Schwarzschild, anholonomic
chart coordinates: $x^\tilde{\mu} = \{t, r, \theta, \phi\}$
chart coordinate basis: $e_\tilde{\mu} = \{e_{\tilde{t}}, e_{\tilde{r}}, e_{\tilde{\theta}}, e_{\tilde{\phi}}\}$
embedding coordinates: $u^I = \{t, x, y, z\}$
embedding basis $e_I = \{e_{t}, e_{x}, e_{y}, e_{z}\}$
flat metric:
${{{ \eta} _I} _J} = {\overset{I\downarrow J\rightarrow}{\left[\begin{array}{cccc} -1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
transform from basis to coordinate:
${{{ e} _{\tilde{a}}} ^a} = {\overset{A\downarrow a\rightarrow}{\left[\begin{array}{cccc} f& 0& 0& 0\\ 0& \frac{1}{f}& 0& 0\\ 0& 0& r& 0\\ 0& 0& 0& {{r}} {{\sin\left( \theta\right)}}\end{array}\right]}}$
transform from coorinate to basis:
${{{ e} ^{\tilde{a}}} _a} = {\overset{A\downarrow a\rightarrow}{\left[\begin{array}{cccc} \frac{1}{f}& 0& 0& 0\\ 0& f& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& \frac{1}{{{r}} {{\sin\left( \theta\right)}}}\end{array}\right]}}$
tensor index associated with coordinate $t$ has operator $e_{t}(\zeta) = $${\frac{1}{f}} {\frac{\partial \zeta}{\partial t}}$
tensor index associated with coordinate $r$ has operator $e_{r}(\zeta) = $${{f}} {{\frac{\partial \zeta}{\partial r}}}$
tensor index associated with coordinate $\theta$ has operator $e_{\theta}(\zeta) = $${\frac{1}{r}} {\frac{\partial \zeta}{\partial \theta}}$
tensor index associated with coordinate $\phi$ has operator $e_{\phi}(\zeta) = $$\frac{\frac{\partial \zeta}{\partial \phi}}{{{r}} {{\sin\left( \theta\right)}}}$
anholonomic coordinates: $x^\hat{\mu} = \{\hat{t}, \hat{r}, \hat{\theta}, \hat{\phi}\}$
basis in embedded coordinates / transform from embedded to basis:
${{{ e} _u} ^I} = {\overset{u\downarrow I\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${{{ e} ^u} _I} = {\overset{u\downarrow I\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^v} _I}}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^u} _J}}} = {\overset{I\downarrow J\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
basis determinant: ${det(e)} = {1}$
${{{{ c} _a} _b} ^c} = {\overset{c\downarrow[{a\downarrow b\rightarrow}]}{\left[\begin{matrix} \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& \frac{\partial f}{\partial r}& 0& 0\\ -{\frac{\partial f}{\partial r}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} {f}}& 0\\ 0& {\frac{1}{r}} {f}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {f}}\\ 0& 0& 0& -{\frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}}\\ 0& {\frac{1}{r}} {f}& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& 0\end{array}\right]}\end{matrix}\right]}}$
coordinate metric ${{{ \tilde{g}} _{\tilde{u}}} _{\tilde{v}}} = {\overset{A\downarrow B\rightarrow}{\left[\begin{array}{cccc} -{{f}^{2}}& 0& 0& 0\\ 0& \frac{1}{{f}^{2}}& 0& 0\\ 0& 0& {r}^{2}& 0\\ 0& 0& 0& {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}\end{array}\right]}}$
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
metric determinant: ${det(g)} = {-{1}}$
${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{{{ g} _a} _b} _{,c}} + {{{{ g} _a} _c} _{,b}}{-{{{{ g} _b} _c} _{,a}}} + {{{{ c} _a} _b} _c} + {{{{ c} _a} _c} _b}{-{{{{ c} _c} _b} _a}}}\right)}}}$
commutation coefficients:
${{{{ c} _a} _b} ^c} = {\overset{c\downarrow[{a\downarrow b\rightarrow}]}{\left[\begin{matrix} \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& \frac{\partial f}{\partial r}& 0& 0\\ -{\frac{\partial f}{\partial r}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} {f}}& 0\\ 0& {\frac{1}{r}} {f}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {f}}\\ 0& 0& 0& -{\frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}}\\ 0& {\frac{1}{r}} {f}& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& 0\end{array}\right]}\end{matrix}\right]}}$
metric:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
metric inverse:
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
metric derivative:
${{{{ g} _a} _b} _{,c}} = {\overset{c\downarrow[{a\downarrow b\rightarrow}]}{\left[\begin{matrix} \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
1st kind Christoffel:
${{{{ \Gamma} _a} _b} _c} = {\overset{b\downarrow[{a\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{\partial f}{\partial r}}& 0& 0\\ \frac{\partial f}{\partial r}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} {f}}& 0\\ 0& {\frac{1}{r}} {f}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {f}}\\ 0& 0& 0& -{\frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}}\\ 0& {\frac{1}{r}} {f}& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& 0\end{array}\right]}\end{matrix}\right]}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma} ^a} _b} _c} = {\overset{b\downarrow[{a\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& \frac{\partial f}{\partial r}& 0& 0\\ \frac{\partial f}{\partial r}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} {f}}& 0\\ 0& {\frac{1}{r}} {f}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {f}}\\ 0& 0& 0& -{\frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}}\\ 0& {\frac{1}{r}} {f}& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& 0\end{array}\right]}\end{matrix}\right]}}$
connection coefficients derivative:
${{{{{ \Gamma} ^a} _b} _c} _{,d}} = {\overset{b\downarrow d\rightarrow[{a\downarrow c\rightarrow}]}{\left[\begin{array}{cccc} \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& {{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}& 0& 0\\ {{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{f}} {{\left({{f}{-{{{r}} {{\frac{\partial f}{\partial r}}}}}}\right)}}}{{r}^{2}}& 0\\ 0& \frac{{{f}} {{\left({{-{f}} + {{{r}} {{\frac{\partial f}{\partial r}}}}}\right)}}}{{r}^{2}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{f}} {{\left({{f}{-{{{r}} {{\frac{\partial f}{\partial r}}}}}}\right)}}}{{r}^{2}}\\ 0& 0& 0& \frac{{{f}} {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}\\ 0& \frac{{{f}} {{\left({{-{f}} + {{{r}} {{\frac{\partial f}{\partial r}}}}}\right)}}}{{r}^{2}}& -{\frac{{{f}} {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}}& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{{\cos\left( \theta\right)}^{2}} + {{\sin\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& -{\frac{{{\cos\left( \theta\right)}^{2}} + {{\sin\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}}& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
connection coefficients squared:
${{{{{{ \Gamma} ^a} _e} _c}} {{{{{ \Gamma} ^e} _b} _d}}} = {\overset{b\downarrow d\rightarrow[{a\downarrow c\rightarrow}]}{\left[\begin{array}{cccc} \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} {\frac{\partial f}{\partial r}}^{2}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0\\ 0& 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ {\frac{\partial f}{\partial r}}^{2}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{{f}^{2}}{{r}^{2}}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{{f}} {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{f}^{2}}{{r}^{2}}}\\ 0& 0& 0& -{\frac{{{f}} {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}}\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& -{\frac{{f}^{2}}{{r}^{2}}}& 0\\ 0& 0& 0& -{\frac{{f}^{2}}{{r}^{2}}}\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{f}} {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}}\\ 0& 0& 0& -{\frac{{\cos\left( \theta\right)}^{2}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}}\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0& 0\\ 0& 0& \frac{{{f}} {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}& 0\\ 0& 0& -{\frac{{f}^{2}}{{r}^{2}}}& 0\\ 0& 0& 0& -{\frac{{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}} + {{\cos\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}}\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\left({{{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}& 0& 0\\ {{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\left({{{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}& 0& 0\\ {{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0\\ 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}\\ 0& 0& 0& 0\\ 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0\\ 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& \frac{{-{1}} + {{f}^{2}}}{{r}^{2}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}\\ 0& 0& 0& 0\\ 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}} + {{f}^{2}}}{{r}^{2}}\\ 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\left({{{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}& 0& 0\\ {{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}& 0& 0\\ -{\left({{{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0\\ 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}\\ 0& 0& 0& 0\\ 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0\\ 0& 0& 0& 0\\ -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0\\ 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& \frac{{-{1}} + {{f}^{2}}}{{r}^{2}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}\\ 0& 0& 0& 0\\ 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}} + {{f}^{2}}}{{r}^{2}}\\ 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Ricci curvature, $\sharp\flat$:
${{{ R} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{r}}{\left({{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{2}} {{f}} {{\frac{\partial f}{\partial r}}}} + {{{f}} {{r}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}}& 0& 0& 0\\ 0& -{{\frac{1}{r}}{\left({{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{2}} {{f}} {{\frac{\partial f}{\partial r}}}} + {{{f}} {{r}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}}& 0& 0\\ 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}{-{{{2}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}{-{{{2}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]}}$
Gaussian curvature:
$\frac{{{2}} {{\left({{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}{-{{{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}}{-{{{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{-{{{4}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}}\right)}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{r}}{\left({{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{2}} {{f}} {{\frac{\partial f}{\partial r}}}} + {{{f}} {{r}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}}& 0& 0& 0\\ 0& -{{\frac{1}{r}}{\left({{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{2}} {{f}} {{\frac{\partial f}{\partial r}}}} + {{{f}} {{r}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}}& 0& 0\\ 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}{-{{{2}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}{-{{{2}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{{\frac{1}{r}}{\left({{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{2}} {{f}} {{\frac{\partial f}{\partial r}}}} + {{{f}} {{r}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}}& 0& 0& 0\\ 0& -{{\frac{1}{r}}{\left({{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{2}} {{f}} {{\frac{\partial f}{\partial r}}}} + {{{f}} {{r}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}}& 0& 0\\ 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}{-{{{2}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}{-{{{2}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]}}$
Schouten, $\sharp\flat$:
${{{ P} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{\frac{{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{2}} {{f}} {{\frac{\partial f}{\partial r}}}} + {{{f}} {{r}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{r}}}}& 0& 0& 0\\ 0& -{\frac{{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{2}} {{f}} {{\frac{\partial f}{\partial r}}}} + {{{f}} {{r}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{r}}}}& 0& 0\\ 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}{-{{{2}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}{-{{{2}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]}}$
Weyl, $\sharp\sharp\flat\flat$:
${{{{{ C} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\left({{{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}& 0& 0\\ {{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}& 0& 0\\ -{\left({{{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0\\ 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}\\ 0& 0& 0& 0\\ 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0\\ 0& 0& 0& 0\\ -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0\\ 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& \frac{{-{1}} + {{f}^{2}}}{{r}^{2}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}\\ 0& 0& 0& 0\\ 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}} + {{f}^{2}}}{{r}^{2}}\\ 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C} _a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}& 0& 0\\ -{\left({{{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0\\ 0& 0& 0& 0\\ -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\left({{{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}\right)}& 0& 0\\ {{\frac{\partial f}{\partial r}}^{2}} + {{{f}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0\\ 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}\\ 0& 0& 0& 0\\ 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0\\ 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& \frac{{-{1}} + {{f}^{2}}}{{r}^{2}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}\\ 0& 0& 0& 0\\ 0& -{{\frac{1}{r}} {{{f}} {{\frac{\partial f}{\partial r}}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}} + {{f}^{2}}}{{r}^{2}}\\ 0& 0& \frac{{-{{{{f}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{\sin\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Plebanski, $\sharp\sharp\flat\flat$:
${{{{{ P} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{{4}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{4}}}} + {{{{f}^{2}}} {{{r}^{2}}} {{{\frac{\partial^ 2 f}{\partial r^ 2}}^{2}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{4}} {{f}} {{r}} {{{\frac{\partial f}{\partial r}}^{3}}}} + {{{4}} {{r}} {{{f}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{2}}}}& 0& 0\\ -{\frac{{{{4}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{4}}}} + {{{{f}^{2}}} {{{r}^{2}}} {{{\frac{\partial^ 2 f}{\partial r^ 2}}^{2}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{4}} {{f}} {{r}} {{{\frac{\partial f}{\partial r}}^{3}}}} + {{{4}} {{r}} {{{f}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{2}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{-{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}{-{{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}} + {{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}} + {{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}} + {{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}}{-{{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}} + {{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}{-{{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}{-{{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}{-{{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{-{{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{-{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}{-{{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}} + {{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}} + {{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}} + {{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}}{-{{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}} + {{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}{-{{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}{-{{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}{-{{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{-{{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{{{{4}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{4}}}} + {{{{f}^{2}}} {{{r}^{2}}} {{{\frac{\partial^ 2 f}{\partial r^ 2}}^{2}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{4}} {{f}} {{r}} {{{\frac{\partial f}{\partial r}}^{3}}}} + {{{4}} {{r}} {{{f}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{2}}}}}& 0& 0\\ \frac{{{{4}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}} + {{{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{4}}}} + {{{{f}^{2}}} {{{r}^{2}}} {{{\frac{\partial^ 2 f}{\partial r^ 2}}^{2}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{4}} {{f}} {{r}} {{{\frac{\partial f}{\partial r}}^{3}}}} + {{{4}} {{r}} {{{f}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{-{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}{-{{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}} + {{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}} + {{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}} + {{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& \frac{{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}}{-{{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}} + {{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}{-{{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}{-{{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}{-{{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{-{{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}{-{{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}} + {{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}} + {{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}} + {{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}}{-{{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}} + {{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}{-{{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}{-{{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}{-{{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{-{{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}} + {{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}{-{{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}{-{{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}{-{{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{-{{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{-{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}{-{{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}} + {{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}} + {{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}} + {{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}} + {{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}{-{{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}{-{{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}{-{{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{-{{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& \frac{{-{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}{-{{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}} + {{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}} + {{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}} + {{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{{{{f}^{4}}} {{{\sin\left( \theta\right)}^{4}}}} + {{\sin\left( \theta\right)}^{4}}{-{{{2}} {{{f}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}} + {{{4}} {{{f}^{2}}} {{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}{-{{{4}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{4}}} {{\frac{\partial f}{\partial r}}}}} + {{{4}} {{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{4}}} {{\frac{\partial f}{\partial r}}}}}{{{2}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{4}}}}\\ 0& 0& \frac{{-{{{{f}^{4}}} {{{\sin\left( \theta\right)}^{4}}}}}{-{{\sin\left( \theta\right)}^{4}}} + {{{2}} {{{f}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}{-{{{4}} {{{f}^{2}}} {{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}} + {{{4}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{4}}} {{\frac{\partial f}{\partial r}}}}{-{{{4}} {{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{4}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{4}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}} + {{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}{-{{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}{-{{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}{-{{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{-{{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{-{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}{-{{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}} + {{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}} + {{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}} + {{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{{r}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}} + {{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}{-{{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}}{-{{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}{-{{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{-{{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{-{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}}}} + {{{2}} {{f}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial f}{\partial r}}^{3}}}}{-{{{2}} {{f}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}}} + {{{2}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}}} + {{{5}} {{r}} {{{f}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{f}} {{r}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}} + {{{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}} + {{{2}} {{{f}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial f}{\partial r}}} {{\frac{\partial^ 2 f}{\partial r^ 2}}}}}{{{2}} {{{r}^{3}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{{f}^{4}}} {{{\sin\left( \theta\right)}^{4}}}}}{-{{\sin\left( \theta\right)}^{4}}} + {{{2}} {{{f}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}{-{{{4}} {{{f}^{2}}} {{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}} + {{{4}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{4}}} {{\frac{\partial f}{\partial r}}}}{-{{{4}} {{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{4}}} {{\frac{\partial f}{\partial r}}}}}}{{{2}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{4}}}}\\ 0& 0& \frac{{{{{f}^{4}}} {{{\sin\left( \theta\right)}^{4}}}} + {{\sin\left( \theta\right)}^{4}}{-{{{2}} {{{f}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}} + {{{4}} {{{f}^{2}}} {{{r}^{2}}} {{{\frac{\partial f}{\partial r}}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}{-{{{4}} {{f}} {{r}} {{{\sin\left( \theta\right)}^{4}}} {{\frac{\partial f}{\partial r}}}}} + {{{4}} {{r}} {{{f}^{3}}} {{{\sin\left( \theta\right)}^{4}}} {{\frac{\partial f}{\partial r}}}}}{{{2}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{4}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
divergence: ${{{{ A} ^i} _{,i}} + {{{{{{ \Gamma} ^i} _i} _j}} {{{ A} ^j}}}} = {{{{\frac{\partial {A^{\hat{\phi}}}}{\partial \phi}}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}} + {{{{A^{\hat{\theta}}}}} \cdot {{\cos\left( \theta\right)}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}} + {{{\frac{\partial {A^{\hat{\theta}}}}{\partial \theta}}} {{\frac{1}{r}}}} + {{{\frac{\partial {A^{\hat{t}}}}{\partial t}}} {{\frac{1}{f}}}} + {{{2}} {{{A^{\hat{r}}}}} \cdot {{f}} {{\frac{1}{r}}}} + {{{f}} {{\frac{\partial {A^{\hat{r}}}}{\partial r}}}} + {{{{A^{\hat{r}}}}} \cdot {{\frac{\partial f}{\partial r}}}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{\hat{t}} \\ \ddot{\hat{r}} \\ \ddot{\hat{\theta}} \\ \ddot{\hat{\phi}}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} {{-1}} {{\dot{\hat{r}}}} \cdot {{\dot{\hat{t}}}} \cdot {{\frac{\partial f}{\partial r}}} \\ {{{f}} {{{\dot{\hat{\phi}}}^{2}}} {{\frac{1}{r}}}} + {{{f}} {{{\dot{\hat{\theta}}}^{2}}} {{\frac{1}{r}}}} + {{{-1}} {{{\dot{\hat{t}}}^{2}}} {{\frac{\partial f}{\partial r}}}} \\ {{{{\dot{\hat{\phi}}}^{2}}} {{\cos\left( \theta\right)}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}} + {{{-1}} {{\dot{\hat{\theta}}}} \cdot {{\dot{\hat{r}}}} \cdot {{f}} {{\frac{1}{r}}}} \\ {{{-1}} {{\dot{\hat{\phi}}}} \cdot {{\dot{\hat{\theta}}}} \cdot {{\cos\left( \theta\right)}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}} + {{{-1}} {{\dot{\hat{\phi}}}} \cdot {{\dot{\hat{r}}}} \cdot {{f}} {{\sin\left( \theta\right)}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}}\end{matrix}\right]}}$
parallel propagators:
${{[\Gamma_t]}} = {\left[\begin{array}{cccc} 0& {{f}} {{\frac{\partial f}{\partial r}}}& 0& 0\\ {{f}} {{\frac{\partial f}{\partial r}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
$\int\limits_{{{t_L}}}^{{{t_R}}}{{\left[\begin{array}{cccc} 0& {{f}} {{\frac{\partial f}{\partial r}}}& 0& 0\\ {{f}} {{\frac{\partial f}{\partial r}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d t$
= $\left[\begin{array}{cccc} 0& {{f}} {{\frac{\partial f}{\partial r}}} {{\left({{-{{t_L}}} + {{t_R}}}\right)}}& 0& 0\\ {{f}} {{\frac{\partial f}{\partial r}}} {{\left({{-{{t_L}}} + {{t_R}}}\right)}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
${ P} _t$ = $\exp\left( -{\int\limits_{{{t_L}}}^{{{t_R}}}{{\left[\begin{array}{cccc} 0& {{f}} {{\frac{\partial f}{\partial r}}}& 0& 0\\ {{f}} {{\frac{\partial f}{\partial r}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d t}\right)$
= $\left[\begin{array}{cccc} {\frac{1}{2}} {{{\exp\left({{{f}} {{\frac{\partial f}{\partial r}}} {{\left({{{t_L}}{-{{t_R}}}}\right)}}}\right)}} {{\left({{1} + {\exp\left({{{2}} {{f}} {{\left({{-{{t_L}}} + {{t_R}}}\right)}} {{\frac{\partial f}{\partial r}}}}\right)}}\right)}}}& {\frac{1}{2}} {{{\exp\left({{{f}} {{\frac{\partial f}{\partial r}}} {{\left({{{t_L}}{-{{t_R}}}}\right)}}}\right)}} {{\left({{1}{-{\exp\left({{{2}} {{f}} {{\left({{-{{t_L}}} + {{t_R}}}\right)}} {{\frac{\partial f}{\partial r}}}}\right)}}}\right)}}}& 0& 0\\ {\frac{1}{2}} {{{\exp\left({{{f}} {{\frac{\partial f}{\partial r}}} {{\left({{{t_L}}{-{{t_R}}}}\right)}}}\right)}} {{\left({{1}{-{\exp\left({{{2}} {{f}} {{\left({{-{{t_L}}} + {{t_R}}}\right)}} {{\frac{\partial f}{\partial r}}}}\right)}}}\right)}}}& {\frac{1}{2}} {{{\exp\left({{{f}} {{\frac{\partial f}{\partial r}}} {{\left({{{t_L}}{-{{t_R}}}}\right)}}}\right)}} {{\left({{1} + {\exp\left({{{2}} {{f}} {{\left({{-{{t_L}}} + {{t_R}}}\right)}} {{\frac{\partial f}{\partial r}}}}\right)}}\right)}}}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
${{ P} _t}^{-1}$ = $\exp\left({\int\limits_{{{t_L}}}^{{{t_R}}}{{\left[\begin{array}{cccc} 0& {{f}} {{\frac{\partial f}{\partial r}}}& 0& 0\\ {{f}} {{\frac{\partial f}{\partial r}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d t}\right)$
= $\left[\begin{array}{cccc} \frac{{1} + {\exp\left({{{2}} {{f}} {{\left({{{t_L}}{-{{t_R}}}}\right)}} {{\frac{\partial f}{\partial r}}}}\right)}}{{{2}} {{\exp\left({{{f}} {{\frac{\partial f}{\partial r}}} {{\left({{{t_L}}{-{{t_R}}}}\right)}}}\right)}}}& \frac{{1}{-{\exp\left({{{2}} {{f}} {{\left({{{t_L}}{-{{t_R}}}}\right)}} {{\frac{\partial f}{\partial r}}}}\right)}}}{{{2}} {{\exp\left({{{f}} {{\frac{\partial f}{\partial r}}} {{\left({{{t_L}}{-{{t_R}}}}\right)}}}\right)}}}& 0& 0\\ \frac{{1}{-{\exp\left({{{2}} {{f}} {{\left({{{t_L}}{-{{t_R}}}}\right)}} {{\frac{\partial f}{\partial r}}}}\right)}}}{{{2}} {{\exp\left({{{f}} {{\frac{\partial f}{\partial r}}} {{\left({{{t_L}}{-{{t_R}}}}\right)}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{f}} {{\left({{{t_L}}{-{{t_R}}}}\right)}} {{\frac{\partial f}{\partial r}}}}\right)}}{{{2}} {{\exp\left({{{f}} {{\frac{\partial f}{\partial r}}} {{\left({{{t_L}}{-{{t_R}}}}\right)}}}\right)}}}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
${{[\Gamma_r]}} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
$\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d r$
= $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
${ P} _r$ = $\exp\left( -{\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d r}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
${{ P} _r}^{-1}$ = $\exp\left({\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d r}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
${{[\Gamma_\theta]}} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{f}& 0\\ 0& f& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
$\int\limits_{{{\theta_L}}}^{{{\theta_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{f}& 0\\ 0& f& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d \theta$
= $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& {{f}} {{\left({{{\theta_L}}{-{{\theta_R}}}}\right)}}& 0\\ 0& {{f}} {{\left({{-{{\theta_L}}} + {{\theta_R}}}\right)}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
${ P} _{\theta}$ = $\exp\left( -{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{f}& 0\\ 0& f& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d \theta}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{1} + {\exp\left({{{2}} {{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({{{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}& \frac{{{\theta_L}}{-{{\theta_R}}}{-{{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}} + {{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}{{{2}} {{\exp\left({{{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ 0& \frac{{{\exp\left({{{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{-{1}} + {\frac{1}{\exp\left({{{2}} {{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}\right)}}}{{{2}} {{\left({{{\theta_L}}{-{{\theta_R}}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({{{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]$
${{ P} _{\theta}}^{-1}$ = $\exp\left({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{f}& 0\\ 0& f& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d \theta}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& \frac{{1} + {\exp\left({{{2}} {{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({{{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}& \frac{{-{{\theta_L}}} + {{\theta_R}} + {{{{\theta_L}}} \cdot {{\exp\left({{{2}} {{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}{-{{{{\theta_R}}} \cdot {{\exp\left({{{2}} {{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}}{{{2}} {{\exp\left({{{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}& 0\\ 0& \frac{{{\exp\left({{{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}} {{\left({{1}{-{\frac{1}{\exp\left({{{2}} {{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{{\theta_L}}{-{{\theta_R}}}}\right)}}}& \frac{{1} + {\exp\left({{{2}} {{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}{{{2}} {{\exp\left({{{f}} {{\sqrt{{-{{{\theta_L}}^{2}}}{-{{{\theta_R}}^{2}}} + {{{2}} {{{\theta_L}}} \cdot {{{\theta_R}}}}}}}}\right)}}}& 0\\ 0& 0& 0& 1\end{array}\right]$
${{[\Gamma_\phi]}} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{{f}} {{\sin\left( \theta\right)}}}\\ 0& 0& 0& -{\cos\left( \theta\right)}\\ 0& {{f}} {{\sin\left( \theta\right)}}& \cos\left( \theta\right)& 0\end{array}\right]}$
$\int\limits_{{{\phi_L}}}^{{{\phi_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{{f}} {{\sin\left( \theta\right)}}}\\ 0& 0& 0& -{\cos\left( \theta\right)}\\ 0& {{f}} {{\sin\left( \theta\right)}}& \cos\left( \theta\right)& 0\end{array}\right]}}d \phi$
= $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {{f}} {{\sin\left( \theta\right)}} {{\left({{-{{\phi_R}}} + {{\phi_L}}}\right)}}\\ 0& 0& 0& {{\left({{{\phi_L}}{-{{\phi_R}}}}\right)}} {{\cos\left( \theta\right)}}\\ 0& {{f}} {{\sin\left( \theta\right)}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}& {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}} {{\cos\left( \theta\right)}}& 0\end{array}\right]$
${ P} _{\phi}$ = $\exp\left( -{\int\limits_{{{\phi_L}}}^{{{\phi_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{{f}} {{\sin\left( \theta\right)}}}\\ 0& 0& 0& -{\cos\left( \theta\right)}\\ 0& {{f}} {{\sin\left( \theta\right)}}& \cos\left( \theta\right)& 0\end{array}\right]}}d \phi}\right)$