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