}{{x}}}}} + { {A_{x}}_{,{{0}}{{z}}}} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{y}}_{,{{0}}{{0}}}}}} - { \phi_{,{{0}}{{y}}}}}\right)} & {-{ {A_{x}}_{,{{0}}{{y}}}}} + { {A_{y}}_{,{{0}}{{x}}}} & 0 & {-{ {A_{z}}_{,{{0}}{{y}}}}} + { {A_{y}}_{,{{0}}{{z}}}} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{z}}_{,{{0}}{{0}}}}}} - { \phi_{,{{0}}{{z}}}}}\right)} & {-{ {A_{x}}_{,{{0}}{{z}}}}} + { {A_{z}}_{,{{0}}{{x}}}} & {-{ {A_{y}}_{,{{0}}{{z}}}}} + { {A_{z}}_{,{{0}}{{y}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{x}}_{,{{0}}{{x}}}}}}} + { \phi_{,{{x}}{{x}}}}}\right)} & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{y}}_{,{{0}}{{x}}}}}}} + { \phi_{,{{x}}{{y}}}}}\right)} & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{z}}_{,{{0}}{{x}}}}}}} + { \phi_{,{{x}}{{z}}}}}\right)} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{x}}_{,{{0}}{{x}}}}}} - { \phi_{,{{x}}{{x}}}}}\right)} & 0 & {-{ {A_{y}}_{,{{x}}{{x}}}}} + { {A_{x}}_{,{{x}}{{y}}}} & {-{ {A_{z}}_{,{{x}}{{x}}}}} + { {A_{x}}_{,{{x}}{{z}}}} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{y}}_{,{{0}}{{x}}}}}} - { \phi_{,{{x}}{{y}}}}}\right)} & {-{ {A_{x}}_{,{{x}}{{y}}}}} + { {A_{y}}_{,{{x}}{{x}}}} & 0 & {-{ {A_{z}}_{,{{x}}{{y}}}}} + { {A_{y}}_{,{{x}}{{z}}}} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{z}}_{,{{0}}{{x}}}}}} - { \phi_{,{{x}}{{z}}}}}\right)} & {-{ {A_{x}}_{,{{x}}{{z}}}}} + { {A_{z}}_{,{{x}}{{x}}}} & {-{ {A_{y}}_{,{{x}}{{z}}}}} + { {A_{z}}_{,{{x}}{{y}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{x}}_{,{{0}}{{y}}}}}}} + { \phi_{,{{x}}{{y}}}}}\right)} & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{y}}_{,{{0}}{{y}}}}}}} + { \phi_{,{{y}}{{y}}}}}\right)} & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{z}}_{,{{0}}{{y}}}}}}} + { \phi_{,{{y}}{{z}}}}}\right)} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{x}}_{,{{0}}{{y}}}}}} - { \phi_{,{{x}}{{y}}}}}\right)} & 0 & {-{ {A_{y}}_{,{{x}}{{y}}}}} + { {A_{x}}_{,{{y}}{{y}}}} & {-{ {A_{z}}_{,{{x}}{{y}}}}} + { {A_{x}}_{,{{y}}{{z}}}} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{y}}_{,{{0}}{{y}}}}}} - { \phi_{,{{y}}{{y}}}}}\right)} & {-{ {A_{x}}_{,{{y}}{{y}}}}} + { {A_{y}}_{,{{x}}{{y}}}} & 0 & {-{ {A_{z}}_{,{{y}}{{y}}}}} + { {A_{y}}_{,{{y}}{{z}}}} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{z}}_{,{{0}}{{y}}}}}} - { \phi_{,{{y}}{{z}}}}}\right)} & {-{ {A_{x}}_{,{{y}}{{z}}}}} + { {A_{z}}_{,{{x}}{{y}}}} & {-{ {A_{y}}_{,{{y}}{{z}}}}} + { {A_{z}}_{,{{y}}{{y}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{x}}_{,{{0}}{{z}}}}}}} + { \phi_{,{{x}}{{z}}}}}\right)} & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{y}}_{,{{0}}{{z}}}}}}} + { \phi_{,{{y}}{{z}}}}}\right)} & {\frac{1}{c}}{\left({{-{{{c}} {{ {A_{z}}_{,{{0}}{{z}}}}}}} + { \phi_{,{{z}}{{z}}}}}\right)} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{x}}_{,{{0}}{{z}}}}}} - { \phi_{,{{x}}{{z}}}}}\right)} & 0 & {-{ {A_{y}}_{,{{x}}{{z}}}}} + { {A_{x}}_{,{{y}}{{z}}}} & {-{ {A_{z}}_{,{{x}}{{z}}}}} + { {A_{x}}_{,{{z}}{{z}}}} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{y}}_{,{{0}}{{z}}}}}} - { \phi_{,{{y}}{{z}}}}}\right)} & {-{ {A_{x}}_{,{{y}}{{z}}}}} + { {A_{y}}_{,{{x}}{{z}}}} & 0 & {-{ {A_{z}}_{,{{y}}{{z}}}}} + { {A_{y}}_{,{{z}}{{z}}}} \\ {\frac{1}{c}}{\left({{{{c}} {{ {A_{z}}_{,{{0}}{{z}}}}}} - { \phi_{,{{z}}{{z}}}}}\right)} & {-{ {A_{x}}_{,{{z}}{{z}}}}} + { {A_{z}}_{,{{x}}{{z}}}} & {-{ {A_{y}}_{,{{z}}{{z}}}}} + { {A_{z}}_{,{{y}}{{z}}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$
$dF = d^2 A$
${{{{ dF} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
$\star d^2 A = \star d F$
${{ \star dF} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ 0\end{matrix} \right]}}$

---

redo the whole thing except use variables for $F_{ab}$
${{{ F} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & {F_{{0x}}} & {F_{{0y}}} & {F_{{0z}}} \\ -{{F_{{0x}}}} & 0 & {F_{{xy}}} & {F_{{xz}}} \\ -{{F_{{0y}}}} & -{{F_{{xy}}}} & 0 & {F_{{yz}}} \\ -{{F_{{0z}}}} & -{{F_{{xz}}}} & -{{F_{{yz}}}} & 0\end{matrix} \right]}}$
${{{{ F} _b} _c} _{,a}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {F_{{0x}}}_{,{{0}}} & {F_{{0y}}}_{,{{0}}} & {F_{{0z}}}_{,{{0}}} \\ -{ {F_{{0x}}}_{,{{0}}}} & 0 & {F_{{xy}}}_{,{{0}}} & {F_{{xz}}}_{,{{0}}} \\ -{ {F_{{0y}}}_{,{{0}}}} & -{ {F_{{xy}}}_{,{{0}}}} & 0 & {F_{{yz}}}_{,{{0}}} \\ -{ {F_{{0z}}}_{,{{0}}}} & -{ {F_{{xz}}}_{,{{0}}}} & -{ {F_{{yz}}}_{,{{0}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {F_{{0x}}}_{,{{x}}} & {F_{{0y}}}_{,{{x}}} & {F_{{0z}}}_{,{{x}}} \\ -{ {F_{{0x}}}_{,{{x}}}} & 0 & {F_{{xy}}}_{,{{x}}} & {F_{{xz}}}_{,{{x}}} \\ -{ {F_{{0y}}}_{,{{x}}}} & -{ {F_{{xy}}}_{,{{x}}}} & 0 & {F_{{yz}}}_{,{{x}}} \\ -{ {F_{{0z}}}_{,{{x}}}} & -{ {F_{{xz}}}_{,{{x}}}} & -{ {F_{{yz}}}_{,{{x}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {F_{{0x}}}_{,{{y}}} & {F_{{0y}}}_{,{{y}}} & {F_{{0z}}}_{,{{y}}} \\ -{ {F_{{0x}}}_{,{{y}}}} & 0 & {F_{{xy}}}_{,{{y}}} & {F_{{xz}}}_{,{{y}}} \\ -{ {F_{{0y}}}_{,{{y}}}} & -{ {F_{{xy}}}_{,{{y}}}} & 0 & {F_{{yz}}}_{,{{y}}} \\ -{ {F_{{0z}}}_{,{{y}}}} & -{ {F_{{xz}}}_{,{{y}}}} & -{ {F_{{yz}}}_{,{{y}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {F_{{0x}}}_{,{{z}}} & {F_{{0y}}}_{,{{z}}} & {F_{{0z}}}_{,{{z}}} \\ -{ {F_{{0x}}}_{,{{z}}}} & 0 & {F_{{xy}}}_{,{{z}}} & {F_{{xz}}}_{,{{z}}} \\ -{ {F_{{0y}}}_{,{{z}}}} & -{ {F_{{xy}}}_{,{{z}}}} & 0 & {F_{{yz}}}_{,{{z}}} \\ -{ {F_{{0z}}}_{,{{z}}}} & -{ {F_{{xz}}}_{,{{z}}}} & -{ {F_{{yz}}}_{,{{z}}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$
$d^2 A = 0$ in terms of $F_{ab}$
${{{{ dF} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & {{2}} {{\left({{{ {F_{{0x}}}_{,{{y}}}} - { {F_{{0y}}}_{,{{x}}}}} + { {F_{{xy}}}_{,{{0}}}}}\right)}} & {{2}} {{\left({{{ {F_{{0x}}}_{,{{z}}}} - { {F_{{0z}}}_{,{{x}}}}} + { {F_{{xz}}}_{,{{0}}}}}\right)}} \\ 0 & {{2}} {{\left({{-{ {F_{{0x}}}_{,{{y}}}}} + {{ {F_{{0y}}}_{,{{x}}}} - { {F_{{xy}}}_{,{{0}}}}}}\right)}} & 0 & {{2}} {{\left({{{ {F_{{0y}}}_{,{{z}}}} - { {F_{{0z}}}_{,{{y}}}}} + { {F_{{yz}}}_{,{{0}}}}}\right)}} \\ 0 & {{2}} {{\left({{-{ {F_{{0x}}}_{,{{z}}}}} + {{ {F_{{0z}}}_{,{{x}}}} - { {F_{{xz}}}_{,{{0}}}}}}\right)}} & {{2}} {{\left({{-{ {F_{{0y}}}_{,{{z}}}}} + {{ {F_{{0z}}}_{,{{y}}}} - { {F_{{yz}}}_{,{{0}}}}}}\right)}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & {{2}} {{\left({{-{ {F_{{0x}}}_{,{{y}}}}} + {{ {F_{{0y}}}_{,{{x}}}} - { {F_{{xy}}}_{,{{0}}}}}}\right)}} & {{2}} {{\left({{-{ {F_{{0x}}}_{,{{z}}}}} + {{ {F_{{0z}}}_{,{{x}}}} - { {F_{{xz}}}_{,{{0}}}}}}\right)}} \\ 0 & 0 & 0 & 0 \\ {{2}} {{\left({{{ {F_{{0x}}}_{,{{y}}}} - { {F_{{0y}}}_{,{{x}}}}} + { {F_{{xy}}}_{,{{0}}}}}\right)}} & 0 & 0 & {{2}} {{\left({{{ {F_{{xy}}}_{,{{z}}}} - { {F_{{xz}}}_{,{{y}}}}} + { {F_{{yz}}}_{,{{x}}}}}\right)}} \\ {{2}} {{\left({{{ {F_{{0x}}}_{,{{z}}}} - { {F_{{0z}}}_{,{{x}}}}} + { {F_{{xz}}}_{,{{0}}}}}\right)}} & 0 & {{2}} {{\left({{-{ {F_{{xy}}}_{,{{z}}}}} + {{ {F_{{xz}}}_{,{{y}}}} - { {F_{{yz}}}_{,{{x}}}}}}\right)}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {{2}} {{\left({{{ {F_{{0x}}}_{,{{y}}}} - { {F_{{0y}}}_{,{{x}}}}} + { {F_{{xy}}}_{,{{0}}}}}\right)}} & 0 & {{2}} {{\left({{-{ {F_{{0y}}}_{,{{z}}}}} + {{ {F_{{0z}}}_{,{{y}}}} - { {F_{{yz}}}_{,{{0}}}}}}\right)}} \\ {{2}} {{\left({{-{ {F_{{0x}}}_{,{{y}}}}} + {{ {F_{{0y}}}_{,{{x}}}} - { {F_{{xy}}}_{,{{0}}}}}}\right)}} & 0 & 0 & {{2}} {{\left({{-{ {F_{{xy}}}_{,{{z}}}}} + {{ {F_{{xz}}}_{,{{y}}}} - { {F_{{yz}}}_{,{{x}}}}}}\right)}} \\ 0 & 0 & 0 & 0 \\ {{2}} {{\left({{{ {F_{{0y}}}_{,{{z}}}} - { {F_{{0z}}}_{,{{y}}}}} + { {F_{{yz}}}_{,{{0}}}}}\right)}} & {{2}} {{\left({{{ {F_{{xy}}}_{,{{z}}}} - { {F_{{xz}}}_{,{{y}}}}} + { {F_{{yz}}}_{,{{x}}}}}\right)}} & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {{2}} {{\left({{{ {F_{{0x}}}_{,{{z}}}} - { {F_{{0z}}}_{,{{x}}}}} + { {F_{{xz}}}_{,{{0}}}}}\right)}} & {{2}} {{\left({{{ {F_{{0y}}}_{,{{z}}}} - { {F_{{0z}}}_{,{{y}}}}} + { {F_{{yz}}}_{,{{0}}}}}\right)}} & 0 \\ {{2}} {{\left({{-{ {F_{{0x}}}_{,{{z}}}}} + {{ {F_{{0z}}}_{,{{x}}}} - { {F_{{xz}}}_{,{{0}}}}}}\right)}} & 0 & {{2}} {{\left({{{ {F_{{xy}}}_{,{{z}}}} - { {F_{{xz}}}_{,{{y}}}}} + { {F_{{yz}}}_{,{{x}}}}}\right)}} & 0 \\ {{2}} {{\left({{-{ {F_{{0y}}}_{,{{z}}}}} + {{ {F_{{0z}}}_{,{{y}}}} - { {F_{{yz}}}_{,{{0}}}}}}\right)}} & {{2}} {{\left({{-{ {F_{{xy}}}_{,{{z}}}}} + {{ {F_{{xz}}}_{,{{y}}}} - { {F_{{yz}}}_{,{{x}}}}}}\right)}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
$\star d^2 A = \star d F$
${{{ \star dF} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} {{2}} {{\left({{-{ {F_{{xy}}}_{,{{z}}}}} + {{ {F_{{xz}}}_{,{{y}}}} - { {F_{{yz}}}_{,{{x}}}}}}\right)}} \\ {{2}} {{\left({{-{ {F_{{0y}}}_{,{{z}}}}} + {{ {F_{{0z}}}_{,{{y}}}} - { {F_{{yz}}}_{,{{0}}}}}}\right)}} \\ {{2}} {{\left({{{ {F_{{0x}}}_{,{{z}}}} - { {F_{{0z}}}_{,{{x}}}}} + { {F_{{xz}}}_{,{{0}}}}}\right)}} \\ {{2}} {{\left({{-{ {F_{{0x}}}_{,{{y}}}}} + {{ {F_{{0y}}}_{,{{x}}}} - { {F_{{xy}}}_{,{{0}}}}}}\right)}}\end{matrix} \right]}}} = {\overset{a\downarrow}{\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ 0\end{matrix} \right]}}$

---

now redo the whole thing except use $E_i$ and $B_i$
${{ E} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} 0 \\ {E_{x}} \\ {E_{y}} \\ {E_{z}}\end{matrix} \right]}}$
${{ B} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} 0 \\ {B_{x}} \\ {B_{y}} \\ {B_{z}}\end{matrix} \right]}}$
${{{ F} _a} _b} = {{{{{\frac{1}{c}}} {{{ n} _a}} {{{ E} _b}}} - {{{\frac{1}{c}}} {{{ E} _a}} {{{ n} _b}}}} + {{{{ n} ^c}} {{{ B} ^d}} {{{{{{ \epsilon} _c} _d} _a} _b}}}}$
${{{ F} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & -{{\frac{1}{c}} {{E_{x}}}} & -{{\frac{1}{c}} {{E_{y}}}} & -{{\frac{1}{c}} {{E_{z}}}} \\ {\frac{1}{c}} {{E_{x}}} & 0 & {B_{z}} & -{{B_{y}}} \\ {\frac{1}{c}} {{E_{y}}} & -{{B_{z}}} & 0 & {B_{x}} \\ {\frac{1}{c}} {{E_{z}}} & {B_{y}} & -{{B_{x}}} & 0\end{matrix} \right]}}$
${{{ F} _a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & -{{\frac{1}{c}} {{E_{x}}}} & -{{\frac{1}{c}} {{E_{y}}}} & -{{\frac{1}{c}} {{E_{z}}}} \\ -{{\frac{1}{c}} {{E_{x}}}} & 0 & {B_{z}} & -{{B_{y}}} \\ -{{\frac{1}{c}} {{E_{y}}}} & -{{B_{z}}} & 0 & {B_{x}} \\ -{{\frac{1}{c}} {{E_{z}}}} & {B_{y}} & -{{B_{x}}} & 0\end{matrix} \right]}}$
${{{ F} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}} {{E_{x}}} & {\frac{1}{c}} {{E_{y}}} & {\frac{1}{c}} {{E_{z}}} \\ {\frac{1}{c}} {{E_{x}}} & 0 & {B_{z}} & -{{B_{y}}} \\ {\frac{1}{c}} {{E_{y}}} & -{{B_{z}}} & 0 & {B_{x}} \\ {\frac{1}{c}} {{E_{z}}} & {B_{y}} & -{{B_{x}}} & 0\end{matrix} \right]}}$
${{{ F} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}} {{E_{x}}} & {\frac{1}{c}} {{E_{y}}} & {\frac{1}{c}} {{E_{z}}} \\ -{{\frac{1}{c}} {{E_{x}}}} & 0 & {B_{z}} & -{{B_{y}}} \\ -{{\frac{1}{c}} {{E_{y}}}} & -{{B_{z}}} & 0 & {B_{x}} \\ -{{\frac{1}{c}} {{E_{z}}}} & {B_{y}} & -{{B_{x}}} & 0\end{matrix} \right]}}$
${{{{ F} _b} _c} _{,a}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & -{{\frac{1}{c}} { {E_{x}}_{,{{0}}}}} & -{{\frac{1}{c}} { {E_{y}}_{,{{0}}}}} & -{{\frac{1}{c}} { {E_{z}}_{,{{0}}}}} \\ {\frac{1}{c}} { {E_{x}}_{,{{0}}}} & 0 & {B_{z}}_{,{{0}}} & -{ {B_{y}}_{,{{0}}}} \\ {\frac{1}{c}} { {E_{y}}_{,{{0}}}} & -{ {B_{z}}_{,{{0}}}} & 0 & {B_{x}}_{,{{0}}} \\ {\frac{1}{c}} { {E_{z}}_{,{{0}}}} & {B_{y}}_{,{{0}}} & -{ {B_{x}}_{,{{0}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & -{{\frac{1}{c}} { {E_{x}}_{,{{x}}}}} & -{{\frac{1}{c}} { {E_{y}}_{,{{x}}}}} & -{{\frac{1}{c}} { {E_{z}}_{,{{x}}}}} \\ {\frac{1}{c}} { {E_{x}}_{,{{x}}}} & 0 & {B_{z}}_{,{{x}}} & -{ {B_{y}}_{,{{x}}}} \\ {\frac{1}{c}} { {E_{y}}_{,{{x}}}} & -{ {B_{z}}_{,{{x}}}} & 0 & {B_{x}}_{,{{x}}} \\ {\frac{1}{c}} { {E_{z}}_{,{{x}}}} & {B_{y}}_{,{{x}}} & -{ {B_{x}}_{,{{x}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & -{{\frac{1}{c}} { {E_{x}}_{,{{y}}}}} & -{{\frac{1}{c}} { {E_{y}}_{,{{y}}}}} & -{{\frac{1}{c}} { {E_{z}}_{,{{y}}}}} \\ {\frac{1}{c}} { {E_{x}}_{,{{y}}}} & 0 & {B_{z}}_{,{{y}}} & -{ {B_{y}}_{,{{y}}}} \\ {\frac{1}{c}} { {E_{y}}_{,{{y}}}} & -{ {B_{z}}_{,{{y}}}} & 0 & {B_{x}}_{,{{y}}} \\ {\frac{1}{c}} { {E_{z}}_{,{{y}}}} & {B_{y}}_{,{{y}}} & -{ {B_{x}}_{,{{y}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & -{{\frac{1}{c}} { {E_{x}}_{,{{z}}}}} & -{{\frac{1}{c}} { {E_{y}}_{,{{z}}}}} & -{{\frac{1}{c}} { {E_{z}}_{,{{z}}}}} \\ {\frac{1}{c}} { {E_{x}}_{,{{z}}}} & 0 & {B_{z}}_{,{{z}}} & -{ {B_{y}}_{,{{z}}}} \\ {\frac{1}{c}} { {E_{y}}_{,{{z}}}} & -{ {B_{z}}_{,{{z}}}} & 0 & {B_{x}}_{,{{z}}} \\ {\frac{1}{c}} { {E_{z}}_{,{{z}}}} & {B_{y}}_{,{{z}}} & -{ {B_{x}}_{,{{z}}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$
$d^2 A = 0$ in terms of $F_{ab}$
${{{{ dF} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{x}}_{,{{y}}}}} + { {E_{y}}_{,{{x}}}} + {{{c}} {{ {B_{z}}_{,{{0}}}}}}}\right)}}} & {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{x}}_{,{{z}}}}} + {{ {E_{z}}_{,{{x}}}} - {{{c}} {{ {B_{y}}_{,{{0}}}}}}}}\right)}}} \\ 0 & {\frac{1}{c}} {{{2}} {{\left({{{ {E_{x}}_{,{{y}}}} - { {E_{y}}_{,{{x}}}}} - {{{c}} {{ {B_{z}}_{,{{0}}}}}}}\right)}}} & 0 & {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{y}}_{,{{z}}}}} + { {E_{z}}_{,{{y}}}} + {{{c}} {{ {B_{x}}_{,{{0}}}}}}}\right)}}} \\ 0 & {\frac{1}{c}} {{{2}} {{\left({{{ {E_{x}}_{,{{z}}}} - { {E_{z}}_{,{{x}}}}} + {{{c}} {{ {B_{y}}_{,{{0}}}}}}}\right)}}} & {\frac{1}{c}} {{{2}} {{\left({{{ {E_{y}}_{,{{z}}}} - { {E_{z}}_{,{{y}}}}} - {{{c}} {{ {B_{x}}_{,{{0}}}}}}}\right)}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & {\frac{1}{c}} {{{2}} {{\left({{{ {E_{x}}_{,{{y}}}} - { {E_{y}}_{,{{x}}}}} - {{{c}} {{ {B_{z}}_{,{{0}}}}}}}\right)}}} & {\frac{1}{c}} {{{2}} {{\left({{{ {E_{x}}_{,{{z}}}} - { {E_{z}}_{,{{x}}}}} + {{{c}} {{ {B_{y}}_{,{{0}}}}}}}\right)}}} \\ 0 & 0 & 0 & 0 \\ {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{x}}_{,{{y}}}}} + { {E_{y}}_{,{{x}}}} + {{{c}} {{ {B_{z}}_{,{{0}}}}}}}\right)}}} & 0 & 0 & {{2}} {{\left({{ {B_{x}}_{,{{x}}}} + { {B_{y}}_{,{{y}}}} + { {B_{z}}_{,{{z}}}}}\right)}} \\ {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{x}}_{,{{z}}}}} + {{ {E_{z}}_{,{{x}}}} - {{{c}} {{ {B_{y}}_{,{{0}}}}}}}}\right)}}} & 0 & -{{{2}} {{\left({{ {B_{x}}_{,{{x}}}} + { {B_{y}}_{,{{y}}}} + { {B_{z}}_{,{{z}}}}}\right)}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{x}}_{,{{y}}}}} + { {E_{y}}_{,{{x}}}} + {{{c}} {{ {B_{z}}_{,{{0}}}}}}}\right)}}} & 0 & {\frac{1}{c}} {{{2}} {{\left({{{ {E_{y}}_{,{{z}}}} - { {E_{z}}_{,{{y}}}}} - {{{c}} {{ {B_{x}}_{,{{0}}}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{{ {E_{x}}_{,{{y}}}} - { {E_{y}}_{,{{x}}}}} - {{{c}} {{ {B_{z}}_{,{{0}}}}}}}\right)}}} & 0 & 0 & -{{{2}} {{\left({{ {B_{x}}_{,{{x}}}} + { {B_{y}}_{,{{y}}}} + { {B_{z}}_{,{{z}}}}}\right)}}} \\ 0 & 0 & 0 & 0 \\ {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{y}}_{,{{z}}}}} + { {E_{z}}_{,{{y}}}} + {{{c}} {{ {B_{x}}_{,{{0}}}}}}}\right)}}} & {{2}} {{\left({{ {B_{x}}_{,{{x}}}} + { {B_{y}}_{,{{y}}}} + { {B_{z}}_{,{{z}}}}}\right)}} & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{x}}_{,{{z}}}}} + {{ {E_{z}}_{,{{x}}}} - {{{c}} {{ {B_{y}}_{,{{0}}}}}}}}\right)}}} & {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{y}}_{,{{z}}}}} + { {E_{z}}_{,{{y}}}} + {{{c}} {{ {B_{x}}_{,{{0}}}}}}}\right)}}} & 0 \\ {\frac{1}{c}} {{{2}} {{\left({{{ {E_{x}}_{,{{z}}}} - { {E_{z}}_{,{{x}}}}} + {{{c}} {{ {B_{y}}_{,{{0}}}}}}}\right)}}} & 0 & {{2}} {{\left({{ {B_{x}}_{,{{x}}}} + { {B_{y}}_{,{{y}}}} + { {B_{z}}_{,{{z}}}}}\right)}} & 0 \\ {\frac{1}{c}} {{{2}} {{\left({{{ {E_{y}}_{,{{z}}}} - { {E_{z}}_{,{{y}}}}} - {{{c}} {{ {B_{x}}_{,{{0}}}}}}}\right)}}} & -{{{2}} {{\left({{ {B_{x}}_{,{{x}}}} + { {B_{y}}_{,{{y}}}} + { {B_{z}}_{,{{z}}}}}\right)}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
$\star d^2 A = \star d F$
${{{ \star dF} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} -{{{2}} {{\left({{ {B_{x}}_{,{{x}}}} + { {B_{y}}_{,{{y}}}} + { {B_{z}}_{,{{z}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{{ {E_{y}}_{,{{z}}}} - { {E_{z}}_{,{{y}}}}} - {{{c}} {{ {B_{x}}_{,{{0}}}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{x}}_{,{{z}}}}} + {{ {E_{z}}_{,{{x}}}} - {{{c}} {{ {B_{y}}_{,{{0}}}}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{{ {E_{x}}_{,{{y}}}} - { {E_{y}}_{,{{x}}}}} - {{{c}} {{ {B_{z}}_{,{{0}}}}}}}\right)}}}\end{matrix} \right]}}} = {\overset{a\downarrow}{\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ 0\end{matrix} \right]}}$
replace $\partial_0 = \frac{1}{c} \partial_t$
${{{ \star dF} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} -{{{2}} {{\left({{ {B_{x}}_{,{{x}}}} + { {B_{y}}_{,{{y}}}} + { {B_{z}}_{,{{z}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{-{ {B_{x}}_{,{{t}}}}} + {{ {E_{y}}_{,{{z}}}} - { {E_{z}}_{,{{y}}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{{-{ {B_{y}}_{,{{t}}}}} - { {E_{x}}_{,{{z}}}}} + { {E_{z}}_{,{{x}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{-{ {B_{z}}_{,{{t}}}}} + {{ {E_{x}}_{,{{y}}}} - { {E_{y}}_{,{{x}}}}}}\right)}}}\end{matrix} \right]}}} = {\overset{a\downarrow}{\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ 0\end{matrix} \right]}}$
collect equations
${{ {B_{x}}_{,{{x}}}} + { {B_{y}}_{,{{y}}}} + { {B_{z}}_{,{{z}}}}} = {0}$
${{{ {B_{x}}_{,{{t}}}} - { {E_{y}}_{,{{z}}}}} + { {E_{z}}_{,{{y}}}}} = {0}$
${{ {B_{y}}_{,{{t}}}} + {{ {E_{x}}_{,{{z}}}} - { {E_{z}}_{,{{x}}}}}} = {0}$
${{{ {B_{z}}_{,{{t}}}} - { {E_{x}}_{,{{y}}}}} + { {E_{y}}_{,{{x}}}}} = {0}$
and you have the Gauss law for the magnetic field and the Faraday law

---

four-current as a one-form:
${{ J} ^a} = {\overset{a\downarrow}{\left[ \begin{matrix} {{c}} {{\rho}} \\ {J^{x}} \\ {J^{y}} \\ {J^{z}}\end{matrix} \right]}}$
now look at the co-differential
$(\star F)_{ab} = (\star dA)_{ab} = \frac{1}{2} \epsilon_{abcd} F^{cd}$
${{{ \star F} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & {B_{x}} & {B_{y}} & {B_{z}} \\ -{{B_{x}}} & 0 & {\frac{1}{c}} {{E_{z}}} & -{{\frac{1}{c}} {{E_{y}}}} \\ -{{B_{y}}} & -{{\frac{1}{c}} {{E_{z}}}} & 0 & {\frac{1}{c}} {{E_{x}}} \\ -{{B_{z}}} & {\frac{1}{c}} {{E_{y}}} & -{{\frac{1}{c}} {{E_{x}}}} & 0\end{matrix} \right]}}$
$(\partial_a \star dA)_{bc}$
${{{{ \star F} _b} _c} _{,a}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {B_{x}}_{,{{0}}} & {B_{y}}_{,{{0}}} & {B_{z}}_{,{{0}}} \\ -{ {B_{x}}_{,{{0}}}} & 0 & {\frac{1}{c}} { {E_{z}}_{,{{0}}}} & -{{\frac{1}{c}} { {E_{y}}_{,{{0}}}}} \\ -{ {B_{y}}_{,{{0}}}} & -{{\frac{1}{c}} { {E_{z}}_{,{{0}}}}} & 0 & {\frac{1}{c}} { {E_{x}}_{,{{0}}}} \\ -{ {B_{z}}_{,{{0}}}} & {\frac{1}{c}} { {E_{y}}_{,{{0}}}} & -{{\frac{1}{c}} { {E_{x}}_{,{{0}}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {B_{x}}_{,{{x}}} & {B_{y}}_{,{{x}}} & {B_{z}}_{,{{x}}} \\ -{ {B_{x}}_{,{{x}}}} & 0 & {\frac{1}{c}} { {E_{z}}_{,{{x}}}} & -{{\frac{1}{c}} { {E_{y}}_{,{{x}}}}} \\ -{ {B_{y}}_{,{{x}}}} & -{{\frac{1}{c}} { {E_{z}}_{,{{x}}}}} & 0 & {\frac{1}{c}} { {E_{x}}_{,{{x}}}} \\ -{ {B_{z}}_{,{{x}}}} & {\frac{1}{c}} { {E_{y}}_{,{{x}}}} & -{{\frac{1}{c}} { {E_{x}}_{,{{x}}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {B_{x}}_{,{{y}}} & {B_{y}}_{,{{y}}} & {B_{z}}_{,{{y}}} \\ -{ {B_{x}}_{,{{y}}}} & 0 & {\frac{1}{c}} { {E_{z}}_{,{{y}}}} & -{{\frac{1}{c}} { {E_{y}}_{,{{y}}}}} \\ -{ {B_{y}}_{,{{y}}}} & -{{\frac{1}{c}} { {E_{z}}_{,{{y}}}}} & 0 & {\frac{1}{c}} { {E_{x}}_{,{{y}}}} \\ -{ {B_{z}}_{,{{y}}}} & {\frac{1}{c}} { {E_{y}}_{,{{y}}}} & -{{\frac{1}{c}} { {E_{x}}_{,{{y}}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {B_{x}}_{,{{z}}} & {B_{y}}_{,{{z}}} & {B_{z}}_{,{{z}}} \\ -{ {B_{x}}_{,{{z}}}} & 0 & {\frac{1}{c}} { {E_{z}}_{,{{z}}}} & -{{\frac{1}{c}} { {E_{y}}_{,{{z}}}}} \\ -{ {B_{y}}_{,{{z}}}} & -{{\frac{1}{c}} { {E_{z}}_{,{{z}}}}} & 0 & {\frac{1}{c}} { {E_{x}}_{,{{z}}}} \\ -{ {B_{z}}_{,{{z}}}} & {\frac{1}{c}} { {E_{y}}_{,{{z}}}} & -{{\frac{1}{c}} { {E_{x}}_{,{{z}}}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$
$d \star dA = d \star F = -2 \star \mu_0 J$
${{{{ d \star F} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & {\frac{1}{c}} {{{2}} {{\left({{ {E_{z}}_{,{{0}}}} + {{{{c}} {{ {B_{x}}_{,{{y}}}}}} - {{{c}} {{ {B_{y}}_{,{{x}}}}}}}}\right)}}} & {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{y}}_{,{{0}}}}} + {{{{c}} {{ {B_{x}}_{,{{z}}}}}} - {{{c}} {{ {B_{z}}_{,{{x}}}}}}}}\right)}}} \\ 0 & {\frac{1}{c}} {{{2}} {{\left({{{-{ {E_{z}}_{,{{0}}}}} - {{{c}} {{ {B_{x}}_{,{{y}}}}}}} + {{{c}} {{ {B_{y}}_{,{{x}}}}}}}\right)}}} & 0 & {\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{0}}}} + {{{{c}} {{ {B_{y}}_{,{{z}}}}}} - {{{c}} {{ {B_{z}}_{,{{y}}}}}}}}\right)}}} \\ 0 & {\frac{1}{c}} {{{2}} {{\left({{{ {E_{y}}_{,{{0}}}} - {{{c}} {{ {B_{x}}_{,{{z}}}}}}} + {{{c}} {{ {B_{z}}_{,{{x}}}}}}}\right)}}} & {\frac{1}{c}} {{{2}} {{\left({{{-{ {E_{x}}_{,{{0}}}}} - {{{c}} {{ {B_{y}}_{,{{z}}}}}}} + {{{c}} {{ {B_{z}}_{,{{y}}}}}}}\right)}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & {\frac{1}{c}} {{{2}} {{\left({{{-{ {E_{z}}_{,{{0}}}}} - {{{c}} {{ {B_{x}}_{,{{y}}}}}}} + {{{c}} {{ {B_{y}}_{,{{x}}}}}}}\right)}}} & {\frac{1}{c}} {{{2}} {{\left({{{ {E_{y}}_{,{{0}}}} - {{{c}} {{ {B_{x}}_{,{{z}}}}}}} + {{{c}} {{ {B_{z}}_{,{{x}}}}}}}\right)}}} \\ 0 & 0 & 0 & 0 \\ {\frac{1}{c}} {{{2}} {{\left({{ {E_{z}}_{,{{0}}}} + {{{{c}} {{ {B_{x}}_{,{{y}}}}}} - {{{c}} {{ {B_{y}}_{,{{x}}}}}}}}\right)}}} & 0 & 0 & {\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{x}}}} + { {E_{y}}_{,{{y}}}} + { {E_{z}}_{,{{z}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{y}}_{,{{0}}}}} + {{{{c}} {{ {B_{x}}_{,{{z}}}}}} - {{{c}} {{ {B_{z}}_{,{{x}}}}}}}}\right)}}} & 0 & -{{\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{x}}}} + { {E_{y}}_{,{{y}}}} + { {E_{z}}_{,{{z}}}}}\right)}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}} {{{2}} {{\left({{ {E_{z}}_{,{{0}}}} + {{{{c}} {{ {B_{x}}_{,{{y}}}}}} - {{{c}} {{ {B_{y}}_{,{{x}}}}}}}}\right)}}} & 0 & {\frac{1}{c}} {{{2}} {{\left({{{-{ {E_{x}}_{,{{0}}}}} - {{{c}} {{ {B_{y}}_{,{{z}}}}}}} + {{{c}} {{ {B_{z}}_{,{{y}}}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{{-{ {E_{z}}_{,{{0}}}}} - {{{c}} {{ {B_{x}}_{,{{y}}}}}}} + {{{c}} {{ {B_{y}}_{,{{x}}}}}}}\right)}}} & 0 & 0 & -{{\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{x}}}} + { {E_{y}}_{,{{y}}}} + { {E_{z}}_{,{{z}}}}}\right)}}}} \\ 0 & 0 & 0 & 0 \\ {\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{0}}}} + {{{{c}} {{ {B_{y}}_{,{{z}}}}}} - {{{c}} {{ {B_{z}}_{,{{y}}}}}}}}\right)}}} & {\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{x}}}} + { {E_{y}}_{,{{y}}}} + { {E_{z}}_{,{{z}}}}}\right)}}} & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{y}}_{,{{0}}}}} + {{{{c}} {{ {B_{x}}_{,{{z}}}}}} - {{{c}} {{ {B_{z}}_{,{{x}}}}}}}}\right)}}} & {\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{0}}}} + {{{{c}} {{ {B_{y}}_{,{{z}}}}}} - {{{c}} {{ {B_{z}}_{,{{y}}}}}}}}\right)}}} & 0 \\ {\frac{1}{c}} {{{2}} {{\left({{{ {E_{y}}_{,{{0}}}} - {{{c}} {{ {B_{x}}_{,{{z}}}}}}} + {{{c}} {{ {B_{z}}_{,{{x}}}}}}}\right)}}} & 0 & {\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{x}}}} + { {E_{y}}_{,{{y}}}} + { {E_{z}}_{,{{z}}}}}\right)}}} & 0 \\ {\frac{1}{c}} {{{2}} {{\left({{{-{ {E_{x}}_{,{{0}}}}} - {{{c}} {{ {B_{y}}_{,{{z}}}}}}} + {{{c}} {{ {B_{z}}_{,{{y}}}}}}}\right)}}} & -{{\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{x}}}} + { {E_{y}}_{,{{y}}}} + { {E_{z}}_{,{{z}}}}}\right)}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
$\star d \star d A = \star d \star F = 2 \mu_0 J$
${{{ \star d \star F} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} -{{\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{x}}}} + { {E_{y}}_{,{{y}}}} + { {E_{z}}_{,{{z}}}}}\right)}}}} \\ {\frac{1}{c}} {{{2}} {{\left({{{-{ {E_{x}}_{,{{0}}}}} - {{{c}} {{ {B_{y}}_{,{{z}}}}}}} + {{{c}} {{ {B_{z}}_{,{{y}}}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{-{ {E_{y}}_{,{{0}}}}} + {{{{c}} {{ {B_{x}}_{,{{z}}}}}} - {{{c}} {{ {B_{z}}_{,{{x}}}}}}}}\right)}}} \\ {\frac{1}{c}} {{{2}} {{\left({{{-{ {E_{z}}_{,{{0}}}}} - {{{c}} {{ {B_{x}}_{,{{y}}}}}}} + {{{c}} {{ {B_{y}}_{,{{x}}}}}}}\right)}}}\end{matrix} \right]}}} = {\overset{a\downarrow}{\left[ \begin{matrix} -{{{2}} {{c}} {{\rho}} \cdot {{{ \mu} _0}}} \\ {{2}} {{{J^{x}}}} \cdot {{{ \mu} _0}} \\ {{2}} {{{J^{y}}}} \cdot {{{ \mu} _0}} \\ {{2}} {{{J^{z}}}} \cdot {{{ \mu} _0}}\end{matrix} \right]}}$
replace $\partial_0 = \frac{1}{c} \partial_t$
${{{ \star d \star F} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} -{{\frac{1}{c}} {{{2}} {{\left({{ {E_{x}}_{,{{x}}}} + { {E_{y}}_{,{{y}}}} + { {E_{z}}_{,{{z}}}}}\right)}}}} \\ \frac{{{2}} {{\left({{{-{ {E_{x}}_{,{{t}}}}} - {{{{c}^{2}}} {{ {B_{y}}_{,{{z}}}}}}} + {{{{c}^{2}}} {{ {B_{z}}_{,{{y}}}}}}}\right)}}}{{c}^{2}} \\ \frac{{{2}} {{\left({{-{ {E_{y}}_{,{{t}}}}} + {{{{{c}^{2}}} {{ {B_{x}}_{,{{z}}}}}} - {{{{c}^{2}}} {{ {B_{z}}_{,{{x}}}}}}}}\right)}}}{{c}^{2}} \\ \frac{{{2}} {{\left({{{-{ {E_{z}}_{,{{t}}}}} - {{{{c}^{2}}} {{ {B_{x}}_{,{{y}}}}}}} + {{{{c}^{2}}} {{ {B_{y}}_{,{{x}}}}}}}\right)}}}{{c}^{2}}\end{matrix} \right]}}} = {\overset{a\downarrow}{\left[ \begin{matrix} -{{{2}} {{c}} {{\rho}} \cdot {{{ \mu} _0}}} \\ {{2}} {{{J^{x}}}} \cdot {{{ \mu} _0}} \\ {{2}} {{{J^{y}}}} \cdot {{{ \mu} _0}} \\ {{2}} {{{J^{z}}}} \cdot {{{ \mu} _0}}\end{matrix} \right]}}$
collect equations
${{ {E_{x}}_{,{{x}}}} + { {E_{y}}_{,{{y}}}} + { {E_{z}}_{,{{z}}}}} = {{{\rho}} \cdot {{{ \mu} _0}} {{{c}^{2}}}}$
${\frac{{ {E_{x}}_{,{{t}}}} + {{{{{c}^{2}}} {{ {B_{y}}_{,{{z}}}}}} - {{{{c}^{2}}} {{ {B_{z}}_{,{{y}}}}}}}}{{c}^{2}}} = {-{{{{J^{x}}}} \cdot {{{ \mu} _0}}}}$
${\frac{{{ {E_{y}}_{,{{t}}}} - {{{{c}^{2}}} {{ {B_{x}}_{,{{z}}}}}}} + {{{{c}^{2}}} {{ {B_{z}}_{,{{x}}}}}}}{{c}^{2}}} = {-{{{{J^{y}}}} \cdot {{{ \mu} _0}}}}$
${\frac{{ {E_{z}}_{,{{t}}}} + {{{{{c}^{2}}} {{ {B_{x}}_{,{{y}}}}}} - {{{{c}^{2}}} {{ {B_{y}}_{,{{x}}}}}}}}{{c}^{2}}} = {-{{{{J^{z}}}} \cdot {{{ \mu} _0}}}}$
and you have the Gauss law for the electic field and the Ampere law

---

side thought: what is $d \star d \star A$?
${{{{ \star A} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -{{A_{z}}} & {A_{y}} \\ 0 & {A_{z}} & 0 & -{{A_{x}}} \\ 0 & -{{A_{y}}} & {A_{x}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & {A_{z}} & -{{A_{y}}} \\ 0 & 0 & 0 & 0 \\ -{{A_{z}}} & 0 & 0 & -{{\frac{1}{c}} {\phi}} \\ {A_{y}} & 0 & {\frac{1}{c}} {\phi} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & -{{A_{z}}} & 0 & {A_{x}} \\ {A_{z}} & 0 & 0 & {\frac{1}{c}} {\phi} \\ 0 & 0 & 0 & 0 \\ -{{A_{x}}} & -{{\frac{1}{c}} {\phi}} & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & {A_{y}} & -{{A_{x}}} & 0 \\ -{{A_{y}}} & 0 & -{{\frac{1}{c}} {\phi}} & 0 \\ {A_{x}} & {\frac{1}{c}} {\phi} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{{{ \partial \star A} _b} _c} _d} _{,a}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -{ {A_{z}}_{,{{0}}}} & {A_{y}}_{,{{0}}} \\ 0 & {A_{z}}_{,{{0}}} & 0 & -{ {A_{x}}_{,{{0}}}} \\ 0 & -{ {A_{y}}_{,{{0}}}} & {A_{x}}_{,{{0}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & {A_{z}}_{,{{0}}} & -{ {A_{y}}_{,{{0}}}} \\ 0 & 0 & 0 & 0 \\ -{ {A_{z}}_{,{{0}}}} & 0 & 0 & -{{\frac{1}{c}} { \phi_{,{{0}}}}} \\ {A_{y}}_{,{{0}}} & 0 & {\frac{1}{c}} { \phi_{,{{0}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{ {A_{z}}_{,{{0}}}} & 0 & {A_{x}}_{,{{0}}} \\ {A_{z}}_{,{{0}}} & 0 & 0 & {\frac{1}{c}} { \phi_{,{{0}}}} \\ 0 & 0 & 0 & 0 \\ -{ {A_{x}}_{,{{0}}}} & -{{\frac{1}{c}} { \phi_{,{{0}}}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {A_{y}}_{,{{0}}} & -{ {A_{x}}_{,{{0}}}} & 0 \\ -{ {A_{y}}_{,{{0}}}} & 0 & -{{\frac{1}{c}} { \phi_{,{{0}}}}} & 0 \\ {A_{x}}_{,{{0}}} & {\frac{1}{c}} { \phi_{,{{0}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -{ {A_{z}}_{,{{x}}}} & {A_{y}}_{,{{x}}} \\ 0 & {A_{z}}_{,{{x}}} & 0 & -{ {A_{x}}_{,{{x}}}} \\ 0 & -{ {A_{y}}_{,{{x}}}} & {A_{x}}_{,{{x}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & {A_{z}}_{,{{x}}} & -{ {A_{y}}_{,{{x}}}} \\ 0 & 0 & 0 & 0 \\ -{ {A_{z}}_{,{{x}}}} & 0 & 0 & -{{\frac{1}{c}} { \phi_{,{{x}}}}} \\ {A_{y}}_{,{{x}}} & 0 & {\frac{1}{c}} { \phi_{,{{x}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{ {A_{z}}_{,{{x}}}} & 0 & {A_{x}}_{,{{x}}} \\ {A_{z}}_{,{{x}}} & 0 & 0 & {\frac{1}{c}} { \phi_{,{{x}}}} \\ 0 & 0 & 0 & 0 \\ -{ {A_{x}}_{,{{x}}}} & -{{\frac{1}{c}} { \phi_{,{{x}}}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {A_{y}}_{,{{x}}} & -{ {A_{x}}_{,{{x}}}} & 0 \\ -{ {A_{y}}_{,{{x}}}} & 0 & -{{\frac{1}{c}} { \phi_{,{{x}}}}} & 0 \\ {A_{x}}_{,{{x}}} & {\frac{1}{c}} { \phi_{,{{x}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -{ {A_{z}}_{,{{y}}}} & {A_{y}}_{,{{y}}} \\ 0 & {A_{z}}_{,{{y}}} & 0 & -{ {A_{x}}_{,{{y}}}} \\ 0 & -{ {A_{y}}_{,{{y}}}} & {A_{x}}_{,{{y}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & {A_{z}}_{,{{y}}} & -{ {A_{y}}_{,{{y}}}} \\ 0 & 0 & 0 & 0 \\ -{ {A_{z}}_{,{{y}}}} & 0 & 0 & -{{\frac{1}{c}} { \phi_{,{{y}}}}} \\ {A_{y}}_{,{{y}}} & 0 & {\frac{1}{c}} { \phi_{,{{y}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{ {A_{z}}_{,{{y}}}} & 0 & {A_{x}}_{,{{y}}} \\ {A_{z}}_{,{{y}}} & 0 & 0 & {\frac{1}{c}} { \phi_{,{{y}}}} \\ 0 & 0 & 0 & 0 \\ -{ {A_{x}}_{,{{y}}}} & -{{\frac{1}{c}} { \phi_{,{{y}}}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {A_{y}}_{,{{y}}} & -{ {A_{x}}_{,{{y}}}} & 0 \\ -{ {A_{y}}_{,{{y}}}} & 0 & -{{\frac{1}{c}} { \phi_{,{{y}}}}} & 0 \\ {A_{x}}_{,{{y}}} & {\frac{1}{c}} { \phi_{,{{y}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -{ {A_{z}}_{,{{z}}}} & {A_{y}}_{,{{z}}} \\ 0 & {A_{z}}_{,{{z}}} & 0 & -{ {A_{x}}_{,{{z}}}} \\ 0 & -{ {A_{y}}_{,{{z}}}} & {A_{x}}_{,{{z}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & {A_{z}}_{,{{z}}} & -{ {A_{y}}_{,{{z}}}} \\ 0 & 0 & 0 & 0 \\ -{ {A_{z}}_{,{{z}}}} & 0 & 0 & -{{\frac{1}{c}} { \phi_{,{{z}}}}} \\ {A_{y}}_{,{{z}}} & 0 & {\frac{1}{c}} { \phi_{,{{z}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{ {A_{z}}_{,{{z}}}} & 0 & {A_{x}}_{,{{z}}} \\ {A_{z}}_{,{{z}}} & 0 & 0 & {\frac{1}{c}} { \phi_{,{{z}}}} \\ 0 & 0 & 0 & 0 \\ -{ {A_{x}}_{,{{z}}}} & -{{\frac{1}{c}} { \phi_{,{{z}}}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {A_{y}}_{,{{z}}} & -{ {A_{x}}_{,{{z}}}} & 0 \\ -{ {A_{y}}_{,{{z}}}} & 0 & -{{\frac{1}{c}} { \phi_{,{{z}}}}} & 0 \\ {A_{x}}_{,{{z}}} & {\frac{1}{c}} { \phi_{,{{z}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{{{ d \star A} _a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} \\ 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} \\ 0 & 0 & 0 & 0 \\ 0 & {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 \\ 0 & {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} \\ 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 \\ 0 & 0 & 0 & 0 \\ {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} \\ 0 & 0 & 0 & 0 \\ 0 & {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0 \\ {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 \\ 0 & {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 \\ 0 & 0 & 0 & 0 \\ {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0 \\ {\frac{1}{c}} {{{6}} {{\left({{-{ \phi_{,{{0}}}}} + {{{c}} {{ {A_{x}}_{,{{x}}}}}} + {{{c}} {{ {A_{y}}_{,{{y}}}}}} + {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
${\star d \star A} = {{\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}}} - {{{c}} {{ {A_{x}}_{,{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}}}}}}\right)}}}}$
...and this would be set to zero by the Lorentz gauge condition
${{ d \star d \star A} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}{{0}}}} - {{{c}} {{ {A_{x}}_{,{{0}}{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{0}}{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{0}}{{z}}}}}}}\right)}}} \\ {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}{{x}}}} - {{{c}} {{ {A_{x}}_{,{{x}}{{x}}}}}}} - {{{c}} {{ {A_{y}}_{,{{x}}{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{x}}{{z}}}}}}}\right)}}} \\ {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}{{y}}}} - {{{c}} {{ {A_{x}}_{,{{x}}{{y}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}{{y}}}}}}} - {{{c}} {{ {A_{z}}_{,{{y}}{{z}}}}}}}\right)}}} \\ {\frac{1}{c}} {{{6}} {{\left({{{{ \phi_{,{{0}}{{z}}}} - {{{c}} {{ {A_{x}}_{,{{x}}{{z}}}}}}} - {{{c}} {{ {A_{y}}_{,{{y}}{{z}}}}}}} - {{{c}} {{ {A_{z}}_{,{{z}}{{z}}}}}}}\right)}}}\end{matrix} \right]}}$
...and this is equal to ${A^\mu}_{,\mu\alpha}$, which means as long as ${A^\mu}_{,\mu} = const$ then the gradient is zero.
This means that $\frac{1}{2} \Delta A = \frac{1}{2} (d \delta + \delta d) A = \frac{1}{2} (d \star d \star + \star d \star d) A = \mu_0 J$ is equal to the Gauss-Ampere laws plus the gradient of the divergence of the four-potential
So the Gauss/Faraday side of Maxwell's laws are summed up in $d^2 A = 0$
And the Gauss/Ampere side of Maxwell's laws are summed up in $\frac{1}{2} \Delta A = \mu_0 J$