${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & {r}^{2} & 0 \\ 0 & 0 & 0 & {{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1} - {\cos\left( \theta\right)}}\right)}}\end{matrix} \right]}}$
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{1}{a} & 0 & 0 & 0 \\ 0 & \frac{1}{b} & 0 & 0 \\ 0 & 0 & \frac{1}{{r}^{2}} & 0 \\ 0 & 0 & 0 & \frac{1}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1} - {\cos\left( \theta\right)}}\right)}}}\end{matrix} \right]}}$
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & \frac{\partial_ {{r}}\left( a\right)}{{{2}} {{a}}} & 0 & 0 \\ \frac{\partial_ {{r}}\left( a\right)}{{{2}} {{a}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} -{\frac{\partial_ {{r}}\left( a\right)}{{{2}} {{b}}}} & 0 & 0 & 0 \\ 0 & \frac{\partial_ {{r}}\left( b\right)}{{{2}} {{b}}} & 0 & 0 \\ 0 & 0 & -{{\frac{1}{b}} {r}} & 0 \\ 0 & 0 & 0 & -{{\frac{1}{b}} {{{r}} {{{\sin\left( \theta\right)}^{2}}}}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & \frac{1}{r} & 0 & 0 \\ 0 & 0 & 0 & -{{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{r} \\ 0 & 0 & 0 & \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)} \\ 0 & \frac{1}{r} & \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)} & 0\end{matrix} \right]}\end{matrix} \right]}}$
Ricci from manual metric
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{{{a}} {{r}} {{\partial_ {{r}}\left( a\right)}} {{\partial_ {{r}}\left( b\right)}}} - {{{2}} {{a}} {{b}} {{r}} {{\partial_{{{r}}{{r}}}\left( a\right)}}}} + {{{{b}} {{r}} {{{\partial_ {{r}}\left( a\right)}^{2}}}} - {{{4}} {{a}} {{b}} {{\partial_ {{r}}\left( a\right)}}}}}{{{4}} {{a}} {{r}} {{{b}^{2}}}} & 0 & 0 & 0 \\ 0 & \frac{{{{{b}} {{r}} {{{\partial_ {{r}}\left( a\right)}^{2}}}} - {{{2}} {{a}} {{b}} {{r}} {{\partial_{{{r}}{{r}}}\left( a\right)}}}} + {{{a}} {{r}} {{\partial_ {{r}}\left( a\right)}} {{\partial_ {{r}}\left( b\right)}}} + {{{4}} {{{a}^{2}}} {{\partial_ {{r}}\left( b\right)}}}}{{{4}} {{b}} {{r}} {{{a}^{2}}}} & 0 & 0 \\ 0 & 0 & \frac{{{-{{{b}} {{r}} {{\partial_ {{r}}\left( a\right)}}}} - {{{2}} {{a}} {{b}}}} + {{{2}} {{a}} {{{b}^{2}}}} + {{{a}} {{r}} {{\partial_ {{r}}\left( b\right)}}}}{{{2}} {{a}} {{{b}^{2}}}} & 0 \\ 0 & 0 & 0 & \frac{{-{{{b}} {{r}} {{\partial_ {{r}}\left( a\right)}}}} + {{{{b}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( a\right)}}} - {{{2}} {{a}} {{b}}}} + {{{2}} {{a}} {{b}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{2}} {{a}} {{{b}^{2}}}} - {{{2}} {{a}} {{{b}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{a}} {{r}} {{\partial_ {{r}}\left( b\right)}}} - {{{a}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( b\right)}}}}}{{{2}} {{a}} {{{b}^{2}}}}\end{matrix} \right]}}$
Faraday in spherical geometry
${{{ F} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & -{{E_{r}}} & -{{E_{\theta}}} & -{{E_{\phi}}} \\ {E_{r}} & 0 & -{\frac{{{{B_{\phi}}}} \cdot {{\frac{1}{\sqrt{a}}}} {{\sqrt{b}}}}{\sin\left( \theta\right)}} & {{{B_{\theta}}}} \cdot {{\frac{1}{\sqrt{a}}}} {{\sqrt{b}}} {{\sin\left( \theta\right)}} \\ {E_{\theta}} & \frac{{{{B_{\phi}}}} \cdot {{\frac{1}{\sqrt{a}}}} {{\sqrt{b}}}}{\sin\left( \theta\right)} & 0 & -{{{{B_{r}}}} \cdot {{\frac{1}{\sqrt{a}}}} {{\frac{1}{\sqrt{b}}}} {{{r}^{2}}} {{\sin\left( \theta\right)}}} \\ {E_{\phi}} & -{{{{B_{\theta}}}} \cdot {{\frac{1}{\sqrt{a}}}} {{\sqrt{b}}} {{\sin\left( \theta\right)}}} & {{{B_{r}}}} \cdot {{\frac{1}{\sqrt{a}}}} {{\frac{1}{\sqrt{b}}}} {{{r}^{2}}} {{\sin\left( \theta\right)}} & 0\end{matrix} \right]}}$