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