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:
${{{{ g} _a} _b} _{,c}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{r}} {\partial_ {{t}}\left( R\right)}& \frac{{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}{{r}^{2}}& {\frac{1}{r}} {\partial_ {{\theta}}\left( R\right)}& {\frac{1}{r}} {\partial_ {{\phi}}\left( R\right)}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ \frac{{{r}} {{\partial_ {{t}}\left( R\right)}}}{{\left({{R}{-{r}}}\right)}^{2}}& \frac{{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}{{\left({{R}{-{r}}}\right)}^{2}}& \frac{{{r}} {{\partial_ {{\theta}}\left( R\right)}}}{{\left({{R}{-{r}}}\right)}^{2}}& \frac{{{r}} {{\partial_ {{\phi}}\left( R\right)}}}{{\left({{R}{-{r}}}\right)}^{2}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}} {{{\sin\left( \theta\right)}^{2}}}& {{2}} {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$
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} \frac{\partial_ {{t}}\left( R\right)}{{{2}} {{r}}}& \frac{{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{{r}^{2}}}}& \frac{\partial_ {{\theta}}\left( R\right)}{{{2}} {{r}}}& \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{r}}}\\ \frac{{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{{r}^{2}}}}& -{\frac{{{r}} {{\partial_ {{t}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}}& 0& 0\\ \frac{\partial_ {{\theta}}\left( R\right)}{{{2}} {{r}}}& 0& 0& 0\\ \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{r}}}& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} \frac{{R}{-{{{r}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{{r}^{2}}}}& \frac{{{r}} {{\partial_ {{t}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0& 0\\ \frac{{{r}} {{\partial_ {{t}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{r}} {{\partial_ {{\theta}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{r}} {{\partial_ {{\phi}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ 0& \frac{{{r}} {{\partial_ {{\theta}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& -{r}& 0\\ 0& \frac{{{r}} {{\partial_ {{\phi}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0& -{{{r}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} -{\frac{\partial_ {{\theta}}\left( R\right)}{{{2}} {{r}}}}& 0& 0& 0\\ 0& -{\frac{{{r}} {{\partial_ {{\theta}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}}& 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} -{\frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{r}}}}& 0& 0& 0\\ 0& -{\frac{{{r}} {{\partial_ {{\phi}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}}& 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} \frac{\partial_ {{t}}\left( R\right)}{{{2}} {{\left({{R}{-{r}}}\right)}}}& \frac{{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{r}} {{\left({{-{r}} + {R}}\right)}}}& \frac{\partial_ {{\theta}}\left( R\right)}{{{2}} {{\left({{R}{-{r}}}\right)}}}& \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{\left({{R}{-{r}}}\right)}}}\\ \frac{{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{r}} {{\left({{-{r}} + {R}}\right)}}}& -{\frac{{{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}}}{{{2}} {{\left({{{R}^{3}}{-{{r}^{3}}} + {{{3}} {{R}} {{{r}^{2}}}}{-{{{3}} {{r}} {{{R}^{2}}}}}}\right)}}}}& 0& 0\\ \frac{\partial_ {{\theta}}\left( R\right)}{{{2}} {{\left({{R}{-{r}}}\right)}}}& 0& 0& 0\\ \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{\left({{R}{-{r}}}\right)}}}& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} \frac{{-{{R}^{2}}} + {{{R}} {{r}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{{r}^{3}}}}& \frac{\partial_ {{t}}\left( R\right)}{{{2}} {{\left({{-{R}} + {r}}\right)}}}& 0& 0\\ \frac{\partial_ {{t}}\left( R\right)}{{{2}} {{\left({{-{R}} + {r}}\right)}}}& \frac{{{R}^{2}}{-{{{R}} {{r}}}} + {{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\left({{-{R}} + {r}}\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{\left({{-{R}} + {r}}\right)}}}\\ 0& \frac{{{\left({{-{R}} + {r}}\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& {R}{-{r}}& 0\\ 0& \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{\left({{-{R}} + {r}}\right)}}}& 0& {R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}\end{array}\right] \\ \left[\begin{array}{cccc} -{\frac{\partial_ {{\theta}}\left( R\right)}{{{2}} {{{r}^{3}}}}}& 0& 0& 0\\ 0& -{\frac{\partial_ {{\theta}}\left( R\right)}{{{2}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}}& \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} -{\frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}}& 0& 0& 0\\ 0& -{\frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{\left({{{r}^{3}} + {{{r}} {{{R}^{2}}}}{-{{{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{R}} {{{r}^{2}}}}} + {{{2}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}}& 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} \frac{{-{{\partial_ {{t}}\left( R\right)}^{2}}} + {{{R}} {{\partial_{{{t}}{{t}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{t}}{{t}}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{\partial_ {{t}}\left( R\right)} + {{{R}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}}{-{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{{R}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}}{-{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{{R}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}}{-{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ \frac{{{{r}^{2}}} {{\left({{\partial_ {{t}}\left( R\right)}{-{{{r}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}} + {{{R}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}{-{{{\partial_ {{r}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& \frac{{{R}^{2}}{-{{{{r}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{{r}^{3}}} {{\partial_{{{r}}{{r}}}\left( R\right)}}}}{-{{{2}} {{R}} {{r}}}} + {{{2}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{R}} {{{r}^{2}}} {{\partial_{{{r}}{{r}}}\left( R\right)}}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& \frac{{{{r}^{2}}} {{\left({{\partial_ {{\theta}}\left( R\right)}{-{{{r}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}} + {{{R}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}{-{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& \frac{{{{r}^{2}}} {{\left({{\partial_ {{\phi}}\left( R\right)}{-{{{r}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}} + {{{R}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}{-{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}\\ \frac{{{{R}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}}{-{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{\partial_ {{\theta}}\left( R\right)} + {{{R}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}}{-{{{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{{\partial_ {{\theta}}\left( R\right)}^{2}}} + {{{R}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{{R}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}}{-{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ \frac{{{{R}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}}{-{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{\partial_ {{\phi}}\left( R\right)} + {{{R}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}}{-{{{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{{R}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}}{-{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{{\partial_ {{\phi}}\left( R\right)}^{2}}} + {{{R}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{{{r}^{2}}} {{\left({{\partial_ {{t}}\left( R\right)}{-{{{r}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}} + {{{R}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}{-{{{\partial_ {{r}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& \frac{{{R}^{2}}{-{{{{r}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{{r}^{3}}} {{\partial_{{{r}}{{r}}}\left( R\right)}}}}{-{{{2}} {{R}} {{r}}}} + {{{2}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{R}} {{{r}^{2}}} {{\partial_{{{r}}{{r}}}\left( R\right)}}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& \frac{{{{r}^{2}}} {{\left({{\partial_ {{\theta}}\left( R\right)}{-{{{r}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}} + {{{R}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}{-{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& \frac{{{{r}^{2}}} {{\left({{\partial_ {{\phi}}\left( R\right)}{-{{{r}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}} + {{{R}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}{-{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}\\ \frac{{{{{r}^{5}}} {{\partial_{{{t}}{{t}}}\left( R\right)}}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}} + {{{3}} {{{r}^{4}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{{R}^{3}}} {{{r}^{2}}} {{\partial_{{{t}}{{t}}}\left( R\right)}}}}{-{{{3}} {{R}} {{{r}^{4}}} {{\partial_{{{t}}{{t}}}\left( R\right)}}}} + {{{3}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_{{{t}}{{t}}}\left( R\right)}}}{-{{{6}} {{R}} {{{r}^{3}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}}}{{{2}} {{\left({{{R}^{6}} + {{r}^{6}}{-{{{20}} {{{R}^{3}}} {{{r}^{3}}}}}{-{{{6}} {{R}} {{{r}^{5}}}}}{-{{{6}} {{r}} {{{R}^{5}}}}} + {{{15}} {{{R}^{2}}} {{{r}^{4}}}} + {{{15}} {{{R}^{4}}} {{{r}^{2}}}}}\right)}}}& \frac{{-{{{{r}^{4}}} {{\partial_ {{t}}\left( R\right)}}}} + {{{{r}^{5}}} {{\partial_{{{r}}{{t}}}\left( R\right)}}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}}}{-{{{{R}^{3}}} {{{r}^{2}}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}}{-{{{2}} {{r}} {{{R}^{3}}} {{\partial_ {{t}}\left( R\right)}}}}{-{{{3}} {{R}} {{{r}^{4}}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}} + {{{3}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_{{{r}}{{t}}}\left( R\right)}}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}} + {{{3}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{-{{{6}} {{R}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{6}} + {{r}^{6}}{-{{{20}} {{{R}^{3}}} {{{r}^{3}}}}}{-{{{6}} {{R}} {{{r}^{5}}}}}{-{{{6}} {{r}} {{{R}^{5}}}}} + {{{15}} {{{R}^{2}}} {{{r}^{4}}}} + {{{15}} {{{R}^{4}}} {{{r}^{2}}}}}\right)}}}& \frac{{{{{r}^{5}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}{-{{{{R}^{3}}} {{{r}^{2}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}}{-{{{3}} {{R}} {{{r}^{4}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}} + {{{3}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}} + {{{3}} {{{r}^{4}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{-{{{6}} {{R}} {{{r}^{3}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{6}} + {{r}^{6}}{-{{{20}} {{{R}^{3}}} {{{r}^{3}}}}}{-{{{6}} {{R}} {{{r}^{5}}}}}{-{{{6}} {{r}} {{{R}^{5}}}}} + {{{15}} {{{R}^{2}}} {{{r}^{4}}}} + {{{15}} {{{R}^{4}}} {{{r}^{2}}}}}\right)}}}& \frac{{{{{r}^{5}}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}{-{{{{R}^{3}}} {{{r}^{2}}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}}{-{{{3}} {{R}} {{{r}^{4}}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}} + {{{3}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}} + {{{3}} {{{r}^{4}}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{-{{{6}} {{R}} {{{r}^{3}}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{6}} + {{r}^{6}}{-{{{20}} {{{R}^{3}}} {{{r}^{3}}}}}{-{{{6}} {{R}} {{{r}^{5}}}}}{-{{{6}} {{r}} {{{R}^{5}}}}} + {{{15}} {{{R}^{2}}} {{{r}^{4}}}} + {{{15}} {{{R}^{4}}} {{{r}^{2}}}}}\right)}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{{{R}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}}{-{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{\partial_ {{\theta}}\left( R\right)} + {{{R}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}}{-{{{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{{\partial_ {{\theta}}\left( R\right)}^{2}}} + {{{R}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{{R}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}}{-{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ 0& 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}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}}{-{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{\partial_ {{\phi}}\left( R\right)} + {{{R}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}}{-{{{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{{R}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}}{-{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{{\partial_ {{\phi}}\left( R\right)}^{2}}} + {{{R}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ 0& 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}} {{\partial_ {{t}}\left( R\right)}}}{-{{{{r}^{2}}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}}{-{{{2}} {{R}} {{\partial_ {{t}}\left( R\right)}}}} + {{{R}} {{r}} {{\partial_{{{r}}{{t}}}\left( R\right)}}} + {{{r}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}{{{2}} {{{r}^{3}}}}& \frac{{{{3}} {{{R}^{2}}} {{{r}^{2}}}} + {{{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{{r}^{5}}} {{\partial_{{{r}}{{r}}}\left( R\right)}}}}{-{{{2}} {{R}} {{{r}^{3}}}}} + {{{2}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}}} + {{{R}} {{{r}^{4}}} {{\partial_{{{r}}{{r}}}\left( R\right)}}}{-{{{4}} {{R}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{{r}^{6}}}}& \frac{{{{r}} {{\partial_ {{\theta}}\left( R\right)}}}{-{{{{r}^{2}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}}{-{{{2}} {{R}} {{\partial_ {{\theta}}\left( R\right)}}}} + {{{R}} {{r}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}} + {{{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{{r}^{3}}}}& \frac{{{{r}} {{\partial_ {{\phi}}\left( R\right)}}}{-{{{{r}^{2}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}}{-{{{2}} {{R}} {{\partial_ {{\phi}}\left( R\right)}}}} + {{{R}} {{r}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}} + {{{r}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{{r}^{3}}}}\\ \frac{{{\partial_ {{t}}\left( R\right)}^{2}}{-{{{R}} {{\partial_{{{t}}{{t}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{t}}{{t}}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{\partial_ {{t}}\left( R\right)}}{-{{{R}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{r}}{{t}}}\left( R\right)}}} + {{{\partial_ {{r}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{{{R}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}} + {{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{{{R}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}} + {{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{{\partial_ {{t}}\left( R\right)}^{2}}{-{{{R}} {{\partial_{{{t}}{{t}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{t}}{{t}}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{\partial_ {{t}}\left( R\right)}}{-{{{R}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{r}}{{t}}}\left( R\right)}}} + {{{\partial_ {{r}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{{{R}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}} + {{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{{{R}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}} + {{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ \frac{{-{{{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}}}}{-{{{{r}^{4}}} {{\partial_ {{t}}\left( R\right)}}}} + {{{{r}^{5}}} {{\partial_{{{r}}{{t}}}\left( R\right)}}} + {{{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{-{{{{R}^{3}}} {{{r}^{2}}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}} + {{{{r}^{4}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}} + {{{2}} {{R}} {{{r}^{3}}} {{\partial_ {{t}}\left( R\right)}}}{-{{{3}} {{R}} {{{r}^{4}}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}} + {{{3}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_{{{r}}{{t}}}\left( R\right)}}}{-{{{2}} {{R}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}& \frac{{-{{R}^{4}}}{-{{{5}} {{{R}^{2}}} {{{r}^{2}}}}} + {{{{R}^{2}}} {{{r}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{{r}^{5}}} {{\partial_{{{r}}{{r}}}\left( R\right)}}} + {{{2}} {{R}} {{{r}^{3}}}}{-{{{2}} {{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{2}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{4}} {{r}} {{{R}^{3}}}}{-{{{{R}^{3}}} {{{r}^{2}}} {{\partial_{{{r}}{{r}}}\left( R\right)}}}}{-{{{2}} {{R}} {{{r}^{3}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{3}} {{R}} {{{r}^{4}}} {{\partial_{{{r}}{{r}}}\left( R\right)}}}} + {{{3}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_{{{r}}{{r}}}\left( R\right)}}} + {{{4}} {{R}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}& \frac{{-{{{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}}}{-{{{{r}^{4}}} {{\partial_ {{\theta}}\left( R\right)}}}} + {{{{r}^{5}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}} + {{{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}{-{{{{R}^{3}}} {{{r}^{2}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}} + {{{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{2}} {{R}} {{{r}^{3}}} {{\partial_ {{\theta}}\left( R\right)}}}{-{{{3}} {{R}} {{{r}^{4}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}} + {{{3}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}{-{{{2}} {{R}} {{{r}^{3}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}& \frac{{-{{{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{\phi}}\left( R\right)}}}}{-{{{{r}^{4}}} {{\partial_ {{\phi}}\left( R\right)}}}} + {{{{r}^{5}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}} + {{{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}{-{{{{R}^{3}}} {{{r}^{2}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}} + {{{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}} + {{{2}} {{R}} {{{r}^{3}}} {{\partial_ {{\phi}}\left( R\right)}}}{-{{{3}} {{R}} {{{r}^{4}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}} + {{{3}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}{-{{{2}} {{R}} {{{r}^{3}}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}\\ \frac{{-{{{{R}^{3}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}} + {{{{r}^{3}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}} + {{{{R}^{2}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{-{{{3}} {{R}} {{{r}^{2}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}} + {{{3}} {{r}} {{{R}^{2}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}{-{{{2}} {{R}} {{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{4}} + {{r}^{4}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}}\right)}}}& \frac{{-{{{{R}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}}}{-{{{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}}}{-{{{{R}^{3}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}} + {{{{r}^{3}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}} + {{{{R}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{2}} {{R}} {{r}} {{\partial_ {{\theta}}\left( R\right)}}}{-{{{3}} {{R}} {{{r}^{2}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}} + {{{3}} {{r}} {{{R}^{2}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}{-{{{2}} {{R}} {{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{4}} + {{r}^{4}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}}\right)}}}& \frac{{{{{R}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{{R}^{3}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}} + {{{{r}^{3}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}{-{{{2}} {{R}} {{r}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}}{-{{{3}} {{R}} {{{r}^{2}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}} + {{{3}} {{r}} {{{R}^{2}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{4}} + {{r}^{4}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}}\right)}}}& \frac{{-{{{{R}^{3}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}} + {{{{r}^{3}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}} + {{{{R}^{2}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}} + {{{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{-{{{3}} {{R}} {{{r}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}} + {{{3}} {{r}} {{{R}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}{-{{{2}} {{R}} {{r}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{4}} + {{r}^{4}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}}\right)}}}\\ \frac{{-{{{R}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}} + {{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{\partial_ {{\phi}}\left( R\right)}}{-{{{R}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}} + {{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{{{R}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}} + {{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}^{2}}{-{{{R}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ \frac{{-{{{{R}^{3}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}} + {{{{r}^{3}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}} + {{{{R}^{2}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{-{{{3}} {{R}} {{{r}^{2}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}} + {{{3}} {{r}} {{{R}^{2}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}{-{{{2}} {{R}} {{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{4}} + {{r}^{4}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}}\right)}}}& \frac{{-{{{{R}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}}}{-{{{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}}}{-{{{{R}^{3}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}} + {{{{r}^{3}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}} + {{{{R}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{2}} {{R}} {{r}} {{\partial_ {{\theta}}\left( R\right)}}}{-{{{3}} {{R}} {{{r}^{2}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}} + {{{3}} {{r}} {{{R}^{2}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}{-{{{2}} {{R}} {{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{4}} + {{r}^{4}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}}\right)}}}& \frac{{{{{R}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{{R}^{3}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}} + {{{{r}^{3}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}{-{{{2}} {{R}} {{r}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}}{-{{{3}} {{R}} {{{r}^{2}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}} + {{{3}} {{r}} {{{R}^{2}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{4}} + {{r}^{4}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}}\right)}}}& \frac{{-{{{{R}^{3}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}} + {{{{r}^{3}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}} + {{{{R}^{2}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}} + {{{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{-{{{3}} {{R}} {{{r}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}} + {{{3}} {{r}} {{{R}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}{-{{{2}} {{R}} {{r}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}}{{{2}} {{\left({{{R}^{4}} + {{r}^{4}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}}\right)}}}\\ \partial_ {{t}}\left( R\right)& {-{1}} + {\partial_ {{r}}\left( R\right)}& \partial_ {{\theta}}\left( R\right)& \partial_ {{\phi}}\left( R\right)\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ \frac{{-{{{R}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}} + {{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{\partial_ {{\phi}}\left( R\right)}}{-{{{R}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}} + {{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{-{{{R}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}} + {{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}^{2}}{-{{{R}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}} + {{{r}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ 0& 0& 0& 0\\ {{{\sin\left( \theta\right)}^{2}}} {{\partial_ {{t}}\left( R\right)}}& {-{1}} + {\partial_ {{r}}\left( R\right)} + {{\cos\left( \theta\right)}^{2}}{-{{{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}& {{\left({{{{\partial_ {{\theta}}\left( R\right)}} {{\sin\left( \theta\right)}}} + {{{2}} {{R}} {{\cos\left( \theta\right)}}}{-{{{2}} {{r}} {{\cos\left( \theta\right)}}}}}\right)}} {{\sin\left( \theta\right)}}& {{{\sin\left( \theta\right)}^{2}}} {{\partial_ {{\phi}}\left( R\right)}}\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{\frac{\partial_{{{\theta}}{{t}}}\left( R\right)}{{{2}} {{{r}^{3}}}}}& \frac{{{{3}} {{\partial_ {{\theta}}\left( R\right)}}}{-{{{r}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}}}{{{2}} {{{r}^{4}}}}& -{\frac{\partial_{{{\theta}}{{\theta}}}\left( R\right)}{{{2}} {{{r}^{3}}}}}& -{\frac{\partial_{{{\phi}}{{\theta}}}\left( R\right)}{{{2}} {{{r}^{3}}}}}\\ 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\\ \frac{{{r}} {{\left({{-{{{{r}^{2}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}}{-{{{{R}^{2}}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}} + {{{2}} {{R}} {{r}} {{\partial_{{{\theta}}{{t}}}\left( R\right)}}}{-{{{2}} {{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}} + {{{2}} {{R}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}}\right)}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}& \frac{{{{{R}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}}{-{{{{r}^{3}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}} + {{{3}} {{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}}{-{{{r}} {{{R}^{2}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}} + {{{2}} {{R}} {{{r}^{2}}} {{\partial_{{{\theta}}{{r}}}\left( R\right)}}}{-{{{2}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}{-{{{4}} {{R}} {{r}} {{\partial_ {{\theta}}\left( R\right)}}}} + {{{2}} {{R}} {{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}& \frac{{-{{{2}} {{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}}{-{{{{r}^{3}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}}{-{{{r}} {{{R}^{2}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}} + {{{2}} {{R}} {{r}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{2}} {{R}} {{{r}^{2}}} {{\partial_{{{\theta}}{{\theta}}}\left( R\right)}}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}& \frac{{{r}} {{\left({{-{{{{r}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}}{-{{{{R}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}} + {{{2}} {{R}} {{r}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}{-{{{2}} {{r}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}} + {{{2}} {{R}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}\right)}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}\\ 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} -{\frac{\partial_{{{\phi}}{{t}}}\left( R\right)}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}}& \frac{{-{{{{r}^{3}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}} + {{{3}} {{{r}^{2}}} {{\partial_ {{\phi}}\left( R\right)}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\phi}}\left( R\right)}}}} + {{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}}{{{2}} {{{r}^{6}}} {{\left({{-{{{{\sin\left( \theta\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}}\right)}}}& \frac{{-{{{\sin\left( \theta\right)}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}} + {{{2}} {{\partial_ {{\phi}}\left( R\right)}} {{\cos\left( \theta\right)}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{3}}}}& -{\frac{\partial_{{{\phi}}{{\phi}}}\left( R\right)}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}}\\ 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\\ \frac{{-{{{{r}^{3}}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}}{-{{{r}} {{{R}^{2}}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}} + {{{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}} + {{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}} + {{{2}} {{R}} {{{r}^{2}}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}{-{{{2}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{t}}}\left( R\right)}}}}{-{{{2}} {{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}} + {{{2}} {{R}} {{r}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{-{{{2}} {{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}} + {{{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{4}}}} + {{{{r}^{6}}} {{{\cos\left( \theta\right)}^{4}}}}{-{{{2}} {{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{5}}}}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{4}}}}} + {{{8}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{4}}}}{-{{{12}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{4}}}}} + {{{8}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \frac{{{{{R}^{2}}} {{\partial_ {{\phi}}\left( R\right)}}}{-{{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\phi}}\left( R\right)}}}}{-{{{{r}^{3}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}} + {{{3}} {{{r}^{2}}} {{\partial_ {{\phi}}\left( R\right)}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\phi}}\left( R\right)}}}}{-{{{r}} {{{R}^{2}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}} + {{{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}} + {{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}} + {{{2}} {{R}} {{{r}^{2}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}{-{{{2}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{r}}}\left( R\right)}}}}{-{{{2}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}}{-{{{4}} {{R}} {{r}} {{\partial_ {{\phi}}\left( R\right)}}}} + {{{2}} {{R}} {{r}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{r}}\left( R\right)}}} + {{{4}} {{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\phi}}\left( R\right)}}}{-{{{2}} {{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}} + {{{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{4}}}} + {{{{r}^{6}}} {{{\cos\left( \theta\right)}^{4}}}}{-{{{2}} {{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{5}}}}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{4}}}}} + {{{8}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{4}}}}{-{{{12}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{4}}}}} + {{{8}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \frac{{-{{{{r}^{3}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}}{-{{{r}} {{{R}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}} + {{{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}} + {{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}} + {{{2}} {{R}} {{{r}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}{-{{{2}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{\theta}}}\left( R\right)}}}}{-{{{2}} {{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{2}} {{R}} {{r}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{2}} {{{r}^{3}}} {{\partial_ {{\phi}}\left( R\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}{-{{{2}} {{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}} + {{{2}} {{r}} {{{R}^{2}}} {{\partial_ {{\phi}}\left( R\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}{-{{{4}} {{R}} {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\partial_ {{\phi}}\left( R\right)}} {{\sin\left( \theta\right)}}}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}} + {{{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{4}}}} + {{{{r}^{6}}} {{{\cos\left( \theta\right)}^{4}}}}{-{{{2}} {{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{5}}}}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{4}}}}} + {{{8}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{4}}}}{-{{{12}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{4}}}}} + {{{8}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \frac{{-{{{2}} {{{r}^{2}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{{r}^{3}}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}} + {{{2}} {{{r}^{2}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{r}} {{{R}^{2}}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}} + {{{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}} + {{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}} + {{{2}} {{R}} {{r}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}{-{{{2}} {{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}} + {{{2}} {{R}} {{{r}^{2}}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}{-{{{2}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_{{{\phi}}{{\phi}}}\left( R\right)}}}}}{{{2}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}} + {{{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{4}}}} + {{{{r}^{6}}} {{{\cos\left( \theta\right)}^{4}}}}{-{{{2}} {{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{5}}}}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{4}}}}} + {{{8}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{4}}}}{-{{{12}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{4}}}}} + {{{8}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}\\ 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}}} {{{r}^{2}}}} + {{{{r}^{2}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{R}^{4}} + {{{{R}^{2}}} {{{r}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{{R}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{R}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{{r}^{4}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}}{-{{{2}} {{r}} {{{R}^{3}}}}}{-{{{R}} {{r}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{R}} {{r}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{4}} {{{R}^{2}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{{R}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{2}} {{R}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{2}} {{R}} {{{r}^{3}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{2}} {{R}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{2}} {{r}} {{{R}^{3}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{2}} {{r}} {{{R}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{{R}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{2}} {{R}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{2}} {{r}} {{{R}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{{{4}} {{{r}^{4}}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{R}} {{r}}}} + {{{2}} {{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& 0& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ 0& \frac{{-{{R}^{6}}} + {{{4}} {{{R}^{3}}} {{{r}^{3}}}}{-{{{{R}^{2}}} {{{r}^{4}}}}}{-{{{{R}^{2}}} {{{r}^{4}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}}{-{{{{R}^{4}}} {{{r}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{{r}^{6}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{{r}^{6}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{4}} {{r}} {{{R}^{5}}}} + {{{4}} {{{R}^{3}}} {{{r}^{3}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{6}} {{{R}^{2}}} {{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{6}} {{{R}^{4}}} {{{r}^{2}}}}} + {{{2}} {{R}} {{{r}^{5}}} {{\partial_ {{r}}\left( R\right)}}} + {{{2}} {{R}} {{{r}^{5}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}} + {{{2}} {{r}} {{{R}^{5}}} {{\partial_ {{r}}\left( R\right)}}} + {{{12}} {{{R}^{3}}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}} + {{{4}} {{R}} {{{r}^{5}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{8}} {{{R}^{2}}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{8}} {{{R}^{4}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}}{{{4}} {{\left({{-{{r}^{8}}}{-{{{{R}^{6}}} {{{r}^{2}}}}}{-{{{15}} {{{R}^{4}}} {{{r}^{4}}}}} + {{{6}} {{R}} {{{r}^{7}}}} + {{{6}} {{{R}^{5}}} {{{r}^{3}}}} + {{{20}} {{{R}^{3}}} {{{r}^{5}}}}{-{{{15}} {{{R}^{2}}} {{{r}^{6}}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ \frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{\partial_ {{\theta}}\left( R\right)}^{2}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{\partial_ {{\phi}}\left( R\right)}^{2}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{-{{{r}} {{{R}^{8}}}}}{-{{{{R}^{2}}} {{{r}^{7}}}}} + {{{{R}^{2}}} {{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{R}^{5}}} {{{r}^{2}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{{R}^{5}}} {{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{{r}^{7}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{{r}^{7}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}}{-{{{{r}^{9}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{{r}^{9}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{{r}^{9}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{9}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}} + {{{r}} {{{R}^{8}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{R}^{4}}} {{{r}^{5}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{{R}^{4}}} {{{r}^{5}}} {{{\partial_ {{t}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{R}^{5}}} {{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{R}^{6}}} {{{r}^{3}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{{R}^{6}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{6}} {{{R}^{2}}} {{{r}^{7}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{6}} {{{R}^{3}}} {{{r}^{6}}}} + {{{6}} {{{R}^{7}}} {{{r}^{2}}}}{-{{{6}} {{{R}^{7}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{10}} {{{R}^{2}}} {{{r}^{5}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{10}} {{{R}^{2}}} {{{r}^{5}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{2}} {{R}} {{{r}^{8}}} {{\partial_ {{r}}\left( R\right)}}} + {{{2}} {{{R}^{7}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{2}} {{{R}^{7}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{15}} {{{R}^{2}}} {{{r}^{7}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{15}} {{{R}^{4}}} {{{r}^{5}}}}}{-{{{15}} {{{R}^{6}}} {{{r}^{3}}}}}{-{{{5}} {{R}} {{{r}^{6}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{5}} {{R}} {{{r}^{6}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{5}} {{R}} {{{r}^{6}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{5}} {{{R}^{4}}} {{{r}^{3}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{5}} {{{R}^{4}}} {{{r}^{3}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{5}} {{{R}^{4}}} {{{r}^{3}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{20}} {{{R}^{5}}} {{{r}^{4}}}} + {{{4}} {{R}} {{{r}^{8}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{4}} {{R}} {{{r}^{8}}} {{{\partial_ {{t}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{{R}^{3}}} {{{r}^{6}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{4}} {{{R}^{3}}} {{{r}^{6}}} {{{\partial_ {{t}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{6}} {{R}} {{{r}^{8}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{6}} {{R}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{7}}} {{{\partial_ {{t}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{6}} {{{R}^{3}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{6}} {{{R}^{5}}} {{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{6}} {{{R}^{5}}} {{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{10}} {{{R}^{2}}} {{{r}^{5}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{10}} {{{R}^{3}}} {{{r}^{4}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{10}} {{{R}^{3}}} {{{r}^{4}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{10}} {{{R}^{3}}} {{{r}^{4}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{2}} {{R}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{15}} {{{R}^{2}}} {{{r}^{7}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{15}} {{{R}^{4}}} {{{r}^{5}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{15}} {{{R}^{4}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{15}} {{{R}^{4}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{15}} {{{R}^{6}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{12}} {{{R}^{2}}} {{{r}^{7}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{12}} {{{R}^{2}}} {{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{12}} {{{R}^{6}}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{20}} {{{R}^{3}}} {{{r}^{6}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{20}} {{{R}^{3}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{20}} {{{R}^{5}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{40}} {{{R}^{4}}} {{{r}^{5}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{30}} {{{R}^{3}}} {{{r}^{6}}} {{\partial_ {{r}}\left( R\right)}}} + {{{30}} {{{R}^{5}}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}}} + {{{12}} {{{R}^{6}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{40}} {{{R}^{4}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{30}} {{{R}^{3}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{30}} {{{R}^{5}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}}{{{4}} {{{r}^{3}}} {{\left({{{r}^{8}} + {{R}^{8}}{-{{{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{8}} {{R}} {{{r}^{7}}}}}{-{{{8}} {{r}} {{{R}^{7}}}}}{-{{{{R}^{8}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{28}} {{{R}^{2}}} {{{r}^{6}}}}{-{{{28}} {{{R}^{2}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{28}} {{{R}^{6}}} {{{r}^{2}}}}{-{{{56}} {{{R}^{3}}} {{{r}^{5}}}}}{-{{{56}} {{{R}^{5}}} {{{r}^{3}}}}} + {{{8}} {{R}} {{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{r}} {{{R}^{7}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{70}} {{{R}^{4}}} {{{r}^{4}}}}{-{{{28}} {{{R}^{6}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{56}} {{{R}^{3}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{56}} {{{R}^{5}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{70}} {{{R}^{4}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{{{2}} {{{r}^{2}}}} + {{{3}} {{{R}^{2}}}} + {{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{5}} {{R}} {{r}}}}{-{{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{\left({{-{{r}^{4}}}{-{{{3}} {{{R}^{2}}} {{{r}^{2}}}}} + {{{r}} {{{R}^{3}}}} + {{{3}} {{R}} {{{r}^{3}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{{{2}} {{r}}}{-{{{3}} {{R}}}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{r}} {{\left({{-{{r}^{2}}}{-{{R}^{2}}} + {{{2}} {{R}} {{r}}}}\right)}}}\\ \frac{{-{{R}^{6}}} + {{{4}} {{{R}^{3}}} {{{r}^{3}}}}{-{{{{R}^{2}}} {{{r}^{4}}}}}{-{{{{R}^{2}}} {{{r}^{4}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}}{-{{{{R}^{4}}} {{{r}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{{r}^{6}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{{r}^{6}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{4}} {{r}} {{{R}^{5}}}} + {{{4}} {{{R}^{3}}} {{{r}^{3}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{6}} {{{R}^{2}}} {{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{6}} {{{R}^{4}}} {{{r}^{2}}}}} + {{{2}} {{R}} {{{r}^{5}}} {{\partial_ {{r}}\left( R\right)}}} + {{{2}} {{R}} {{{r}^{5}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}} + {{{2}} {{r}} {{{R}^{5}}} {{\partial_ {{r}}\left( R\right)}}} + {{{12}} {{{R}^{3}}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}} + {{{4}} {{R}} {{{r}^{5}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{8}} {{{R}^{2}}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{8}} {{{R}^{4}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}}{{{4}} {{\left({{-{{r}^{8}}}{-{{{{R}^{6}}} {{{r}^{2}}}}}{-{{{15}} {{{R}^{4}}} {{{r}^{4}}}}} + {{{6}} {{R}} {{{r}^{7}}}} + {{{6}} {{{R}^{5}}} {{{r}^{3}}}} + {{{20}} {{{R}^{3}}} {{{r}^{5}}}}{-{{{15}} {{{R}^{2}}} {{{r}^{6}}}}}}\right)}}}& 0& \frac{{{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{r}} + {R}}\right)}}}{{{4}} {{\left({{{R}^{5}}{-{{r}^{5}}} + {{{5}} {{R}} {{{r}^{4}}}}{-{{{5}} {{r}} {{{R}^{4}}}}}{-{{{10}} {{{R}^{2}}} {{{r}^{3}}}}} + {{{10}} {{{R}^{3}}} {{{r}^{2}}}}}\right)}}}& -{\frac{{{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{-{{R}^{4}}}{-{{r}^{4}}}{-{{{6}} {{{R}^{2}}} {{{r}^{2}}}}} + {{{4}} {{R}} {{{r}^{3}}}} + {{{4}} {{r}} {{{R}^{3}}}}}\right)}}}}\\ \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& -{\frac{{{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{R}^{4}} + {{r}^{4}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}}\right)}}}}& 0& 0\\ \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& -{\frac{{{{r}^{2}}} {{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{R}^{4}} + {{r}^{4}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}}\right)}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{{{2}} {{{r}^{2}}}} + {{{3}} {{{R}^{2}}}} + {{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{5}} {{R}} {{r}}}}{-{{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{\left({{-{{r}^{4}}}{-{{{3}} {{{R}^{2}}} {{{r}^{2}}}}} + {{{r}} {{{R}^{3}}}} + {{{3}} {{R}} {{{r}^{3}}}}}\right)}}}& \frac{{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{r}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\sin\left( \theta\right)}} {{\left({{R}{-{r}}}\right)}}}\\ \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{r}} + {R}}\right)}}}{{{4}} {{\left({{{R}^{5}}{-{{r}^{5}}} + {{{5}} {{R}} {{{r}^{4}}}}{-{{{5}} {{r}} {{{R}^{4}}}}}{-{{{10}} {{{R}^{2}}} {{{r}^{3}}}}} + {{{10}} {{{R}^{3}}} {{{r}^{2}}}}}\right)}}}& \frac{{{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}}}{{{2}} {{\left({{-{{R}^{2}}}{-{{r}^{2}}} + {{{2}} {{R}} {{r}}}}\right)}}}& 0\\ \frac{{\partial_ {{\theta}}\left( R\right)}^{2}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0& 0& 0\\ \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{{{2}} {{r}}}{-{{{3}} {{R}}}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{r}} {{\left({{-{{r}^{2}}}{-{{R}^{2}}} + {{{2}} {{R}} {{r}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\sin\left( \theta\right)}} {{\left({{R}{-{r}}}\right)}}}& \frac{{-{{R}^{2}}} + {{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{R}} {{r}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}{-{{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{r}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} {{\partial_ {{\theta}}\left( R\right)}}}}}{{{2}} {{r}} {{\left({{-{r}} + {R}}\right)}}}\\ \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{-{R}} + {{{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& -{\frac{{{{r}^{2}}} {{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{-{{R}^{4}}}{-{{r}^{4}}}{-{{{6}} {{{R}^{2}}} {{{r}^{2}}}}} + {{{4}} {{R}} {{{r}^{3}}}} + {{{4}} {{r}} {{{R}^{3}}}}}\right)}}}}& 0& \frac{{{\left({{{r}^{3}}{-{{{R}} {{{r}^{2}}}}} + {{{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{2}} {{\left({{{R}^{3}}{-{{r}^{3}}} + {{{3}} {{R}} {{{r}^{2}}}}{-{{{3}} {{r}} {{{R}^{2}}}}}}\right)}}}\\ \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0& 0& 0\\ \frac{{\partial_ {{\phi}}\left( R\right)}^{2}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{R}^{5}}{-{{{{R}^{2}}} {{{r}^{3}}}}} + {{{{R}^{3}}} {{{r}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{{r}^{5}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{{r}^{5}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}}{-{{{3}} {{r}} {{{R}^{4}}}}}{-{{{3}} {{{R}^{2}}} {{{r}^{3}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{3}} {{{R}^{3}}} {{{r}^{2}}}} + {{{R}} {{{r}^{4}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}} + {{{2}} {{R}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{2}} {{r}} {{{R}^{4}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{3}} {{R}} {{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{6}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{6}} {{{R}^{3}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{{{4}} {{{r}^{4}}} {{\left({{-{{r}^{3}}} + {{R}^{3}} + {{{3}} {{R}} {{{r}^{2}}}}{-{{{3}} {{r}} {{{R}^{2}}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{{R}^{2}}} + {{{R}} {{r}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{-{{R}^{2}}} + {{{R}} {{r}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R}}\right)}}}\\ \frac{{{R}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{2}}} {{{r}^{4}}}} + {{{{R}^{2}}} {{{r}^{4}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{R}^{4}}} {{{r}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{{r}^{4}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{{r}^{4}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{{r}^{4}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{R}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{r}^{6}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{{r}^{6}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{{r}^{6}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{6}}} {{{\partial_ {{t}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{r}} {{{R}^{3}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{r}} {{{R}^{3}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{r}} {{{R}^{3}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{r}} {{{R}^{5}}}}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{4}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}}{-{{{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{6}} {{{R}^{4}}} {{{r}^{2}}}}{-{{{6}} {{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{R}} {{{r}^{5}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{2}} {{R}} {{{r}^{5}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{2}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{2}} {{r}} {{{R}^{5}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{3}} {{R}} {{{r}^{3}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{3}} {{R}} {{{r}^{3}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{3}} {{R}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{3}} {{{R}^{2}}} {{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{12}} {{{R}^{3}}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{4}} {{R}} {{{r}^{5}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{4}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{4}} {{r}} {{{R}^{5}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{{R}^{3}}} {{{r}^{3}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{R}^{2}}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{8}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{8}} {{{R}^{4}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{8}} {{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{6}} {{{R}^{2}}} {{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{2}} {{r}} {{{R}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{12}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{{{4}} {{\left({{-{{{{R}^{4}}} {{{r}^{4}}}}}{-{{r}^{8}}} + {{{{R}^{4}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{R}} {{{r}^{7}}}} + {{{4}} {{{R}^{3}}} {{{r}^{5}}}}{-{{{6}} {{{R}^{2}}} {{{r}^{6}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{4}} {{R}} {{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{{R}^{3}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}& 0& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{-{{R}^{2}}}{-{{r}^{2}}} + {{{2}} {{R}} {{r}}}}\right)}}}& \frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{-{{R}^{2}}}{-{{r}^{2}}} + {{{2}} {{R}} {{r}}}}\right)}}}\\ \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{{R}^{3}}} + {{{2}} {{{r}^{3}}}}{-{{{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{5}} {{R}} {{{r}^{2}}}}}{-{{{r}} {{{R}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{4}} {{r}} {{{R}^{2}}}} + {{{2}} {{R}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{{r}^{3}}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0& 0\\ \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{{R}^{2}}{-{{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{2}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{3}} {{R}} {{r}}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}} + {{{3}} {{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{{r}^{3}}} {{\left({{r}{-{R}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{R}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{{R}^{5}}{-{{{{R}^{2}}} {{{r}^{3}}}}} + {{{{R}^{3}}} {{{r}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{{r}^{5}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{{r}^{5}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}}{-{{{3}} {{r}} {{{R}^{4}}}}}{-{{{3}} {{{R}^{2}}} {{{r}^{3}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{3}} {{{R}^{3}}} {{{r}^{2}}}} + {{{R}} {{{r}^{4}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}} + {{{2}} {{R}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{2}} {{r}} {{{R}^{4}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{3}} {{R}} {{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{6}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{6}} {{{R}^{3}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{{{4}} {{{r}^{4}}} {{\left({{-{{r}^{3}}} + {{R}^{3}} + {{{3}} {{R}} {{{r}^{2}}}}{-{{{3}} {{r}} {{{R}^{2}}}}}}\right)}}}& 0& \frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ 0& \frac{{-{{{r}} {{{R}^{10}}}}}{-{{{{R}^{2}}} {{{r}^{9}}}}} + {{{{R}^{2}}} {{{r}^{9}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{R}^{7}}} {{{r}^{2}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{{R}^{7}}} {{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{{r}^{9}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{{r}^{9}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{{r}^{9}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{11}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{{r}^{11}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{{r}^{11}}} {{{\partial_ {{t}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{11}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{r}} {{{R}^{10}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{8}} {{{R}^{3}}} {{{r}^{8}}}} + {{{8}} {{{R}^{9}}} {{{r}^{2}}}}{-{{{8}} {{{R}^{9}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{R}^{6}}} {{{r}^{5}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{{R}^{6}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}} + {{{{R}^{7}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{{R}^{8}}} {{{r}^{3}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{{R}^{8}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{2}} {{R}} {{{r}^{10}}} {{\partial_ {{r}}\left( R\right)}}} + {{{2}} {{{R}^{9}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{2}} {{{R}^{9}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{15}} {{{R}^{2}}} {{{r}^{9}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{21}} {{{R}^{2}}} {{{r}^{7}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{21}} {{{R}^{2}}} {{{r}^{7}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{7}} {{R}} {{{r}^{8}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{7}} {{R}} {{{r}^{8}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{7}} {{R}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{7}} {{{R}^{6}}} {{{r}^{3}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{7}} {{{R}^{6}}} {{{r}^{3}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{7}} {{{R}^{6}}} {{{r}^{3}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{28}} {{{R}^{2}}} {{{r}^{9}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}}{-{{{28}} {{{R}^{4}}} {{{r}^{7}}}}}{-{{{28}} {{{R}^{8}}} {{{r}^{3}}}}} + {{{56}} {{{R}^{5}}} {{{r}^{6}}}} + {{{56}} {{{R}^{7}}} {{{r}^{4}}}} + {{{8}} {{R}} {{{r}^{10}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{8}} {{R}} {{{r}^{10}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{8}} {{{R}^{3}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{8}} {{{R}^{7}}} {{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{8}} {{{R}^{7}}} {{{r}^{4}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{16}} {{{R}^{2}}} {{{r}^{9}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{16}} {{{R}^{2}}} {{{r}^{9}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{16}} {{{R}^{8}}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{6}} {{R}} {{{r}^{10}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{6}} {{R}} {{{r}^{10}}} {{{\partial_ {{t}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{6}} {{{R}^{5}}} {{{r}^{6}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{6}} {{{R}^{5}}} {{{r}^{6}}} {{{\partial_ {{t}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{70}} {{{R}^{6}}} {{{r}^{5}}}}}{-{{{2}} {{R}} {{{r}^{10}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{15}} {{{R}^{2}}} {{{r}^{9}}} {{{\partial_ {{t}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{15}} {{{R}^{4}}} {{{r}^{7}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{15}} {{{R}^{4}}} {{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{35}} {{{R}^{3}}} {{{r}^{6}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{35}} {{{R}^{3}}} {{{r}^{6}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{35}} {{{R}^{3}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{35}} {{{R}^{4}}} {{{r}^{5}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{35}} {{{R}^{4}}} {{{r}^{5}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{35}} {{{R}^{4}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}}{-{{{21}} {{{R}^{2}}} {{{r}^{7}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{21}} {{{R}^{5}}} {{{r}^{4}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}}{-{{{21}} {{{R}^{5}}} {{{r}^{4}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{21}} {{{R}^{5}}} {{{r}^{4}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{20}} {{{R}^{3}}} {{{r}^{8}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}{-{{{20}} {{{R}^{3}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{t}}\left( R\right)}^{2}}}}} + {{{28}} {{{R}^{2}}} {{{r}^{9}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{28}} {{{R}^{4}}} {{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{28}} {{{R}^{6}}} {{{r}^{5}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{28}} {{{R}^{6}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}} + {{{28}} {{{R}^{8}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{56}} {{{R}^{3}}} {{{r}^{8}}} {{\partial_ {{r}}\left( R\right)}}} + {{{56}} {{{R}^{3}}} {{{r}^{8}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{56}} {{{R}^{3}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{56}} {{{R}^{5}}} {{{r}^{6}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}{-{{{56}} {{{R}^{5}}} {{{r}^{6}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{56}} {{{R}^{5}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{56}} {{{R}^{7}}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{56}} {{{R}^{7}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{112}} {{{R}^{4}}} {{{r}^{7}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{112}} {{{R}^{6}}} {{{r}^{5}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{16}} {{{R}^{8}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{70}} {{{R}^{4}}} {{{r}^{7}}} {{{\partial_ {{r}}\left( R\right)}^{2}}}}} + {{{70}} {{{R}^{4}}} {{{r}^{7}}} {{{\partial_ {{r}}\left( R\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{70}} {{{R}^{6}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{140}} {{{R}^{5}}} {{{r}^{6}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{56}} {{{R}^{3}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{56}} {{{R}^{7}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{112}} {{{R}^{4}}} {{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{112}} {{{R}^{6}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{140}} {{{R}^{5}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}}{{{4}} {{{r}^{3}}} {{\left({{-{{r}^{10}}}{-{{R}^{10}}} + {{{{r}^{10}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{R}^{10}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{10}} {{R}} {{{r}^{9}}}} + {{{10}} {{r}} {{{R}^{9}}}}{-{{{45}} {{{R}^{2}}} {{{r}^{8}}}}} + {{{45}} {{{R}^{2}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{45}} {{{R}^{8}}} {{{r}^{2}}}}}{-{{{10}} {{R}} {{{r}^{9}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{10}} {{r}} {{{R}^{9}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{252}} {{{R}^{5}}} {{{r}^{5}}}} + {{{120}} {{{R}^{3}}} {{{r}^{7}}}} + {{{120}} {{{R}^{7}}} {{{r}^{3}}}} + {{{45}} {{{R}^{8}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{210}} {{{R}^{4}}} {{{r}^{6}}}}}{-{{{210}} {{{R}^{6}}} {{{r}^{4}}}}}{-{{{252}} {{{R}^{5}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{120}} {{{R}^{3}}} {{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{120}} {{{R}^{7}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{210}} {{{R}^{4}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{210}} {{{R}^{6}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{{{2}} {{{r}^{3}}}}{-{{{3}} {{{R}^{3}}}}} + {{{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{7}} {{R}} {{{r}^{2}}}}} + {{{r}} {{{R}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{8}} {{r}} {{{R}^{2}}}}{-{{{2}} {{R}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{4}} + {{R}^{4}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{{{2}} {{{r}^{2}}}} + {{{3}} {{{R}^{2}}}} + {{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{5}} {{R}} {{r}}}}{-{{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{\left({{{r}^{4}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{r}} {{{R}^{3}}}}}{-{{{3}} {{R}} {{{r}^{3}}}}}}\right)}}}\\ \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{{{2}} {{{r}^{3}}}}{-{{{3}} {{{R}^{3}}}}} + {{{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{7}} {{R}} {{{r}^{2}}}}} + {{{r}} {{{R}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{8}} {{r}} {{{R}^{2}}}}{-{{{2}} {{R}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{4}} + {{R}^{4}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}}\right)}}}& \frac{{{{r}} {{{R}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{4}} {{{R}^{5}}}}{-{{{4}} {{{r}^{5}}}}} + {{{{r}^{3}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{2}} {{R}} {{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{20}} {{R}} {{{r}^{4}}}}{-{{{20}} {{r}} {{{R}^{4}}}}}{-{{{40}} {{{R}^{2}}} {{{r}^{3}}}}} + {{{40}} {{{R}^{3}}} {{{r}^{2}}}}}{{{4}} {{r}} {{\left({{{r}^{4}} + {{R}^{4}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{{{2}} {{{r}^{5}}}} + {{{{r}^{5}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{2}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{3}} {{r}} {{{R}^{4}}}}{-{{{11}} {{{R}^{3}}} {{{r}^{2}}}}} + {{{11}} {{{R}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{9}} {{R}} {{{r}^{4}}}}}{-{{{{R}^{3}}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{{R}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{3}} {{R}} {{{r}^{4}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{3}} {{r}} {{{R}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{3}} {{{R}^{2}}} {{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{3}} {{{R}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{15}} {{{R}^{2}}} {{{r}^{3}}}}{-{{{15}} {{{R}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{9}} {{R}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{R}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{\left({{{r}^{7}}{-{{{{R}^{5}}} {{{r}^{2}}}}} + {{{{R}^{5}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{5}} {{R}} {{{r}^{6}}}}} + {{{5}} {{{R}^{4}}} {{{r}^{3}}}} + {{{10}} {{{R}^{2}}} {{{r}^{5}}}}{-{{{10}} {{{R}^{2}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{10}} {{{R}^{3}}} {{{r}^{4}}}}} + {{{5}} {{R}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{5}} {{{R}^{4}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{10}} {{{R}^{3}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{{{{r}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{4}} {{{R}^{3}}}}{-{{{4}} {{{r}^{3}}}}}{-{{{4}} {{{R}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{12}} {{R}} {{{r}^{2}}}}{-{{{12}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{12}} {{r}} {{{R}^{2}}}}} + {{{12}} {{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{{{4}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{{R}^{2}}} + {{{R}} {{r}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R}}\right)}}}& \frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& -{{\frac{1}{2}} {\partial_ {{t}}\left( R\right)}}& 0\\ \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{-{{R}^{2}}}{-{{r}^{2}}} + {{{2}} {{R}} {{r}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{{{2}} {{{r}^{3}}}}{-{{{3}} {{{R}^{3}}}}} + {{{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{7}} {{R}} {{{r}^{2}}}}} + {{{r}} {{{R}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{8}} {{r}} {{{R}^{2}}}}{-{{{2}} {{R}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{r}} {{\left({{{r}^{4}} + {{R}^{4}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}}\right)}}}& \frac{{R}{-{{{r}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{r}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\sin\left( \theta\right)}} {{\left({{-{R}} + {r}}\right)}}}\\ 0& \frac{{{{r}} {{{R}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{4}} {{{R}^{5}}}}{-{{{4}} {{{r}^{5}}}}} + {{{{r}^{3}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}{-{{{2}} {{R}} {{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{20}} {{R}} {{{r}^{4}}}}{-{{{20}} {{r}} {{{R}^{4}}}}}{-{{{40}} {{{R}^{2}}} {{{r}^{3}}}}} + {{{40}} {{{R}^{3}}} {{{r}^{2}}}}}{{{4}} {{r}} {{\left({{{r}^{4}} + {{R}^{4}}{-{{{4}} {{R}} {{{r}^{3}}}}}{-{{{4}} {{r}} {{{R}^{3}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{2}}}}}\right)}}}& \frac{{{\left({{-{{R}^{2}}}{-{{r}^{2}}} + {{{2}} {{R}} {{r}}}}\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0\\ 0& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& -{{\frac{1}{2}} {\partial_ {{\phi}}\left( R\right)}}& {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}} {{\left({{-{r}} + {R}}\right)}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{-{{R}^{2}}} + {{{R}} {{r}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R}}\right)}}}& \frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0& -{{\frac{1}{2}} {{{{\sin\left( \theta\right)}^{2}}} {{\partial_ {{t}}\left( R\right)}}}}\\ \frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{-{{R}^{2}}}{-{{r}^{2}}} + {{{2}} {{R}} {{r}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{{{2}} {{{r}^{2}}}} + {{{3}} {{{R}^{2}}}} + {{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{5}} {{R}} {{r}}}}{-{{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{\left({{{r}^{4}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{r}} {{{R}^{3}}}}}{-{{{3}} {{R}} {{{r}^{3}}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\sin\left( \theta\right)}} {{\left({{-{R}} + {r}}\right)}}}& \frac{{{R}^{3}} + {{{R}} {{{r}^{2}}}}{-{{{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{R}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{2}} {{r}} {{{R}^{2}}}}} + {{{2}} {{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{r}} {{{R}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{2}} {{R}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{2}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}{-{{{{r}^{2}}} {{\cos\left( \theta\right)}} {{\partial_ {{\theta}}\left( R\right)}} {{\sin\left( \theta\right)}}}} + {{{R}} {{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}{{{2}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ 0& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0& \frac{{-{{{{R}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}}}{-{{{{r}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}}} + {{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}} + {{{2}} {{R}} {{r}} {{\partial_ {{\theta}}\left( R\right)}}}{-{{{2}} {{{R}^{3}}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}}} + {{{2}} {{{r}^{3}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}{-{{{2}} {{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{\theta}}\left( R\right)}}}}{-{{{6}} {{R}} {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}} + {{{6}} {{r}} {{{R}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ 0& \frac{{{{r}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}} + {{{4}} {{{R}^{3}}}}{-{{{4}} {{{r}^{3}}}}}{-{{{4}} {{{R}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{12}} {{R}} {{{r}^{2}}}}{-{{{12}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{12}} {{r}} {{{R}^{2}}}}} + {{{12}} {{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{{{4}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}} {{\left({{-{r}} + {R}}\right)}}& -{{\frac{1}{2}} {{{{\sin\left( \theta\right)}^{2}}} {{\partial_ {{\phi}}\left( R\right)}}}}\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{\frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R}}\right)}}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{R}{-{{{r}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{{r}^{4}}} {{\left({{-{r}} + {R}}\right)}}}& -{\frac{{\partial_ {{\theta}}\left( R\right)}^{2}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R}}\right)}}}}& -{\frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R}}\right)}}}}\\ \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{{R}^{2}}}{-{{{2}} {{{r}^{2}}}}} + {{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}} + {{{3}} {{R}} {{r}}}{-{{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{{r}^{4}}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& -{\frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{r}^{4}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{r}} {{{R}^{3}}}}}{-{{{3}} {{R}} {{{r}^{3}}}}}}\right)}}}}& 0& 0\\ \frac{{-{{R}^{2}}} + {{{R}} {{r}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}{{{2}} {{{r}^{4}}}}& \frac{\partial_ {{t}}\left( R\right)}{{{2}} {{r}} {{\left({{r}{-{R}}}\right)}}}& 0& 0\\ \frac{{{\cos\left( \theta\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{2}} {{{r}^{3}}} {{\sin\left( \theta\right)}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{R}{-{{{r}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{{r}^{4}}} {{\left({{-{r}} + {R}}\right)}}}& \frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{-{{r}^{4}}}{-{{{3}} {{{R}^{2}}} {{{r}^{2}}}}} + {{{r}} {{{R}^{3}}}} + {{{3}} {{R}} {{{r}^{3}}}}}\right)}}}& 0& 0\\ -{\frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{r}^{4}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{r}} {{{R}^{3}}}}}{-{{{3}} {{R}} {{{r}^{3}}}}}}\right)}}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\left({{-{{{2}} {{{r}^{2}}}}}{-{{{3}} {{{R}^{2}}}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{5}} {{R}} {{r}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}& \frac{{-{{{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{4}} {{{R}^{4}}}} + {{{4}} {{{r}^{4}}}} + {{{R}} {{r}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{24}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{16}} {{R}} {{{r}^{3}}}}}{-{{{16}} {{r}} {{{R}^{3}}}}}}{{{4}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}& -{\frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{r}^{4}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{r}} {{{R}^{3}}}}}{-{{{3}} {{R}} {{{r}^{3}}}}}}\right)}}}}\\ \frac{\partial_ {{t}}\left( R\right)}{{{2}} {{r}} {{\left({{r}{-{R}}}\right)}}}& \frac{{{R}^{2}}{-{{{R}} {{r}}}} + {{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& \frac{{{\left({{-{R}} + {r}}\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{2}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{r}} {{\left({{r}{-{R}}}\right)}}}\\ 0& \frac{{{\sin\left( \theta\right)}} {{\partial_ {{\phi}}\left( R\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\left({{{r}^{3}} + {{{r}} {{{R}^{2}}}}{-{{{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{R}} {{{r}^{2}}}}} + {{{2}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& 0& -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{\frac{{\partial_ {{\theta}}\left( R\right)}^{2}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R}}\right)}}}}& 0& 0& 0\\ 0& \frac{{-{{{{r}^{2}}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}}} + {{{4}} {{{R}^{4}}}} + {{{4}} {{{r}^{4}}}} + {{{R}} {{r}} {{{\partial_ {{\theta}}\left( R\right)}^{2}}}} + {{{24}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{16}} {{R}} {{{r}^{3}}}}}{-{{{16}} {{r}} {{{R}^{3}}}}}}{{{4}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}}\right)}}}& \frac{{{\left({{-{R}} + {r}}\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{2}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& 0\\ 0& \frac{{{\left({{-{R}} + {r}}\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{2}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}& {\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} -{\frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R}}\right)}}}}& 0& 0& 0\\ 0& -{\frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{\theta}}\left( R\right)}}}{{{4}} {{\left({{{r}^{4}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{r}} {{{R}^{3}}}}}{-{{{3}} {{R}} {{{r}^{3}}}}}}\right)}}}}& 0& \frac{{{\left({{-{{{R}} {{\partial_ {{\theta}}\left( R\right)}} {{\sin\left( \theta\right)}}}} + {{{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\sin\left( \theta\right)}}}{-{{{2}} {{{R}^{2}}} {{\cos\left( \theta\right)}}}}{-{{{2}} {{{r}^{2}}} {{\cos\left( \theta\right)}}}} + {{{4}} {{R}} {{r}} {{\cos\left( \theta\right)}}}}\right)}} {{\sin\left( \theta\right)}}}{{{2}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}\\ 0& \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{r}} {{\left({{r}{-{R}}}\right)}}}& 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} -{\frac{{{\partial_ {{t}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{R}{-{{{r}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{{r}^{4}}} {{\left({{-{r}} + {R} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}& -{\frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}}& -{\frac{{\partial_ {{\phi}}\left( R\right)}^{2}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}}\\ \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{-{{{2}} {{{r}^{3}}}}}{-{{{r}} {{{R}^{2}}}}} + {{{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{3}}} {{\partial_ {{r}}\left( R\right)}}} + {{{2}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{R}} {{{r}^{2}}}}{-{{{3}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{R}} {{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}{-{{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{{r}^{5}}} {{\left({{{{{R}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{{R}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{2}} {{R}} {{r}} {{{\sin\left( \theta\right)}^{2}}}}}}\right)}}}& -{\frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{r}^{4}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{r}} {{{R}^{3}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{R}} {{{r}^{3}}}}}{-{{{3}} {{{R}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{R}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{R}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}}& 0& 0\\ -{\frac{{{\cos\left( \theta\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{3}}}}}& 0& 0& 0\\ \frac{{-{{{{R}^{2}}} {{\sin\left( \theta\right)}}}} + {{{R}} {{r}} {{\sin\left( \theta\right)}}}{-{{{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\cos\left( \theta\right)}}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\sin\left( \theta\right)}}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}} {{\sin\left( \theta\right)}}}}{{{2}} {{{r}^{4}}} {{\sin\left( \theta\right)}}}& \frac{\partial_ {{t}}\left( R\right)}{{{2}} {{r}} {{\left({{r}{-{R}}}\right)}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{R}{-{{{r}} {{\partial_ {{r}}\left( R\right)}}}}}\right)}}}{{{4}} {{{r}^{4}}} {{\left({{-{r}} + {R} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{-{{r}^{4}}}{-{{{3}} {{{R}^{2}}} {{{r}^{2}}}}} + {{{r}} {{{R}^{3}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{R}} {{{r}^{3}}}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{r}} {{{R}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{R}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}& 0& 0\\ -{\frac{{{\partial_ {{\phi}}\left( R\right)}} {{\partial_ {{t}}\left( R\right)}}}{{{4}} {{\left({{{r}^{4}} + {{{3}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{r}} {{{R}^{3}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{R}} {{{r}^{3}}}}}{-{{{3}} {{{R}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{R}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{R}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}}& \frac{{{\partial_ {{\phi}}\left( R\right)}} {{\left({{-{{{2}} {{{r}^{2}}}}}{-{{{3}} {{{R}^{2}}}}}{-{{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}}}} + {{{5}} {{R}} {{r}}} + {{{R}} {{r}} {{\partial_ {{r}}\left( R\right)}}}}\right)}}}{{{4}} {{\left({{{r}^{6}}{-{{{4}} {{{R}^{3}}} {{{r}^{3}}}}} + {{{{R}^{4}}} {{{r}^{2}}}}{-{{{{R}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{5}}}}} + {{{4}} {{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{6}} {{{R}^{2}}} {{{r}^{4}}}}{-{{{6}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}} {{\left({{-{r}} + {R}}\right)}}}{{{4}} {{\left({{{r}^{5}} + {{{r}} {{{R}^{4}}}}{-{{{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{4}}}}}{-{{{4}} {{{R}^{3}}} {{{r}^{2}}}}} + {{{4}} {{{R}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{r}} {{{R}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{3}}}}{-{{{6}} {{{R}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{R}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \frac{{-{{{{r}^{2}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}} + {{{4}} {{{r}^{4}}}} + {{{12}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}{-{{{4}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{12}} {{R}} {{{r}^{3}}}}}{-{{{12}} {{{R}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{r}} {{{R}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{12}} {{R}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}{{{4}} {{\left({{-{{{{R}^{3}}} {{{r}^{3}}}}} + {{r}^{6}} + {{{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{R}} {{{r}^{5}}}}} + {{{3}} {{{R}^{2}}} {{{r}^{4}}}}{-{{{3}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{3}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}\\ 0& -{\frac{{{\cos\left( \theta\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{2}} {{\sin\left( \theta\right)}} {{\left({{{r}^{3}} + {{{r}} {{{R}^{2}}}}{-{{{r}} {{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{R}} {{{r}^{2}}}}} + {{{2}} {{R}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}}& 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}\\ \frac{\partial_ {{t}}\left( R\right)}{{{2}} {{r}} {{\left({{r}{-{R}}}\right)}}}& \frac{{{{{R}^{2}}} {{\sin\left( \theta\right)}}}{-{{{R}} {{r}} {{\sin\left( \theta\right)}}}}{-{{{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\cos\left( \theta\right)}}}} + {{{{r}^{2}}} {{\partial_ {{r}}\left( R\right)}} {{\sin\left( \theta\right)}}}{-{{{R}} {{r}} {{\sin\left( \theta\right)}} {{\partial_ {{r}}\left( R\right)}}}}}{{{2}} {{\sin\left( \theta\right)}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& \frac{{{{2}} {{{R}^{2}}} {{\cos\left( \theta\right)}}} + {{{2}} {{{r}^{2}}} {{\cos\left( \theta\right)}}}{-{{{R}} {{\partial_ {{\theta}}\left( R\right)}} {{\sin\left( \theta\right)}}}} + {{{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\sin\left( \theta\right)}}}{-{{{4}} {{R}} {{r}} {{\cos\left( \theta\right)}}}}}{{{2}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}} {{\sin\left( \theta\right)}}}& \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{r}} {{\left({{r}{-{R}}}\right)}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{\frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}}& 0& 0& 0\\ 0& \frac{{{\partial_ {{\theta}}\left( R\right)}} {{\partial_ {{\phi}}\left( R\right)}} {{\left({{-{r}} + {R}}\right)}}}{{{4}} {{\left({{{r}^{5}} + {{{r}} {{{R}^{4}}}}{-{{{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{4}} {{R}} {{{r}^{4}}}}}{-{{{4}} {{{R}^{3}}} {{{r}^{2}}}}} + {{{4}} {{{R}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{r}} {{{R}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{6}} {{{R}^{2}}} {{{r}^{3}}}}{-{{{6}} {{{R}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{R}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{\left({{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{R}} {{r}}}} + {{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \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{{{{2}} {{{R}^{2}}} {{\cos\left( \theta\right)}}} + {{{2}} {{{r}^{2}}} {{\cos\left( \theta\right)}}}{-{{{R}} {{\partial_ {{\theta}}\left( R\right)}} {{\sin\left( \theta\right)}}}} + {{{r}} {{\partial_ {{\theta}}\left( R\right)}} {{\sin\left( \theta\right)}}}{-{{{4}} {{R}} {{r}} {{\cos\left( \theta\right)}}}}}{{{2}} {{r}} {{\left({{{r}^{2}} + {{R}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}} {{\sin\left( \theta\right)}}}& {\frac{1}{r}}{\left({{R}{-{r}}}\right)}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} -{\frac{{\partial_ {{\phi}}\left( R\right)}^{2}}{{{4}} {{{r}^{3}}} {{\left({{-{r}} + {R} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}}& 0& 0& 0\\ 0& \frac{{-{{{{r}^{2}}} {{{\partial_ {{\phi}}\left( R\right)}^{2}}}}} + {{{4}} {{{r}^{4}}}} + {{{12}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{4}} {{r}} {{{R}^{3}}}}}{-{{{4}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{12}} {{R}} {{{r}^{3}}}}}{-{{{12}} {{{R}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{r}} {{{R}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{12}} {{R}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}{{{4}} {{\left({{-{{{{R}^{3}}} {{{r}^{3}}}}} + {{r}^{6}} + {{{{R}^{3}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{R}} {{{r}^{5}}}}} + {{{3}} {{{R}^{2}}} {{{r}^{4}}}}{-{{{3}} {{{R}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{3}} {{R}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{r}} {{\left({{r}{-{R}}}\right)}}}\\ 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& \frac{{\cos\left( \theta\right)}^{2}}{{\sin\left( \theta\right)}^{2}}& 0\\ 0& \frac{\partial_ {{\phi}}\left( R\right)}{{{2}} {{r}} {{\left({{r}{-{R}}}\right)}}}& 0& {\frac{1}{r}}{\left({{R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}\end{array}\right]}\end{array}\right]}}$