metric:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{A}& 0& 0& 0\\ 0& B& 0& 0\\ 0& 0& {r}^{2}& 0\\ 0& 0& 0& {{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}\end{array}\right]}}$
metric inverse:
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{\frac{1}{A}}& 0& 0& 0\\ 0& \frac{1}{B}& 0& 0\\ 0& 0& \frac{1}{{r}^{2}}& 0\\ 0& 0& 0& \frac{1}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}\end{array}\right]}}$
metric derivative:
${{{{ g} _a} _b} _{,c}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& -{ A_{,{{r}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& B_{,{{r}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}} {{{\sin\left( \theta\right)}^{2}}}& {{2}} {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$
1st kind Christoffel:
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& -{{\frac{1}{2}} { A_{,{{r}}}}}& 0& 0\\ -{{\frac{1}{2}} { A_{,{{r}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{2}} { A_{,{{r}}}}& 0& 0& 0\\ 0& {\frac{1}{2}} { B_{,{{r}}}}& 0& 0\\ 0& 0& -{r}& 0\\ 0& 0& 0& -{{{r}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& r& 0\\ 0& r& 0& 0\\ 0& 0& 0& -{{{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {{r}} {{{\sin\left( \theta\right)}^{2}}}\\ 0& 0& 0& {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}\\ 0& {{r}} {{{\sin\left( \theta\right)}^{2}}}& {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} 0& \frac{ A_{,{{r}}}}{{{2}} {{A}}}& 0& 0\\ \frac{ A_{,{{r}}}}{{{2}} {{A}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} \frac{ A_{,{{r}}}}{{{2}} {{B}}}& 0& 0& 0\\ 0& \frac{ B_{,{{r}}}}{{{2}} {{B}}}& 0& 0\\ 0& 0& -{{\frac{1}{B}} {r}}& 0\\ 0& 0& 0& -{{\frac{1}{B}} {{{r}} {{{\sin\left( \theta\right)}^{2}}}}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& \frac{1}{r}& 0& 0\\ 0& 0& 0& -{{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{r}\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}\\ 0& \frac{1}{r}& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}& 0\end{array}\right]\end{matrix}\right]}}$
connection coefficients derivative:
${{{{{ \Gamma} ^a} _b} _c} _{,d}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{{-{{ A_{,{{r}}}}^{2}}} + {{{A}} {{ A_{,{{r}}{{r}}}}}}}{{{2}} {{{A}^{2}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{-{{ A_{,{{r}}}}^{2}}} + {{{A}} {{ A_{,{{r}}{{r}}}}}}}{{{2}} {{{A}^{2}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{{B}} {{ A_{,{{r}}{{r}}}}}}{-{{{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}}}{{{2}} {{{B}^{2}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{{-{{ B_{,{{r}}}}^{2}}} + {{{B}} {{ B_{,{{r}}{{r}}}}}}}{{{2}} {{{B}^{2}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& \frac{{-{B}} + {{{r}} {{ B_{,{{r}}}}}}}{{B}^{2}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& \frac{{-{B}} + {{{B}} {{{\cos\left( \theta\right)}^{2}}}} + {{{r}} {{ B_{,{{r}}}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}}{{B}^{2}}& -{{\frac{1}{B}} {{{2}} {{r}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& {1}{-{{{2}} {{{\cos\left( \theta\right)}^{2}}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& -{\frac{1}{{\sin\left( \theta\right)}^{2}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& -{\frac{1}{{\sin\left( \theta\right)}^{2}}}& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
connection coefficients squared:
${{{{{{ \Gamma} ^a} _e} _c}} {{{{{ \Gamma} ^e} _b} _d}}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{ A_{,{{r}}}}^{2}}{{{4}} {{A}} {{B}}}& 0& 0& 0\\ 0& \frac{{ A_{,{{r}}}}^{2}}{{{4}} {{{A}^{2}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{{{4}} {{A}} {{B}}}& 0& 0\\ \frac{{ A_{,{{r}}}}^{2}}{{{4}} {{{A}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{\frac{{{r}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{ A_{,{{r}}}}^{2}}{{{4}} {{A}} {{B}}}& 0& 0\\ \frac{{{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}{{{4}} {{{B}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{ A_{,{{r}}}}^{2}}{{{4}} {{A}} {{B}}}& 0& 0& 0\\ 0& \frac{{ B_{,{{r}}}}^{2}}{{{4}} {{{B}^{2}}}}& 0& 0\\ 0& 0& -{\frac{1}{B}}& 0\\ 0& 0& 0& -{{\frac{1}{B}} {{\sin\left( \theta\right)}^{2}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{{{r}} {{ B_{,{{r}}}}}}{{{2}} {{{B}^{2}}}}}& 0\\ 0& -{\frac{1}{B}}& 0& 0\\ 0& 0& 0& -{{\frac{1}{B}} {{{r}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}{{{2}} {{{B}^{2}}}}}\\ 0& 0& 0& {\frac{1}{B}} {{{r}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\\ 0& -{{\frac{1}{B}} {{\sin\left( \theta\right)}^{2}}}& -{{\frac{1}{B}} {{{r}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{1}{{r}^{2}}& 0\\ 0& \frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}}& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{1}{{r}^{2}}& 0& 0\\ 0& 0& -{\frac{1}{B}}& 0\\ 0& 0& 0& -{{\cos\left( \theta\right)}^{2}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}\\ 0& 0& 0& -{{\frac{1}{B}} {{\sin\left( \theta\right)}^{2}}}\\ 0& -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}& -{{\cos\left( \theta\right)}^{2}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{{r}^{2}}\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}\\ 0& \frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}}& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}\\ 0& 0& 0& \frac{{\cos\left( \theta\right)}^{2}}{{\sin\left( \theta\right)}^{2}}\\ 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& -{\frac{1}{B}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{1}{{r}^{2}}& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& 0\\ 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& \frac{{\cos\left( \theta\right)}^{2}}{{\sin\left( \theta\right)}^{2}}& 0\\ 0& 0& 0& {\frac{1}{B}}{\left({{-{1}} + {{\cos\left( \theta\right)}^{2}}{-{{{B}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{{B}} {{{ A_{,{{r}}}}^{2}}}} + {{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}}}{{{4}} {{B}} {{{A}^{2}}}}& 0& 0\\ \frac{{-{{{B}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}}{{{4}} {{B}} {{{A}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{\frac{{{r}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{r}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{{B}} {{{ A_{,{{r}}}}^{2}}}} + {{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}}}{{{4}} {{A}} {{{B}^{2}}}}& 0& 0\\ \frac{{-{{{B}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}}{{{4}} {{A}} {{{B}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{r}} {{ B_{,{{r}}}}}}{{{2}} {{{B}^{2}}}}& 0\\ 0& -{\frac{{{r}} {{ B_{,{{r}}}}}}{{{2}} {{{B}^{2}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}{{{2}} {{{B}^{2}}}}\\ 0& 0& 0& 0\\ 0& -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}{{{2}} {{{B}^{2}}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{\frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}}}& 0\\ 0& 0& 0& 0\\ \frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}}}& 0\\ 0& \frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{B}}{\left({{-{1}} + {B} + {{\cos\left( \theta\right)}^{2}}{-{{{B}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}\\ 0& 0& {\frac{1}{B}}{\left({{1}{-{B}}{-{{\cos\left( \theta\right)}^{2}}} + {{{B}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}}}\\ 0& 0& 0& 0\\ 0& \frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{B}}{\left({{1}{-{B}}}\right)}\\ 0& 0& {\frac{1}{B}}{\left({{-{1}} + {B}}\right)}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& \frac{{{{B}} {{{ A_{,{{r}}}}^{2}}}} + {{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}}& 0& 0\\ \frac{{-{{{B}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& -{\frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}}& 0\\ 0& 0& 0& 0\\ \frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& -{\frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& \frac{{-{{{B}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}}& 0& 0\\ \frac{{{{B}} {{{ A_{,{{r}}}}^{2}}}} + {{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}& 0\\ 0& -{\frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}\\ 0& 0& 0& 0\\ 0& -{\frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}}& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& \frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}& 0\\ 0& 0& 0& 0\\ -{\frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}}& 0\\ 0& \frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}} + {B}}{{{B}} {{{r}^{2}}}}\\ 0& 0& \frac{{1}{-{B}}}{{{B}} {{{r}^{2}}}}& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& \frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}}\\ 0& 0& 0& 0\\ 0& \frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{1}{-{B}}}{{{B}} {{{r}^{2}}}}\\ 0& 0& \frac{{-{1}} + {B}}{{{B}} {{{r}^{2}}}}& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$
Ricci curvature, $\sharp\flat$:
${{{ R} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{{{B}} {{r}} {{{ A_{,{{r}}}}^{2}}}} + {{{A}} {{r}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}{-{{{4}} {{A}} {{B}} {{ A_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}{{r}}}}}}}}{{{4}} {{r}} {{{A}^{2}}} {{{B}^{2}}}}& 0& 0& 0\\ 0& \frac{{{{B}} {{r}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{ B_{,{{r}}}}}} + {{{A}} {{r}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}{{r}}}}}}}}{{{4}} {{r}} {{{A}^{2}}} {{{B}^{2}}}}& 0& 0\\ 0& 0& \frac{{-{{{2}} {{A}} {{B}}}} + {{{2}} {{A}} {{{B}^{2}}}} + {{{A}} {{r}} {{ B_{,{{r}}}}}}{-{{{B}} {{r}} {{ A_{,{{r}}}}}}}}{{{2}} {{A}} {{{B}^{2}}} {{{r}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{{{2}} {{A}} {{B}}}} + {{{2}} {{A}} {{{B}^{2}}}} + {{{A}} {{r}} {{ B_{,{{r}}}}}}{-{{{B}} {{r}} {{ A_{,{{r}}}}}}}}{{{2}} {{A}} {{{B}^{2}}} {{{r}^{2}}}}\end{array}\right]}}$
Gaussian curvature:
${R} = {\frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{4}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}}}{{{2}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}}$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}{-{{{4}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}}}{{{8}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0& 0\\ 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{4}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{4}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}}{{{8}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0\\ 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{8}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{8}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\end{array}\right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{B}{-{{B}^{2}}}{-{{{r}} {{ B_{,{{r}}}}}}}}{{{{B}^{2}}} {{{r}^{2}}}}& 0& 0& 0\\ 0& \frac{{-{{{{A}^{2}}} {{{B}^{2}}}}} + {{{B}} {{{A}^{2}}}} + {{{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}}{{{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0\\ 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\end{array}\right]}}$
Schouten, $\sharp\flat$:
${{{ P} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{-{{{2}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{2}} {{B}} {{{A}^{2}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0& 0\\ 0& \frac{{-{{{2}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{2}} {{B}} {{{A}^{2}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{4}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0\\ 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{8}} {{B}} {{{A}^{2}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{8}} {{B}} {{{A}^{2}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\end{array}\right]}}$
Weyl, $\sharp\sharp\flat\flat$:
${{{{{ C} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0\\ \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0\\ \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0\\ 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0\\ 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\\ 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\\ 0& 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C} _a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{12}} {{A}} {{B}} {{{r}^{2}}}}& 0& 0\\ \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{12}} {{A}} {{B}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{A}} {{{B}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{A}} {{{B}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{B}} {{{A}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}}}{{{24}} {{A}} {{{B}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{B}} {{{A}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{-{{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}}{{{24}} {{A}} {{{B}^{2}}}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{12}} {{A}} {{B}} {{{r}^{2}}}}& 0& 0\\ \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{12}} {{A}} {{B}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{B}} {{{A}^{2}}}}& 0\\ 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{B}} {{{A}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{B}} {{{A}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{-{{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}}{{{24}} {{B}} {{{A}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{B}} {{{A}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}}}{{{24}} {{B}} {{{A}^{2}}}}& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{A}} {{{B}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{A}} {{{B}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{24}} {{B}} {{{A}^{2}}}}& 0\\ 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{24}} {{B}} {{{A}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}}}{-{{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}{-{{{2}} {{{A}^{2}}} {{{r}^{3}}} {{ B_{,{{r}}}}}}} + {{{2}} {{{A}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}} + {{{A}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{A}} {{B}} {{{r}^{3}}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{A}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}}}\\ 0& 0& \frac{{-{{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}} + {{{2}} {{{A}^{2}}} {{{r}^{3}}} {{ B_{,{{r}}}}}}{-{{{2}} {{{A}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{-{{{A}} {{{r}^{4}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{2}} {{A}} {{B}} {{{r}^{3}}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}}} + {{{A}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}}}& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{B}} {{{A}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{-{{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}}{{{24}} {{A}} {{{B}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{B}} {{{A}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}}}{{{24}} {{A}} {{{B}^{2}}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}}{-{{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{B}} {{{A}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}}}{{{24}} {{B}} {{{A}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{-{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{B}} {{{A}^{2}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}{-{{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{-{{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}}{{{24}} {{B}} {{{A}^{2}}}}& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}} + {{{2}} {{{A}^{2}}} {{{r}^{3}}} {{ B_{,{{r}}}}}}{-{{{2}} {{{A}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{-{{{A}} {{{r}^{4}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{2}} {{A}} {{B}} {{{r}^{3}}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}}} + {{{A}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}}}\\ 0& 0& \frac{{{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}}}{-{{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}{-{{{2}} {{{A}^{2}}} {{{r}^{3}}} {{ B_{,{{r}}}}}}} + {{{2}} {{{A}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}} + {{{A}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{A}} {{B}} {{{r}^{3}}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{A}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}}}& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$
Plebanski, $\sharp\sharp\flat\flat$:
${{{{{ P} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{96}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{96}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{96}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{96}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{96}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{96}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{192}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}}}}}{{{96}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& \frac{{{{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{4}}}} + {{{16}} {{{A}^{4}}} {{{B}^{4}}}}{-{{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}}}} + {{{{A}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{ B_{,{{r}}}}^{2}}}}{-{{{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{4}}}}} + {{{16}} {{{A}^{4}}} {{{B}^{2}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{A}^{4}}} {{{B}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{32}} {{{A}^{4}}} {{{B}^{3}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}{{r}}}}^{2}}}} + {{{4}} {{{A}^{4}}} {{{r}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{r}}}}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{32}} {{{A}^{4}}} {{{B}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}}{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}{{r}}}}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{3}}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{16}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}{-{{{16}} {{{A}^{3}}} {{{B}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}} + {{{4}} {{A}} {{{B}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{8}} {{{A}^{3}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{3}}} {{ B_{,{{r}}}}}}}{-{{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{4}} {{B}} {{{A}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}{{r}}}}}}}{{{96}} {{{A}^{4}}} {{{B}^{4}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
vaccuum constraint:
${{{ R} ^u} _v} = {0}$
${{{{ R} ^t} _t} = {\frac{{{{B}} {{r}} {{{ A_{,{{r}}}}^{2}}}} + {{{A}} {{r}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}{-{{{4}} {{A}} {{B}} {{ A_{,{{r}}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}{{r}}}}}}}}{{{4}} {{r}} {{{A}^{2}}} {{{B}^{2}}}}}} = {0}$
${{{{ R} ^r} _r} = {\frac{{{{B}} {{r}} {{{ A_{,{{r}}}}^{2}}}} + {{{4}} {{{A}^{2}}} {{ B_{,{{r}}}}}} + {{{A}} {{r}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}{{r}}}}}}}}{{{4}} {{r}} {{{A}^{2}}} {{{B}^{2}}}}}} = {0}$
${{{{ R} ^{\theta}} _{\theta}} = {\frac{{-{{{2}} {{A}} {{B}}}} + {{{2}} {{A}} {{{B}^{2}}}} + {{{A}} {{r}} {{ B_{,{{r}}}}}}{-{{{B}} {{r}} {{ A_{,{{r}}}}}}}}{{{2}} {{A}} {{{B}^{2}}} {{{r}^{2}}}}}} = {0}$
${{{{ R} ^{\phi}} _{\phi}} = {\frac{{-{{{2}} {{A}} {{B}}}} + {{{2}} {{A}} {{{B}^{2}}}} + {{{A}} {{r}} {{ B_{,{{r}}}}}}{-{{{B}} {{r}} {{ A_{,{{r}}}}}}}}{{{2}} {{A}} {{{B}^{2}}} {{{r}^{2}}}}}} = {0}$
${{{{ R} ^t} _t}{-{{{ R} ^r} _r}}} = {-{\frac{{{{A}} {{ B_{,{{r}}}}}} + {{{B}} {{ A_{,{{r}}}}}}}{{{A}} {{r}} {{{B}^{2}}}}}}$
${-{\frac{{{{A}} {{ B_{,{{r}}}}}} + {{{B}} {{ A_{,{{r}}}}}}}{{{A}} {{r}} {{{B}^{2}}}}}} = {0}$
${\frac{{{{A}} {{ B_{,{{r}}}}}} + {{{B}} {{ A_{,{{r}}}}}}}{{{A}} {{r}} {{{B}^{2}}}}} = {0}$
${{\left( {{A}} {{B}}\right)} _{,r}} = {0}$
${{{A}} {{B}}} = {K}$
${{{{A}} {{ B_{,{{r}}}}}} + {{{B}} {{ A_{,{{r}}}}}}} = {0}$
substitute into ${{{ R} ^{\theta}} _{\theta}} = {0}$
${{-{B}} + {{B}^{2}} + {{{r}} {{ B_{,{{r}}}}}}} = {0}$
${A} = {\frac{{-{1}} + {{{S}} {{r}}}}{{{S}} {{r}}}}$
${B} = {\frac{{{S}} {{r}}}{{-{1}} + {{{S}} {{r}}}}}$
${A} = {{\frac{1}{r}}{\left({{-{R}} + {r}}\right)}}$
${B} = {\frac{r}{{-{R}} + {r}}}$
now substitute to find:
metric:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{r}}{\left({{R}{-{r}}}\right)}& 0& 0& 0\\ 0& \frac{r}{{-{R}} + {r}}& 0& 0\\ 0& 0& {r}^{2}& 0\\ 0& 0& 0& {{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}\end{array}\right]}}$
metric inverse:
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{r}{{R}{-{r}}}& 0& 0& 0\\ 0& {\frac{1}{r}}{\left({{-{R}} + {r}}\right)}& 0& 0\\ 0& 0& \frac{1}{{r}^{2}}& 0\\ 0& 0& 0& \frac{1}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}\end{array}\right]}}$
metric derivative:
${{{{ g} _a} _b} _{,c}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{R}{{r}^{2}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{R}{{\left({{R}{-{r}}}\right)}^{2}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}} {{{\sin\left( \theta\right)}^{2}}}& {{2}} {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$
1st kind Christoffel:
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{R}{{{2}} {{{r}^{2}}}}}& 0& 0\\ -{\frac{R}{{{2}} {{{r}^{2}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} \frac{R}{{{2}} {{{r}^{2}}}}& 0& 0& 0\\ 0& -{\frac{R}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}}& 0& 0\\ 0& 0& -{r}& 0\\ 0& 0& 0& -{{{r}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& r& 0\\ 0& r& 0& 0\\ 0& 0& 0& -{{{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {{r}} {{{\sin\left( \theta\right)}^{2}}}\\ 0& 0& 0& {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}\\ 0& {{r}} {{{\sin\left( \theta\right)}^{2}}}& {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} 0& -{\frac{R}{{{2}} {{r}} {{\left({{-{r}} + {R}}\right)}}}}& 0& 0\\ -{\frac{R}{{{2}} {{r}} {{\left({{-{r}} + {R}}\right)}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{3}}}}& 0& 0& 0\\ 0& \frac{R}{{{2}} {{r}} {{\left({{-{r}} + {R}}\right)}}}& 0& 0\\ 0& 0& {R}{-{r}}& 0\\ 0& 0& 0& {R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& \frac{1}{r}& 0& 0\\ 0& 0& 0& -{{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{r}\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}\\ 0& \frac{1}{r}& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}& 0\end{array}\right]\end{matrix}\right]}}$
connection coefficients derivative:
${{{{{ \Gamma} ^a} _b} _c} _{,d}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{{{R}} {{\left({{R}{-{{{2}} {{r}}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{R}} {{\left({{R}{-{{{2}} {{r}}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{{3}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{2}} {{R}} {{{r}^{3}}}}}}{{{2}} {{{r}^{6}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{{{R}} {{\left({{-{R}} + {{{2}} {{r}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{1}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{{\sin\left( \theta\right)}^{2}}& {{2}} {{\left({{R}{-{r}}}\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& {1}{-{{{2}} {{{\cos\left( \theta\right)}^{2}}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& -{\frac{1}{{\sin\left( \theta\right)}^{2}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& -{\frac{1}{{\sin\left( \theta\right)}^{2}}}& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
connection coefficients squared:
${{{{{{ \Gamma} ^a} _e} _c}} {{{{{ \Gamma} ^e} _b} _d}}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{R}^{2}}{{{4}} {{{r}^{4}}}}& 0& 0& 0\\ 0& \frac{{R}^{2}}{{{4}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{{R}^{2}}{{{4}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}}& 0& 0\\ \frac{{R}^{2}}{{{4}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{\frac{R}{{{2}} {{r}}}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{r}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{R}^{2}}{{{4}} {{{r}^{4}}}}& 0& 0\\ -{\frac{{R}^{2}}{{{4}} {{{r}^{4}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{R}^{2}}{{{4}} {{{r}^{4}}}}& 0& 0& 0\\ 0& \frac{{R}^{2}}{{{4}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& 0& 0\\ 0& 0& {\frac{1}{r}}{\left({{R}{-{r}}}\right)}& 0\\ 0& 0& 0& {\frac{1}{r}}{\left({{R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{R}{{{2}} {{r}}}& 0\\ 0& {\frac{1}{r}}{\left({{R}{-{r}}}\right)}& 0& 0\\ 0& 0& 0& {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}} {{\left({{-{r}} + {R}}\right)}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{r}}}\\ 0& 0& 0& {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} {{\left({{-{R}} + {r}}\right)}}\\ 0& {\frac{1}{r}}{\left({{R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}& {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}} {{\left({{-{r}} + {R}}\right)}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{4}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{1}{{r}^{2}}& 0\\ 0& \frac{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{1}{{r}^{2}}& 0& 0\\ 0& 0& {\frac{1}{r}}{\left({{R}{-{r}}}\right)}& 0\\ 0& 0& 0& -{{\cos\left( \theta\right)}^{2}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}\\ 0& 0& 0& {\frac{1}{r}}{\left({{R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}\\ 0& -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}& -{{\cos\left( \theta\right)}^{2}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{4}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{{r}^{2}}\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}\\ 0& \frac{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}\\ 0& 0& 0& \frac{{\cos\left( \theta\right)}^{2}}{{\sin\left( \theta\right)}^{2}}\\ 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& {\frac{1}{r}}{\left({{R}{-{r}}}\right)}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{1}{{r}^{2}}& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& 0\\ 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& \frac{{\cos\left( \theta\right)}^{2}}{{\sin\left( \theta\right)}^{2}}& 0\\ 0& 0& 0& {\frac{1}{r}}{\left({{R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{R}{{{{r}^{2}}} {{\left({{r}{-{R}}}\right)}}}& 0& 0\\ \frac{R}{{{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{\frac{R}{{{2}} {{r}}}}& 0\\ 0& 0& 0& 0\\ \frac{R}{{{2}} {{r}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{r}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{r}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{-{{{{R}^{2}}} {{{r}^{2}}}}} + {{{R}} {{{r}^{3}}}}}{{r}^{6}}& 0& 0\\ \frac{{{{{R}^{2}}} {{{r}^{2}}}}{-{{{R}} {{{r}^{3}}}}}}{{r}^{6}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{R}{{{2}} {{r}}}}& 0\\ 0& \frac{R}{{{2}} {{r}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{r}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{r}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{{R}} {{\left({{R}{-{r}}}\right)}}}{{{2}} {{{r}^{4}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{4}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}}& 0\\ 0& \frac{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{r}} {{{R}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& -{{\frac{1}{r}} {{{R}} {{{\sin\left( \theta\right)}^{2}}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{{R}} {{\left({{R}{-{r}}}\right)}}}{{{2}} {{{r}^{4}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{4}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}}\\ 0& 0& 0& 0\\ 0& \frac{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {R}}\\ 0& 0& {\frac{1}{r}} {R}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& \frac{R}{{r}^{3}}& 0& 0\\ -{\frac{R}{{r}^{3}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0\\ 0& 0& 0& 0\\ \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& -{\frac{R}{{r}^{3}}}& 0& 0\\ \frac{R}{{r}^{3}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0\\ 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}\\ 0& 0& 0& 0\\ 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0\\ 0& 0& 0& 0\\ -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0\\ 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{R}{{r}^{3}}\\ 0& 0& -{\frac{R}{{r}^{3}}}& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}\\ 0& 0& 0& 0\\ 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -{\frac{R}{{r}^{3}}}\\ 0& 0& \frac{R}{{r}^{3}}& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$
Ricci curvature, $\sharp\flat$:
${{{ R} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Gaussian curvature:
${R} = {0}$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Schouten, $\sharp\flat$:
${{{ P} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Weyl, $\sharp\sharp\flat\flat$:
${{{{{ C} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{R}{{r}^{3}}& 0& 0\\ -{\frac{R}{{r}^{3}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0\\ 0& 0& 0& 0\\ \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{R}{{r}^{3}}}& 0& 0\\ \frac{R}{{r}^{3}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0\\ 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}\\ 0& 0& 0& 0\\ 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0\\ 0& 0& 0& 0\\ -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0\\ 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{R}{{r}^{3}}\\ 0& 0& -{\frac{R}{{r}^{3}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}\\ 0& 0& 0& 0\\ 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -{\frac{R}{{r}^{3}}}\\ 0& 0& \frac{R}{{r}^{3}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C} _a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& -{\frac{R}{{r}^{3}}}& 0& 0\\ \frac{R}{{r}^{3}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{R}{-{r}}}\right)}}}{{{2}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& \frac{{-{{R}^{2}}} + {{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{R}} {{r}}}{-{{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}}{{{2}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{R}^{2}}{-{{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{R}} {{r}}}} + {{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}{{{2}} {{{r}^{2}}}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& \frac{R}{{r}^{3}}& 0& 0\\ -{\frac{R}{{r}^{3}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{R}{{{2}} {{\left({{-{R}} + {r}}\right)}}}}& 0\\ 0& \frac{R}{{{2}} {{\left({{-{R}} + {r}}\right)}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{\left({{-{R}} + {r}}\right)}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{\left({{-{R}} + {r}}\right)}}}& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& \frac{{{R}} {{\left({{R}{-{r}}}\right)}}}{{{2}} {{{r}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{R}{{{2}} {{\left({{-{R}} + {r}}\right)}}}& 0\\ 0& -{\frac{R}{{{2}} {{\left({{-{R}} + {r}}\right)}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& {{R}} {{r}} {{{\sin\left( \theta\right)}^{2}}}\\ 0& 0& -{{{R}} {{r}} {{{\sin\left( \theta\right)}^{2}}}}& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& \frac{{{R}^{2}}{-{{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{R}} {{r}}}} + {{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}{{{2}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{-{{R}^{2}}} + {{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{R}} {{r}}}{-{{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}}{{{2}} {{{r}^{2}}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{\left({{-{R}} + {r}}\right)}}}\\ 0& 0& 0& 0\\ 0& -{\frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{\left({{-{R}} + {r}}\right)}}}}& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -{{{R}} {{r}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& {{R}} {{r}} {{{\sin\left( \theta\right)}^{2}}}& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$
Plebanski, $\sharp\sharp\flat\flat$:
${{{{{ P} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$