metric:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{A} & C & 0 & 0 \\ C & B & 0 & 0 \\ 0 & 0 & {r}^{2} & 0 \\ 0 & 0 & 0 & {{{r}^{2}}} {{\left({{1} + {\cos\left( \phi\right)}}\right)}} {{\left({{1} - {\cos\left( \phi\right)}}\right)}}\end{matrix} \right]}}$
metric inverse:
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{B} & C & 0 & 0 \\ C & A & 0 & 0 \\ 0 & 0 & \frac{1}{{r}^{2}} & 0 \\ 0 & 0 & 0 & \frac{1}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \phi\right)}}\right)}} {{\left({{1} - {\cos\left( \phi\right)}}\right)}}}\end{matrix} \right]}}$
metric derivative:
${{{{ g} _a} _b} _{,c}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} -{ A_{,{{t}}}} & -{ A_{,{{r}}}} & 0 & 0 \\ C_{,{{t}}} & C_{,{{r}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} C_{,{{t}}} & C_{,{{r}}} & 0 & 0 \\ B_{,{{t}}} & B_{,{{r}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & {{2}} {{r}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & {{2}} {{r}} {{{\sin\left( \phi\right)}^{2}}} & 0 & {{2}} {{{r}^{2}}} {{\sin\left( \phi\right)}} {{\cos\left( \phi\right)}}\end{matrix} \right]}\end{matrix} \right]}}$
1st kind Christoffel:
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} -{{\frac{1}{2}} { A_{,{{t}}}}} & -{{\frac{1}{2}} { A_{,{{r}}}}} & 0 & 0 \\ -{{\frac{1}{2}} { A_{,{{r}}}}} & {\frac{1}{2}}{\left({{{{2}} {{ C_{,{{r}}}}}} - { B_{,{{t}}}}}\right)} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}}{\left({{{{2}} {{ C_{,{{t}}}}}} + { A_{,{{r}}}}}\right)} & {\frac{1}{2}} { B_{,{{t}}}} & 0 & 0 \\ {\frac{1}{2}} { B_{,{{t}}}} & {\frac{1}{2}} { B_{,{{r}}}} & 0 & 0 \\ 0 & 0 & -{r} & 0 \\ 0 & 0 & 0 & -{{{r}} {{{\sin\left( \phi\right)}^{2}}}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & r & 0 \\ 0 & r & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{r}} {{{\sin\left( \phi\right)}^{2}}} \\ 0 & 0 & 0 & 0 \\ 0 & {{r}} {{{\sin\left( \phi\right)}^{2}}} & 0 & {{{r}^{2}}} {{\sin\left( \phi\right)}} {{\cos\left( \phi\right)}}\end{matrix} \right]}\end{matrix} \right]}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} {\frac{1}{2}}{\left({{{{C}} {{ A_{,{{r}}}}}} - {{{A}} {{ B_{,{{t}}}}}}}\right)} & {\frac{1}{2}}{\left({{{{C}} {{ B_{,{{t}}}}}} + {{{B}} {{ A_{,{{r}}}}}}}\right)} & 0 & 0 \\ {\frac{1}{2}}{\left({{{{C}} {{ B_{,{{t}}}}}} + {{{B}} {{ A_{,{{r}}}}}}}\right)} & {\frac{1}{2}}{\left({{{{{C}} {{ B_{,{{r}}}}}} - {{{2}} {{B}} {{ C_{,{{r}}}}}}} + {{{B}} {{ B_{,{{t}}}}}}}\right)} & 0 & 0 \\ 0 & 0 & -{{{C}} {{r}}} & 0 \\ 0 & 0 & 0 & -{{{C}} {{r}} {{{\sin\left( \phi\right)}^{2}}}}\end{matrix} \right] \\ \left[ \begin{matrix} {\frac{1}{2}}{\left({{{{2}} {{A}} {{ C_{,{{t}}}}}} + {{{{A}} {{ A_{,{{r}}}}}} - {{{C}} {{ A_{,{{t}}}}}}}}\right)} & {\frac{1}{2}}{\left({{{{A}} {{ B_{,{{t}}}}}} - {{{C}} {{ A_{,{{r}}}}}}}\right)} & 0 & 0 \\ {\frac{1}{2}}{\left({{{{A}} {{ B_{,{{t}}}}}} - {{{C}} {{ A_{,{{r}}}}}}}\right)} & -{{\frac{1}{2}}{\left({{{{B}} {{ A_{,{{r}}}}}} + {{{C}} {{ B_{,{{t}}}}}}}\right)}} & 0 & 0 \\ 0 & 0 & -{{{A}} {{r}}} & 0 \\ 0 & 0 & 0 & -{{{A}} {{r}} {{{\sin\left( \phi\right)}^{2}}}}\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & \frac{1}{r} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{r} \\ 0 & 0 & 0 & 0 \\ 0 & \frac{1}{r} & 0 & \frac{\cos\left( \phi\right)}{\sin\left( \phi\right)}\end{matrix} \right]\end{matrix} \right]}}$
connection coefficients derivative:
${{{{{ \Gamma} ^a} _b} _c} _{,d}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}}{\left({{{{ A_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{{{C}} {{ A_{,{{r}}{{t}}}}}} - {{{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{A}} {{ B_{,{{t}}{{t}}}}}}}}\right)} & {\frac{1}{2}}{\left({{{{ A_{,{{r}}}}} {{ C_{,{{r}}}}}} + {{{{{C}} {{ A_{,{{r}}{{r}}}}}} - {{{ A_{,{{r}}}}} {{ B_{,{{t}}}}}}} - {{{A}} {{ B_{,{{r}}{{t}}}}}}}}\right)} & 0 & 0 \\ {\frac{1}{2}}{\left({{{{ B_{,{{t}}}}} {{ C_{,{{t}}}}}} + {{{C}} {{ B_{,{{t}}{{t}}}}}} + {{{ A_{,{{r}}}}} {{ B_{,{{t}}}}}} + {{{B}} {{ A_{,{{r}}{{t}}}}}}}\right)} & {\frac{1}{2}}{\left({{{{ B_{,{{t}}}}} {{ C_{,{{r}}}}}} + {{{C}} {{ B_{,{{r}}{{t}}}}}} + {{{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{B}} {{ A_{,{{r}}{{r}}}}}}}\right)} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}}{\left({{{{ B_{,{{t}}}}} {{ C_{,{{t}}}}}} + {{{C}} {{ B_{,{{t}}{{t}}}}}} + {{{ A_{,{{r}}}}} {{ B_{,{{t}}}}}} + {{{B}} {{ A_{,{{r}}{{t}}}}}}}\right)} & {\frac{1}{2}}{\left({{{{ B_{,{{t}}}}} {{ C_{,{{r}}}}}} + {{{C}} {{ B_{,{{r}}{{t}}}}}} + {{{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{B}} {{ A_{,{{r}}{{r}}}}}}}\right)} & 0 & 0 \\ {\frac{1}{2}}{\left({{{{ B_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{{{C}} {{ B_{,{{r}}{{t}}}}}} - {{{2}} {{ B_{,{{t}}}}} {{ C_{,{{r}}}}}}} - {{{2}} {{B}} {{ C_{,{{r}}{{t}}}}}}} + {{ B_{,{{t}}}}^{2}} + {{{B}} {{ B_{,{{t}}{{t}}}}}}}\right)} & {\frac{1}{2}}{\left({{-{{{ C_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{{C}} {{ B_{,{{r}}{{r}}}}}} - {{{2}} {{B}} {{ C_{,{{r}}{{r}}}}}}} + {{{ B_{,{{r}}}}} {{ B_{,{{t}}}}}} + {{{B}} {{ B_{,{{r}}{{t}}}}}}}\right)} & 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 \\ -{{{r}} {{ C_{,{{t}}}}}} & -{\left({{C} + {{{r}} {{ C_{,{{r}}}}}}}\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 & 0 \\ -{{{r}} {{ C_{,{{t}}}}} {{{\sin\left( \phi\right)}^{2}}}} & {{-{C}} - {{{r}} {{ C_{,{{r}}}}}}} + {{{C}} {{{\cos\left( \phi\right)}^{2}}}} + {{{r}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{r}}}}}} & 0 & -{{{2}} {{C}} {{r}} {{\sin\left( \phi\right)}} {{\cos\left( \phi\right)}}}\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}}{\left({{{{ A_{,{{t}}}}} {{ C_{,{{t}}}}}} + {{{2}} {{A}} {{ C_{,{{t}}{{t}}}}}} + {{{ A_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{{A}} {{ A_{,{{r}}{{t}}}}}} - {{{C}} {{ A_{,{{t}}{{t}}}}}}}}\right)} & {\frac{1}{2}}{\left({{{{2}} {{ A_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{2}} {{A}} {{ C_{,{{r}}{{t}}}}}} + {{ A_{,{{r}}}}^{2}} + {{{{{A}} {{ A_{,{{r}}{{r}}}}}} - {{{ C_{,{{r}}}}} {{ A_{,{{t}}}}}}} - {{{C}} {{ A_{,{{r}}{{t}}}}}}}}\right)} & 0 & 0 \\ {\frac{1}{2}}{\left({{{{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} + {{{{{A}} {{ B_{,{{t}}{{t}}}}}} - {{{ C_{,{{t}}}}} {{ A_{,{{r}}}}}}} - {{{C}} {{ A_{,{{r}}{{t}}}}}}}}\right)} & {\frac{1}{2}}{\left({{{{ B_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{{{A}} {{ B_{,{{r}}{{t}}}}}} - {{{ C_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{C}} {{ A_{,{{r}}{{r}}}}}}}}\right)} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}}{\left({{{{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} + {{{{{A}} {{ B_{,{{t}}{{t}}}}}} - {{{ C_{,{{t}}}}} {{ A_{,{{r}}}}}}} - {{{C}} {{ A_{,{{r}}{{t}}}}}}}}\right)} & {\frac{1}{2}}{\left({{{{ B_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{{{A}} {{ B_{,{{r}}{{t}}}}}} - {{{ C_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{C}} {{ A_{,{{r}}{{r}}}}}}}}\right)} & 0 & 0 \\ -{{\frac{1}{2}}{\left({{{{B}} {{ A_{,{{r}}{{t}}}}}} + {{{C}} {{ B_{,{{t}}{{t}}}}}} + {{{ A_{,{{r}}}}} {{ B_{,{{t}}}}}} + {{{ B_{,{{t}}}}} {{ C_{,{{t}}}}}}}\right)}} & -{{\frac{1}{2}}{\left({{{{B}} {{ A_{,{{r}}{{r}}}}}} + {{{C}} {{ B_{,{{r}}{{t}}}}}} + {{{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{ B_{,{{t}}}}} {{ C_{,{{r}}}}}}}\right)}} & 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 \\ -{{{r}} {{ A_{,{{t}}}}}} & -{\left({{A} + {{{r}} {{ A_{,{{r}}}}}}}\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 & 0 \\ -{{{r}} {{ A_{,{{t}}}}} {{{\sin\left( \phi\right)}^{2}}}} & {{-{A}} - {{{r}} {{ A_{,{{r}}}}}}} + {{{A}} {{{\cos\left( \phi\right)}^{2}}}} + {{{r}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{r}}}}}} & 0 & -{{{2}} {{A}} {{r}} {{\sin\left( \phi\right)}} {{\cos\left( \phi\right)}}}\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 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & -{\frac{1}{{r}^{2}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & -{\frac{1}{{r}^{2}}} & 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]} \\ \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 & 0 \\ 0 & -{\frac{1}{{r}^{2}}} & 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 & 0 & 0 & 0 \\ 0 & -{\frac{1}{{r}^{2}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -{\frac{1}{{\sin\left( \phi\right)}^{2}}}\end{matrix} \right]}\end{matrix} \right]}}$
connection coefficients squared:
${{{{{{ \Gamma} ^a} _e} _c}} {{{{{ \Gamma} ^e} _b} _d}}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {\frac{1}{4}}{\left({{{{{{C}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{A}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{t}}}}}}} + {{{{A}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{2}} {{A}} {{C}} {{ C_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{A}} {{B}} {{ C_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{{{A}} {{B}} {{{ A_{,{{r}}}}^{2}}}} - {{{{C}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{B}} {{C}} {{ A_{,{{t}}}}} {{ A_{,{{r}}}}}}}}\right)} & 0 & 0 & 0 \\ {\frac{1}{4}}{\left({{{{2}} {{A}} {{C}} {{ B_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{{{{A}} {{C}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{{C}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{t}}}}}}} - {{{4}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ C_{,{{t}}}}}}} - {{{2}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{2}} {{B}} {{C}} {{ C_{,{{r}}}}} {{ A_{,{{t}}}}}} + {{{{2}} {{A}} {{B}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}} - {{{B}} {{C}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{{C}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{{B}} {{C}} {{{ A_{,{{r}}}}^{2}}}} - {{{A}} {{C}} {{{ B_{,{{t}}}}^{2}}}}}}\right)} & {\frac{1}{4}}{\left({{{{{C}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{B}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{t}}}}}} + {{{{B}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{A}} {{C}} {{ B_{,{{t}}}}} {{ B_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{ B_{,{{t}}}}} {{ C_{,{{r}}}}}}} + {{{{A}} {{B}} {{{ B_{,{{t}}}}^{2}}}} - {{{{C}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{B}} {{C}} {{ A_{,{{r}}}}} {{ C_{,{{r}}}}}}}\right)} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{4}}{\left({{{{-{{{{B}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} - {{{B}} {{C}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}}} - {{{{C}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{{{C}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{A}} {{C}} {{ B_{,{{r}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{B}} {{C}} {{ C_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ B_{,{{t}}}}}} - {{{A}} {{B}} {{{ B_{,{{t}}}}^{2}}}}}}\right)} & 0 & 0 \\ {\frac{1}{4}}{\left({{{{{C}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{B}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{t}}}}}} + {{{{B}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{A}} {{C}} {{ B_{,{{t}}}}} {{ B_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{ B_{,{{t}}}}} {{ C_{,{{r}}}}}}} + {{{{A}} {{B}} {{{ B_{,{{t}}}}^{2}}}} - {{{{C}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{B}} {{C}} {{ A_{,{{r}}}}} {{ C_{,{{r}}}}}}}\right)} & 0 & 0 & 0 \\ 0 & 0 & -{C} & 0 \\ 0 & 0 & 0 & -{{{C}} {{{\sin\left( \phi\right)}^{2}}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & -{{\frac{1}{2}} {{{r}} {{\left({{{{A}} {{B}}} + {{C}^{2}}}\right)}} {{ A_{,{{r}}}}}}} & 0 \\ 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{-{{{A}} {{B}} {{ B_{,{{t}}}}}}} + {{{{{{2}} {{A}} {{B}} {{ C_{,{{r}}}}}} - {{{A}} {{C}} {{ B_{,{{r}}}}}}} - {{{B}} {{C}} {{ A_{,{{r}}}}}}} - {{{{C}^{2}}} {{ B_{,{{t}}}}}}}}\right)}}} & 0 \\ 0 & -{C} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{-{{{A}} {{B}}}} + {{{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}}} + {{{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}}} - {{C}^{2}}}}\right)}} {{ A_{,{{r}}}}}} \\ 0 & 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{-{{{A}} {{B}} {{ B_{,{{t}}}}}}} + {{{2}} {{A}} {{B}} {{ C_{,{{r}}}}}} + {{{{{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{t}}}}}} - {{{2}} {{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{r}}}}}}} - {{{A}} {{C}} {{ B_{,{{r}}}}}}} + {{{{A}} {{C}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{r}}}}}} - {{{B}} {{C}} {{ A_{,{{r}}}}}}} + {{{B}} {{C}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{r}}}}}} + {{{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{t}}}}}} - {{{{C}^{2}}} {{ B_{,{{t}}}}}}}}\right)}}} \\ 0 & 0 & 0 & 0 \\ 0 & -{{{C}} {{{\sin\left( \phi\right)}^{2}}}} & 0 & -{{{C}} {{r}} {{\cos\left( \phi\right)}} {{\sin\left( \phi\right)}}}\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{4}}{\left({{{{{{A}^{2}}} {{{ B_{,{{t}}}}^{2}}}} - {{{A}} {{C}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}}} + {{{{C}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{2}} {{A}} {{C}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}} - {{{{C}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{2}} {{A}} {{B}} {{ A_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{{A}} {{B}} {{{ A_{,{{r}}}}^{2}}}} - {{{B}} {{C}} {{ A_{,{{r}}}}} {{ A_{,{{t}}}}}}}}\right)} & 0 & 0 \\ {\frac{1}{4}}{\left({{{-{{{2}} {{A}} {{B}} {{ A_{,{{r}}}}} {{ C_{,{{t}}}}}}} - {{{A}} {{B}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}} {{C}} {{ A_{,{{r}}}}} {{ A_{,{{t}}}}}} - {{{2}} {{A}} {{C}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}}} + {{{A}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{t}}}}}} + {{{{{{C}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} - {{{{C}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} - {{{{A}^{2}}} {{{ B_{,{{t}}}}^{2}}}}}}\right)} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {\frac{1}{4}}{\left({{{{{{A}^{2}}} {{{ B_{,{{t}}}}^{2}}}} - {{{A}} {{C}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}}} + {{{{C}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{2}} {{A}} {{C}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}} - {{{{C}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{2}} {{A}} {{B}} {{ A_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{{A}} {{B}} {{{ A_{,{{r}}}}^{2}}}} - {{{B}} {{C}} {{ A_{,{{r}}}}} {{ A_{,{{t}}}}}}}}\right)} & {\frac{1}{4}}{\left({{-{{{A}} {{C}} {{{ B_{,{{t}}}}^{2}}}}} + {{{B}} {{C}} {{{ A_{,{{r}}}}^{2}}}} + {{{{C}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{C}} {{ B_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{{{{A}} {{C}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{{C}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{t}}}}}}} - {{{4}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ C_{,{{t}}}}}}} - {{{2}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{2}} {{B}} {{C}} {{ C_{,{{r}}}}} {{ A_{,{{t}}}}}} + {{{{2}} {{A}} {{B}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}} - {{{B}} {{C}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}}}\right)} & 0 & 0 \\ 0 & {\frac{1}{4}}{\left({{{{{B}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{B}} {{C}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{{C}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{A}} {{C}} {{ B_{,{{r}}}}} {{ B_{,{{t}}}}}} - {{{{C}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ B_{,{{t}}}}}}} + {{{2}} {{B}} {{C}} {{ C_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{A}} {{B}} {{{ B_{,{{t}}}}^{2}}}}}\right)} & 0 & 0 \\ 0 & 0 & -{A} & 0 \\ 0 & 0 & 0 & -{{{A}} {{{\sin\left( \phi\right)}^{2}}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{{-{{{2}} {{A}} {{C}} {{ C_{,{{t}}}}}}} - {{{{A}^{2}}} {{ B_{,{{t}}}}}}} + {{{{C}^{2}}} {{ A_{,{{t}}}}}}}\right)}}} & 0 \\ 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{{{A}} {{B}}} + {{C}^{2}}}\right)}} {{ A_{,{{r}}}}}} & 0 \\ 0 & -{A} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{-{{{2}} {{A}} {{C}} {{ C_{,{{t}}}}}}} + {{{2}} {{A}} {{C}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{t}}}}}} + {{{{{{A}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{t}}}}}} - {{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{t}}}}}}} - {{{{A}^{2}}} {{ B_{,{{t}}}}}}} + {{{{C}^{2}}} {{ A_{,{{t}}}}}}}\right)}}} \\ 0 & 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{{{{{A}} {{B}}} - {{{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}}}} - {{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}}}} + {{C}^{2}}}\right)}} {{ A_{,{{r}}}}}} \\ 0 & 0 & 0 & 0 \\ 0 & -{{{A}} {{{\sin\left( \phi\right)}^{2}}}} & 0 & -{{{A}} {{r}} {{\cos\left( \phi\right)}} {{\sin\left( \phi\right)}}}\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{{{2}} {{A}} {{ C_{,{{t}}}}}} + {{{{A}} {{ A_{,{{r}}}}}} - {{{C}} {{ A_{,{{t}}}}}}}}{{{2}} {{r}}} & \frac{{{{A}} {{ B_{,{{t}}}}}} - {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} & 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}{{r}^{2}} & 0 \\ \frac{{{{A}} {{ B_{,{{t}}}}}} - {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} & -{\frac{{{{B}} {{ A_{,{{r}}}}}} + {{{C}} {{ B_{,{{t}}}}}}}{{{2}} {{r}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & \frac{1}{{r}^{2}} & 0 & 0 \\ 0 & 0 & -{A} & 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 & -{{{A}} {{{\sin\left( \phi\right)}^{2}}}} \\ 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 \\ \frac{{{{2}} {{A}} {{ C_{,{{t}}}}}} + {{{{A}} {{ A_{,{{r}}}}}} - {{{C}} {{ A_{,{{t}}}}}}}}{{{2}} {{r}}} & \frac{{{{A}} {{ B_{,{{t}}}}}} - {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{{r}^{2}} \\ 0 & 0 & 0 & 0 \\ \frac{{{{A}} {{ B_{,{{t}}}}}} - {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} & -{\frac{{{{B}} {{ A_{,{{r}}}}}} + {{{C}} {{ B_{,{{t}}}}}}}{{{2}} {{r}}}} & 0 & \frac{\cos\left( \phi\right)}{{{r}} {{\sin\left( \phi\right)}}}\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 & -{A} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & \frac{1}{{r}^{2}} & 0 & \frac{\cos\left( \phi\right)}{{{r}} {{\sin\left( \phi\right)}}} \\ 0 & 0 & 0 & 0 \\ 0 & \frac{\cos\left( \phi\right)}{{{r}} {{\sin\left( \phi\right)}}} & 0 & \frac{{-{{{A}} {{{\sin\left( \phi\right)}^{2}}}}} + {{{A}} {{{\cos\left( \phi\right)}^{2}}} {{{\sin\left( \phi\right)}^{2}}}} + {{\cos\left( \phi\right)}^{2}}}{{\sin\left( \phi\right)}^{2}}\end{matrix} \right]}\end{matrix} \right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R} ^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 & {\frac{1}{4}}{\left({{{-{{{2}} {{ A_{,{{r}}}}} {{ C_{,{{r}}}}}}} - {{{2}} {{C}} {{ A_{,{{r}}{{r}}}}}}} + {{{4}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{ B_{,{{r}}{{t}}}}}} + {{{2}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}} + {{{2}} {{C}} {{ B_{,{{t}}{{t}}}}}} + {{{{{2}} {{B}} {{ A_{,{{r}}{{t}}}}}} - {{{2}} {{A}} {{C}} {{ B_{,{{r}}}}} {{ C_{,{{t}}}}}}} - {{{A}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{{C}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{{{2}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{2}} {{B}} {{C}} {{ C_{,{{r}}}}} {{ A_{,{{t}}}}}}} - {{{2}} {{A}} {{B}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}}} + {{{{{B}} {{C}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} - {{{{C}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{t}}}}}}} - {{{B}} {{C}} {{{ A_{,{{r}}}}^{2}}}}} + {{{A}} {{C}} {{{ B_{,{{t}}}}^{2}}}}}\right)} & 0 & 0 \\ {\frac{1}{4}}{\left({{{{{-{{{2}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}}} - {{{2}} {{C}} {{ B_{,{{t}}{{t}}}}}}} - {{{4}} {{ A_{,{{r}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{B}} {{ A_{,{{r}}{{t}}}}}}} + {{{2}} {{ A_{,{{r}}}}} {{ C_{,{{r}}}}}} + {{{{2}} {{C}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{ B_{,{{r}}{{t}}}}}}} + {{{2}} {{A}} {{C}} {{ B_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{{{{A}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{{C}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{r}}}}}}} - {{{4}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ C_{,{{t}}}}}}} - {{{2}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{2}} {{B}} {{C}} {{ C_{,{{r}}}}} {{ A_{,{{t}}}}}} + {{{{2}} {{A}} {{B}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}} - {{{B}} {{C}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{{C}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{{B}} {{C}} {{{ A_{,{{r}}}}^{2}}}} - {{{A}} {{C}} {{{ B_{,{{t}}}}^{2}}}}}}\right)} & 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}{2}}{\left({{{{-{{{{B}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} - {{{{C}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{A}} {{B}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{{{{2}} {{A}} {{B}} {{ B_{,{{t}}}}} {{ C_{,{{r}}}}}} - {{{A}} {{C}} {{ B_{,{{r}}}}} {{ B_{,{{t}}}}}}} - {{{B}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{B}} {{C}} {{ A_{,{{r}}}}} {{ C_{,{{r}}}}}}} - {{{B}} {{ A_{,{{r}}{{r}}}}}}} + {{{{B}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{B}} {{ C_{,{{r}}{{t}}}}}}} + {{{{{C}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{ B_{,{{t}}}}^{2}} - {{{3}} {{ B_{,{{t}}}}} {{ C_{,{{r}}}}}}} + {{{ B_{,{{r}}}}} {{ C_{,{{t}}}}}}}\right)} & 0 & 0 \\ {\frac{1}{2}}{\left({{{{{B}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{C}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{A}} {{B}} {{{ B_{,{{t}}}}^{2}}}} - {{{2}} {{A}} {{B}} {{ B_{,{{t}}}}} {{ C_{,{{r}}}}}}} + {{{A}} {{C}} {{ B_{,{{r}}}}} {{ B_{,{{t}}}}}} + {{{B}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{B}} {{C}} {{ A_{,{{r}}}}} {{ C_{,{{r}}}}}} + {{{{B}} {{ A_{,{{r}}{{r}}}}}} - {{{B}} {{ B_{,{{t}}{{t}}}}}}} + {{{{2}} {{B}} {{ C_{,{{r}}{{t}}}}}} - {{{{C}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{ B_{,{{t}}}}^{2}}} + {{{{3}} {{ B_{,{{t}}}}} {{ C_{,{{r}}}}}} - {{{ B_{,{{r}}}}} {{ C_{,{{t}}}}}}}}\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 & -{{\frac{1}{2}} {{{r}} {{\left({{{{A}} {{B}} {{ A_{,{{r}}}}}} + {{{{C}^{2}}} {{ A_{,{{r}}}}}} + {{{2}} {{ C_{,{{t}}}}}}}\right)}}}} & 0 \\ 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{-{{{A}} {{B}} {{ B_{,{{t}}}}}}} + {{{{{{{2}} {{A}} {{B}} {{ C_{,{{r}}}}}} - {{{A}} {{C}} {{ B_{,{{r}}}}}}} - {{{B}} {{C}} {{ A_{,{{r}}}}}}} - {{{{C}^{2}}} {{ B_{,{{t}}}}}}} - {{{2}} {{ C_{,{{r}}}}}}}}\right)}}} & 0 \\ {\frac{1}{2}} {{{r}} {{\left({{{{A}} {{B}} {{ A_{,{{r}}}}}} + {{{{C}^{2}}} {{ A_{,{{r}}}}}} + {{{2}} {{ C_{,{{t}}}}}}}\right)}}} & {\frac{1}{2}} {{{r}} {{\left({{{{{A}} {{B}} {{ B_{,{{t}}}}}} - {{{2}} {{A}} {{B}} {{ C_{,{{r}}}}}}} + {{{A}} {{C}} {{ B_{,{{r}}}}}} + {{{B}} {{C}} {{ A_{,{{r}}}}}} + {{{{C}^{2}}} {{ B_{,{{t}}}}}} + {{{2}} {{ C_{,{{r}}}}}}}\right)}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{-{{{A}} {{B}} {{ A_{,{{r}}}}}}} + {{{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{r}}}}}} + {{{{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{r}}}}}} - {{{{C}^{2}}} {{ A_{,{{r}}}}}}} - {{{2}} {{ C_{,{{t}}}}}}} + {{{2}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{t}}}}}}}\right)}}} \\ 0 & 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{-{{{A}} {{B}} {{ B_{,{{t}}}}}}} + {{{2}} {{A}} {{B}} {{ C_{,{{r}}}}}} + {{{{{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{t}}}}}} - {{{2}} {{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{r}}}}}}} - {{{A}} {{C}} {{ B_{,{{r}}}}}}} + {{{{A}} {{C}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{r}}}}}} - {{{B}} {{C}} {{ A_{,{{r}}}}}}} + {{{B}} {{C}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{r}}}}}} + {{{{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{t}}}}}} - {{{{C}^{2}}} {{ B_{,{{t}}}}}}} - {{{2}} {{ C_{,{{r}}}}}}} + {{{2}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{r}}}}}}}\right)}}} \\ 0 & 0 & 0 & 0 \\ {\frac{1}{2}} {{{r}} {{\left({{{{{{A}} {{B}} {{ A_{,{{r}}}}}} - {{{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{r}}}}}}} - {{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{r}}}}}}} + {{{{C}^{2}}} {{ A_{,{{r}}}}}} + {{{{2}} {{ C_{,{{t}}}}}} - {{{2}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{t}}}}}}}}\right)}}} & {\frac{1}{2}} {{{r}} {{\left({{{{{{A}} {{B}} {{ B_{,{{t}}}}}} - {{{2}} {{A}} {{B}} {{ C_{,{{r}}}}}}} - {{{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{t}}}}}}} + {{{2}} {{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{r}}}}}} + {{{{A}} {{C}} {{ B_{,{{r}}}}}} - {{{A}} {{C}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{r}}}}}}} + {{{{{B}} {{C}} {{ A_{,{{r}}}}}} - {{{B}} {{C}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{r}}}}}}} - {{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{t}}}}}}} + {{{{C}^{2}}} {{ B_{,{{t}}}}}} + {{{{2}} {{ C_{,{{r}}}}}} - {{{2}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{r}}}}}}}}\right)}}} & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{2}}{\left({{{{{A}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{C}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{A}} {{B}} {{{ A_{,{{r}}}}^{2}}}} + {{{{2}} {{A}} {{B}} {{ A_{,{{r}}}}} {{ C_{,{{t}}}}}} - {{{A}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{t}}}}}}} + {{{{2}} {{A}} {{C}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}} - {{{A}} {{ A_{,{{r}}{{r}}}}}}} + {{{{{{{{A}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{A}} {{ C_{,{{r}}{{t}}}}}}} - {{{B}} {{C}} {{ A_{,{{r}}}}} {{ A_{,{{t}}}}}}} - {{{{C}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{ A_{,{{r}}}}^{2}}} - {{{3}} {{ A_{,{{r}}}}} {{ C_{,{{t}}}}}}} + {{{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{ A_{,{{t}}}}} {{ C_{,{{r}}}}}}}\right)} & 0 & 0 \\ {\frac{1}{2}}{\left({{{{{-{{{{A}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{{C}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} - {{{A}} {{B}} {{{ A_{,{{r}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{ A_{,{{r}}}}} {{ C_{,{{t}}}}}}} + {{{{A}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{t}}}}}} - {{{2}} {{A}} {{C}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}}} + {{{{A}} {{ A_{,{{r}}{{r}}}}}} - {{{A}} {{ B_{,{{t}}{{t}}}}}}} + {{{2}} {{A}} {{ C_{,{{r}}{{t}}}}}} + {{{B}} {{C}} {{ A_{,{{r}}}}} {{ A_{,{{t}}}}}} + {{{{C}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{ A_{,{{r}}}}^{2}} + {{{{{3}} {{ A_{,{{r}}}}} {{ C_{,{{t}}}}}} - {{{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{ A_{,{{t}}}}} {{ C_{,{{r}}}}}}}}\right)} & 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}{4}}{\left({{{-{{{4}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{ B_{,{{r}}{{t}}}}}}} + {{{2}} {{ A_{,{{r}}}}} {{ C_{,{{r}}}}}} + {{{{{{{2}} {{C}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{B}} {{ A_{,{{r}}{{t}}}}}}} - {{{2}} {{C}} {{ B_{,{{t}}{{t}}}}}}} - {{{2}} {{ C_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{A}} {{C}} {{{ B_{,{{t}}}}^{2}}}}} + {{{B}} {{C}} {{{ A_{,{{r}}}}^{2}}}} + {{{{C}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{C}} {{ B_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{{{{A}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{{C}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{r}}}}}}} - {{{4}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ C_{,{{t}}}}}}} - {{{2}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{2}} {{B}} {{C}} {{ C_{,{{r}}}}} {{ A_{,{{t}}}}}} + {{{{2}} {{A}} {{B}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}} - {{{B}} {{C}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}}}\right)} & 0 & 0 \\ {\frac{1}{4}}{\left({{{{2}} {{B}} {{ A_{,{{r}}{{t}}}}}} + {{{2}} {{C}} {{ B_{,{{t}}{{t}}}}}} + {{{4}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{ C_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{{{2}} {{A}} {{ B_{,{{r}}{{t}}}}}} - {{{2}} {{ C_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{2}} {{C}} {{ A_{,{{r}}{{r}}}}}}} + {{{{{{{A}} {{C}} {{{ B_{,{{t}}}}^{2}}}} - {{{B}} {{C}} {{{ A_{,{{r}}}}^{2}}}}} - {{{{C}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{C}} {{ B_{,{{r}}}}} {{ C_{,{{t}}}}}}} - {{{A}} {{C}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{{C}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ C_{,{{t}}}}}} + {{{{{2}} {{A}} {{B}} {{ C_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{2}} {{B}} {{C}} {{ C_{,{{r}}}}} {{ A_{,{{t}}}}}}} - {{{2}} {{A}} {{B}} {{ B_{,{{t}}}}} {{ C_{,{{t}}}}}}} + {{{B}} {{C}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}}\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 & {\frac{1}{2}} {{{r}} {{\left({{{-{{{2}} {{A}} {{C}} {{ C_{,{{t}}}}}}} - {{{{A}^{2}}} {{ B_{,{{t}}}}}}} + {{{{{C}^{2}}} {{ A_{,{{t}}}}}} - {{{2}} {{ A_{,{{t}}}}}}}}\right)}}} & 0 \\ 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{-{2}} + {{{A}} {{B}}} + {{C}^{2}}}\right)}} {{ A_{,{{r}}}}}} & 0 \\ {\frac{1}{2}} {{{r}} {{\left({{{{2}} {{A}} {{C}} {{ C_{,{{t}}}}}} + {{{{{A}^{2}}} {{ B_{,{{t}}}}}} - {{{{C}^{2}}} {{ A_{,{{t}}}}}}} + {{{2}} {{ A_{,{{t}}}}}}}\right)}}} & {\frac{1}{2}} {{{r}} {{\left({{{2} - {{{A}} {{B}}}} - {{C}^{2}}}\right)}} {{ A_{,{{r}}}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{-{{{2}} {{A}} {{C}} {{ C_{,{{t}}}}}}} + {{{2}} {{A}} {{C}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{t}}}}}} + {{{{{{A}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{t}}}}}} - {{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{t}}}}}}} - {{{{A}^{2}}} {{ B_{,{{t}}}}}}} + {{{{{C}^{2}}} {{ A_{,{{t}}}}}} - {{{2}} {{ A_{,{{t}}}}}}} + {{{2}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{t}}}}}}}\right)}}} \\ 0 & 0 & 0 & {\frac{1}{2}} {{{r}} {{\left({{-{2}} + {{{{{A}} {{B}}} - {{{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}}}} - {{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}}}} + {{C}^{2}} + {{{2}} {{{\cos\left( \phi\right)}^{2}}}}}\right)}} {{ A_{,{{r}}}}}} \\ 0 & 0 & 0 & 0 \\ {\frac{1}{2}} {{{r}} {{\left({{{{{{2}} {{A}} {{C}} {{ C_{,{{t}}}}}} - {{{2}} {{A}} {{C}} {{{\cos\left( \phi\right)}^{2}}} {{ C_{,{{t}}}}}}} - {{{{A}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ B_{,{{t}}}}}}} + {{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{t}}}}}} + {{{{{A}^{2}}} {{ B_{,{{t}}}}}} - {{{{C}^{2}}} {{ A_{,{{t}}}}}}} + {{{{2}} {{ A_{,{{t}}}}}} - {{{2}} {{{\cos\left( \phi\right)}^{2}}} {{ A_{,{{t}}}}}}}}\right)}}} & {\frac{1}{2}} {{{r}} {{\left({{{2} - {{{A}} {{B}}}} + {{{A}} {{B}} {{{\cos\left( \phi\right)}^{2}}}} + {{{{{{C}^{2}}} {{{\cos\left( \phi\right)}^{2}}}} - {{C}^{2}}} - {{{2}} {{{\cos\left( \phi\right)}^{2}}}}}}\right)}} {{ A_{,{{r}}}}}} & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & \frac{{{-{{{2}} {{A}} {{ C_{,{{t}}}}}}} - {{{A}} {{ A_{,{{r}}}}}}} + {{{C}} {{ A_{,{{t}}}}}}}{{{2}} {{r}}} & 0 \\ 0 & 0 & \frac{{-{{{A}} {{ B_{,{{t}}}}}}} + {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} & 0 \\ \frac{{{{2}} {{A}} {{ C_{,{{t}}}}}} + {{{{A}} {{ A_{,{{r}}}}}} - {{{C}} {{ A_{,{{t}}}}}}}}{{{2}} {{r}}} & \frac{{{{A}} {{ B_{,{{t}}}}}} - {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & \frac{{-{{{A}} {{ B_{,{{t}}}}}}} + {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} & 0 \\ 0 & 0 & \frac{{{{B}} {{ A_{,{{r}}}}}} + {{{C}} {{ B_{,{{t}}}}}}}{{{2}} {{r}}} & 0 \\ \frac{{{{A}} {{ B_{,{{t}}}}}} - {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} & -{\frac{{{{B}} {{ A_{,{{r}}}}}} + {{{C}} {{ B_{,{{t}}}}}}}{{{2}} {{r}}}} & 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]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -{{{A}} {{{\sin\left( \phi\right)}^{2}}}} \\ 0 & 0 & {{A}} {{{\sin\left( \phi\right)}^{2}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & \frac{{{-{{{2}} {{A}} {{ C_{,{{t}}}}}}} - {{{A}} {{ A_{,{{r}}}}}}} + {{{C}} {{ A_{,{{t}}}}}}}{{{2}} {{r}}} \\ 0 & 0 & 0 & \frac{{-{{{A}} {{ B_{,{{t}}}}}}} + {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} \\ 0 & 0 & 0 & 0 \\ \frac{{{{2}} {{A}} {{ C_{,{{t}}}}}} + {{{{A}} {{ A_{,{{r}}}}}} - {{{C}} {{ A_{,{{t}}}}}}}}{{{2}} {{r}}} & \frac{{{{A}} {{ B_{,{{t}}}}}} - {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & \frac{{-{{{A}} {{ B_{,{{t}}}}}}} + {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} \\ 0 & 0 & 0 & \frac{{{{B}} {{ A_{,{{r}}}}}} + {{{C}} {{ B_{,{{t}}}}}}}{{{2}} {{r}}} \\ 0 & 0 & 0 & 0 \\ \frac{{{{A}} {{ B_{,{{t}}}}}} - {{{C}} {{ A_{,{{r}}}}}}}{{{2}} {{r}}} & -{\frac{{{{B}} {{ A_{,{{r}}}}}} + {{{C}} {{ B_{,{{t}}}}}}}{{{2}} {{r}}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & A \\ 0 & 0 & -{A} & 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]}}$