spherical, anholonomic, orthonormal
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 $e_I = \{e_{x}, e_{y}, e_{z}\}$
transform from basis to coordinate:
${{{ \tilde{e}}_r}^r} = {1}$;
${{{ \tilde{e}}_{\theta}}^{\theta}} = {r}$;
${{{ \tilde{e}}_{\phi}}^{\phi}} = {{{r}} {{sin\left( \theta\right)}}}$
transform from coorinate to basis:
${{{ \tilde{e}}^r}_r} = {1}$;
${{{ \tilde{e}}^{\theta}}_{\theta}} = {{\frac{1}{r}}{({1})}}$;
${{{ \tilde{e}}^{\phi}}_{\phi}} = {\frac{1}{{{r}} {{sin\left( \theta\right)}}}}$
tensor index associated with coordinate $r$ is index $r$ with operator $e_{r}(\zeta) = $$\frac{\partial \zeta}{\partial r}$
tensor index associated with coordinate $\theta$ is index $\hat{\theta}$ with operator $e_{\hat{\theta}}(\zeta) = $${\frac{1}{r}}{({\frac{\partial \zeta}{\partial \theta}})}$
tensor index associated with coordinate $\phi$ is index $\hat{\phi}$ with operator $e_{\hat{\phi}}(\zeta) = $$\frac{\frac{\partial \zeta}{\partial \phi}}{{{r}} {{sin\left( \theta\right)}}}$
flat metric:
${{{ \eta}_x}_x} = {1}$;
${{{ \eta}_y}_y} = {1}$;
${{{ \eta}_z}_z} = {1}$
chart in embedded coordinates:
${{ u}^x} = {{{r}} {{sin\left( \theta\right)}} {{cos\left( \phi\right)}}}$;
${{ u}^y} = {{{r}} {{sin\left( \theta\right)}} {{sin\left( \phi\right)}}}$;
${{ u}^z} = {{{r}} {{cos\left( \theta\right)}}}$
basis operators applied to chart:
${{{ e}_u}^I} = {{{ u}^I}_{,u}}$
${{{ e}_r}^x} = {{{sin\left( \theta\right)}} {{cos\left( \phi\right)}}}$;
${{{ e}_r}^y} = {{{sin\left( \theta\right)}} {{sin\left( \phi\right)}}}$;
${{{ e}_r}^z} = {cos\left( \theta\right)}$;
${{{ e}_{\hat{\theta}}}^x} = {{{cos\left( \phi\right)}} {{cos\left( \theta\right)}}}$;
${{{ e}_{\hat{\theta}}}^y} = {{{sin\left( \phi\right)}} {{cos\left( \theta\right)}}}$;
${{{ e}_{\hat{\theta}}}^z} = {-{sin\left( \theta\right)}}$;
${{{ e}_{\hat{\phi}}}^x} = {-{sin\left( \phi\right)}}$;
${{{ e}_{\hat{\phi}}}^y} = {cos\left( \phi\right)}$
${{{ e}^r}_x} = {{{cos\left( \phi\right)}} {{sin\left( \theta\right)}}}$;
${{{ e}^r}_y} = {{{sin\left( \phi\right)}} {{sin\left( \theta\right)}}}$;
${{{ e}^r}_z} = {cos\left( \theta\right)}$;
${{{ e}^{\hat{\theta}}}_x} = {{{cos\left( \phi\right)}} {{cos\left( \theta\right)}}}$;
${{{ e}^{\hat{\theta}}}_y} = {{{sin\left( \phi\right)}} {{cos\left( \theta\right)}}}$;
${{{ e}^{\hat{\theta}}}_z} = {-{sin\left( \theta\right)}}$;
${{{ e}^{\hat{\phi}}}_x} = {-{sin\left( \phi\right)}}$;
${{{ e}^{\hat{\phi}}}_y} = {cos\left( \phi\right)}$
${{{{{ e}_u}^I}} {{{{ e}^v}_I}}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]}}$
${{{{{ e}_u}^I}} {{{{ e}^u}_J}}} = {\overset{I\downarrow J\rightarrow}{\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]}}$
basis determinant: ${det(e)} = {1}$
${{{{ c}_r}_{\hat{\theta}}}^{\hat{\theta}}} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{ c}_r}_{\hat{\phi}}}^{\hat{\phi}}} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{ c}_{\hat{\theta}}}_r}^{\hat{\theta}}} = {{\frac{1}{r}}{({1})}}$;
${{{{ c}_{\hat{\theta}}}_{\hat{\phi}}}^{\hat{\phi}}} = {\frac{-{cos\left( \theta\right)}}{{{r}} {{sin\left( \theta\right)}}}}$;
${{{{ c}_{\hat{\phi}}}_r}^{\hat{\phi}}} = {{\frac{1}{r}}{({1})}}$;
${{{{ c}_{\hat{\phi}}}_{\hat{\theta}}}^{\hat{\phi}}} = {\frac{cos\left( \theta\right)}{{{r}} {{sin\left( \theta\right)}}}}$
${{{ g}_u}_v} = {{{{{ e}_u}^I}} {{{{ e}_v}^J}} {{{{ \eta}_I}_J}}}$
${{{ g}_r}_r} = {1}$;
${{{ g}_{\hat{\theta}}}_{\hat{\theta}}} = {1}$;
${{{ g}_{\hat{\phi}}}_{\hat{\phi}}} = {1}$
${{{ g}_u}_v} = {{{{{ e}_u}^I}} {{{{ e}_v}^J}} {{{{ \eta}_I}_J}}}$
metric determinant: ${det(g)} = {1}$
${{{{ \Gamma}_a}_b}_c} = {{{{\frac{1}{2}}{({1})}}} {{({{{{{{{{ g}_a}_b}_{,c}} + {{{{ g}_a}_c}_{,b}}} - {{{{ g}_b}_c}_{,a}}} + {{{{ c}_a}_b}_c} + {{{{ c}_a}_c}_b}} - {{{{ c}_c}_b}_a}})}}}$
commutation coefficients:
${{{{ c}_r}_{\hat{\theta}}}^{\hat{\theta}}} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{ c}_r}_{\hat{\phi}}}^{\hat{\phi}}} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{ c}_{\hat{\theta}}}_r}^{\hat{\theta}}} = {{\frac{1}{r}}{({1})}}$;
${{{{ c}_{\hat{\theta}}}_{\hat{\phi}}}^{\hat{\phi}}} = {\frac{-{cos\left( \theta\right)}}{{{r}} {{sin\left( \theta\right)}}}}$;
${{{{ c}_{\hat{\phi}}}_r}^{\hat{\phi}}} = {{\frac{1}{r}}{({1})}}$;
${{{{ c}_{\hat{\phi}}}_{\hat{\theta}}}^{\hat{\phi}}} = {\frac{cos\left( \theta\right)}{{{r}} {{sin\left( \theta\right)}}}}$
metric:
${{{ g}_r}_r} = {1}$;
${{{ g}_{\hat{\theta}}}_{\hat{\theta}}} = {1}$;
${{{ g}_{\hat{\phi}}}_{\hat{\phi}}} = {1}$
metric inverse:
${{{ g}^r}^r} = {1}$;
${{{ g}^{\hat{\theta}}}^{\hat{\theta}}} = {1}$;
${{{ g}^{\hat{\phi}}}^{\hat{\phi}}} = {1}$
metric derivative:
${{{{ {\partial g}}_a}_b}_c} = {0}$
1st kind Christoffel:
${{{{ \Gamma}_r}_{\hat{\theta}}}_{\hat{\theta}}} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{ \Gamma}_r}_{\hat{\phi}}}_{\hat{\phi}}} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{ \Gamma}_{\hat{\theta}}}_{\hat{\theta}}}_r} = {{\frac{1}{r}}{({1})}}$;
${{{{ \Gamma}_{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{cos\left( \theta\right)}}{{{r}} {{sin\left( \theta\right)}}}}$;
${{{{ \Gamma}_{\hat{\phi}}}_{\hat{\phi}}}_r} = {{\frac{1}{r}}{({1})}}$;
${{{{ \Gamma}_{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}} = {\frac{cos\left( \theta\right)}{{{r}} {{sin\left( \theta\right)}}}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma}^r}_{\hat{\theta}}}_{\hat{\theta}}} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{ \Gamma}^r}_{\hat{\phi}}}_{\hat{\phi}}} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{ \Gamma}^{\hat{\theta}}}_{\hat{\theta}}}_r} = {{\frac{1}{r}}{({1})}}$;
${{{{ \Gamma}^{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{cos\left( \theta\right)}}{{{r}} {{sin\left( \theta\right)}}}}$;
${{{{ \Gamma}^{\hat{\phi}}}_{\hat{\phi}}}_r} = {{\frac{1}{r}}{({1})}}$;
${{{{ \Gamma}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}} = {\frac{cos\left( \theta\right)}{{{r}} {{sin\left( \theta\right)}}}}$
connection coefficients derivative:
${{{{{ {\partial \Gamma}}^r}_{\hat{\theta}}}_{\hat{\theta}}}_r} = {\frac{1}{{r}^{2}}}$;
${{{{{ {\partial \Gamma}}^r}_{\hat{\phi}}}_{\hat{\phi}}}_r} = {\frac{1}{{r}^{2}}}$;
${{{{{ {\partial \Gamma}}^{\hat{\theta}}}_{\hat{\theta}}}_r}_r} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {\partial \Gamma}}^{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}}_r} = {\frac{cos\left( \theta\right)}{{{{r}^{2}}} {{sin\left( \theta\right)}}}}$;
${{{{{ {\partial \Gamma}}^{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}} = {\frac{1}{{{{r}^{2}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}}$;
${{{{{ {\partial \Gamma}}^{\hat{\phi}}}_{\hat{\phi}}}_r}_r} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {\partial \Gamma}}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}}_r} = {\frac{-{cos\left( \theta\right)}}{{{{r}^{2}}} {{sin\left( \theta\right)}}}}$;
${{{{{ {\partial \Gamma}}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}}_{\hat{\theta}}} = {-{\frac{1}{{{{r}^{2}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}}}$
connection coefficients squared:
${{{{{ {(\Gamma^2)}}^r}_r}_{\hat{\theta}}}_{\hat{\theta}}} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {(\Gamma^2)}}^r}_r}_{\hat{\phi}}}_{\hat{\phi}}} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {(\Gamma^2)}}^r}_{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{cos\left( \theta\right)}}{{{{r}^{2}}} {{sin\left( \theta\right)}}}}$;
${{{{{ {(\Gamma^2)}}^r}_{\hat{\phi}}}_{\hat{\theta}}}_{\hat{\phi}}} = {\frac{cos\left( \theta\right)}{{{{r}^{2}}} {{sin\left( \theta\right)}}}}$;
${{{{{ {(\Gamma^2)}}^{\hat{\theta}}}_r}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{cos\left( \theta\right)}}{{{{r}^{2}}} {{sin\left( \theta\right)}}}}$;
${{{{{ {(\Gamma^2)}}^{\hat{\theta}}}_{\hat{\theta}}}_{\hat{\theta}}}_{\hat{\theta}}} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {(\Gamma^2)}}^{\hat{\theta}}}_{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{{cos\left( \theta\right)}^{2}}}{{{{r}^{2}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}}$;
${{{{{ {(\Gamma^2)}}^{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\theta}}}_{\hat{\phi}}} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {(\Gamma^2)}}^{\hat{\phi}}}_r}_{\hat{\phi}}}_{\hat{\theta}}} = {\frac{cos\left( \theta\right)}{{{{r}^{2}}} {{sin\left( \theta\right)}}}}$;
${{{{{ {(\Gamma^2)}}^{\hat{\phi}}}_{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\theta}}} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {(\Gamma^2)}}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\phi}}} = {-{\frac{1}{{{{r}^{2}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R}^a}_b}_c}_d} = {0}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R}^a}^b}_c}_d} = {0}$
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, $\sharp\sharp\flat\flat$:
${{{{{ C}^a}^b}_c}_d} = {0}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C}_a}_b}_c}_d} = {0}$
Plebanski, $\sharp\sharp\flat\flat$:
${{{{{ P}^a}^b}_c}_d} = {0}$
divergence: ${{{{ A}^i}_{,i}} + {{{{{{ \Gamma}^i}_i}_j}} {{{ A}^j}}}} = {{{{2}} {{{A^{r}}}} \cdot {{{\frac{1}{r}}{({1})}}}} + {{{{A^{\hat{\theta}}}}} \cdot {{cos\left( \theta\right)}} {{{\frac{1}{r}}{({1})}}} {{\frac{1}{sin\left( \theta\right)}}}} + {{{\frac{\partial {A^{\hat{\theta}}}}{\partial \theta}}} {{{\frac{1}{r}}{({1})}}}} + {\frac{\partial {A^{r}}}{\partial r}} + {{{\frac{\partial {A^{\hat{\phi}}}}{\partial \phi}}} {{{\frac{1}{r}}{({1})}}} {{\frac{1}{sin\left( \theta\right)}}}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{r} \\ \ddot{\hat{\theta}} \\ \ddot{\hat{\phi}}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} {{{{\dot{\hat{\phi}}}^{2}}} {{{\frac{1}{r}}{({1})}}}} + {{{{\dot{\hat{\theta}}}^{2}}} {{{\frac{1}{r}}{({1})}}}} \\ {{{-1}} {{\dot{\hat{\theta}}}} \cdot {{\dot{r}}} \cdot {{{\frac{1}{r}}{({1})}}}} + {{{{\dot{\hat{\phi}}}^{2}}} {{cos\left( \theta\right)}} {{{\frac{1}{r}}{({1})}}} {{\frac{1}{sin\left( \theta\right)}}}} \\ {{{-1}} {{\dot{\hat{\phi}}}} \cdot {{\dot{\hat{\theta}}}} \cdot {{cos\left( \theta\right)}} {{{\frac{1}{r}}{({1})}}} {{\frac{1}{sin\left( \theta\right)}}}} + {{{-1}} {{\dot{\hat{\phi}}}} \cdot {{\dot{r}}} \cdot {{sin\left( \theta\right)}} {{{\frac{1}{r}}{({1})}}} {{\frac{1}{sin\left( \theta\right)}}}}\end{matrix}\right]}}$
parallel propagators:
${{[\Gamma_r]}} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
$\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)$
= $\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]$
${ P}_r$ = ${ⅇ}^{( -{({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)})})}$
= $\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]$
${{ P}_r}^{-1}$ = ${ⅇ}^{({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)})}$
= $\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]$
${{[\Gamma_\theta]}} = {\left[\begin{matrix} 0 & -{1} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
$\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left( \left[\begin{matrix} 0 & -{1} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)$
= $\left[\begin{matrix} 0 & {-{{\theta_R}}} + {{\theta_L}} & 0 \\ -{({{{\theta_L}} - {{\theta_R}}})} & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]$
${ P}_{\theta}$ = ${ⅇ}^{( -{({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left( \left[\begin{matrix} 0 & -{1} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)})})}$
= $\left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]$
${{ P}_{\theta}}^{-1}$ = ${ⅇ}^{({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left( \left[\begin{matrix} 0 & -{1} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)})}$
= $\left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]$
${{[\Gamma_\phi]}} = {\left[\begin{matrix} 0 & 0 & -{sin\left( \theta\right)} \\ 0 & 0 & -{cos\left( \theta\right)} \\ sin\left( \theta\right) & cos\left( \theta\right) & 0\end{matrix}\right]}$
$\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \phi\left( \left[\begin{matrix} 0 & 0 & -{sin\left( \theta\right)} \\ 0 & 0 & -{cos\left( \theta\right)} \\ sin\left( \theta\right) & cos\left( \theta\right) & 0\end{matrix}\right]\right)$
= $\left[\begin{matrix} 0 & 0 & {{sin\left( \theta\right)}} {{({{{\phi_L}} - {{\phi_R}}})}} \\ 0 & 0 & {{cos\left( \theta\right)}} {{({{{\phi_L}} - {{\phi_R}}})}} \\ -{{{sin\left( \theta\right)}} {{({{{\phi_L}} - {{\phi_R}}})}}} & -{{{cos\left( \theta\right)}} {{({{{\phi_L}} - {{\phi_R}}})}}} & 0\end{matrix}\right]$
${ P}_{\phi}$ = ${ⅇ}^{( -{({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \phi\left( \left[\begin{matrix} 0 & 0 & -{sin\left( \theta\right)} \\ 0 & 0 & -{cos\left( \theta\right)} \\ sin\left( \theta\right) & cos\left( \theta\right) & 0\end{matrix}\right]\right)})})}$
= ${\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( \theta\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( \theta\right)}} {{cos\left( \theta\right)}}}})} & {\frac{1}{6}}{({{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( \theta\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}$
${{ P}_{\phi}}^{-1}$ = ${ⅇ}^{({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \phi\left( \left[\begin{matrix} 0 & 0 & -{sin\left( \theta\right)} \\ 0 & 0 & -{cos\left( \theta\right)} \\ sin\left( \theta\right) & cos\left( \theta\right) & 0\end{matrix}\right]\right)})}$
= ${\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( \theta\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( \theta\right)}} {{cos\left( \theta\right)}}}})} & {\frac{1}{6}}{({-{{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( \theta\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({-{{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} \\ {\frac{1}{6}}{({{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} & {\frac{1}{6}}{({{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}$
propagator commutation:
[ ${ P}_r$ , ${ P}_{\theta}$ ] = ${ {\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]} {\left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]}} - { {\left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]} {\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]}}$ = $\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]$
[ ${ P}_r$ , ${ P}_{\phi}$ ] = ${ {\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]} {{({{\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}})} & {\frac{1}{6}}{({{{sin\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( {\theta_L}\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({{{cos\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}})}}} - {{{({{\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}})} & {\frac{1}{6}}{({{{sin\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( {\theta_L}\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({{{cos\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}})}} {\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]}}$ = $\left[\begin{matrix} \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right] & 0 & 0 \\ 0 & \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right] & 0 \\ 0 & 0 & \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\end{matrix}\right]$
[ ${ P}_{\theta}$ , ${ P}_{\phi}$ ] = ${ {\left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]} {{({{\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}})} & {\frac{1}{6}}{({{{sin\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( {\theta_L}\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({{{cos\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( {\theta_L}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}})}}} - {{{({{\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( {\theta_R}\right)}} {{cos\left( {\theta_R}\right)}}}})} & {\frac{1}{6}}{({{{sin\left( {\theta_R}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( {\theta_R}\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({{{cos\left( {\theta_R}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( {\theta_R}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( {\theta_R}\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}})}} {\left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]}}$ = $\left[\begin{matrix} \left[\begin{matrix} {\frac{1}{2}}{({-{{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}})}}}})} & {\frac{1}{2}}{({{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}})}}})} & {\frac{1}{6}}{({{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_L}\right)}}}} + {{{{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{3}}} {{sin\left( {\theta_R}\right)}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_R}\right)}}} + {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_R}\right)}}} + {{{{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_L}\right)}}}} - {{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}}}}})}}})} \\ {\frac{1}{2}}{({{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}})}}})} & {\frac{1}{2}}{({{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}})}}})} & {\frac{1}{6}}{({{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{cos\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_L}\right)}}}} + {{{{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_R}\right)}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}}} + {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_R}\right)}}} + {{{{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_L}\right)}}}} - {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}}}}})}}})} \\ {\frac{1}{6}}{({-{{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{3}}} {{sin\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_R}\right)}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}}}} + {{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_R}\right)}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{cos\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_R}\right)}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}}}} + {{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_R}\right)}}}})}}}})} & 0\end{matrix}\right] & \left[\begin{matrix} {\frac{1}{2}}{({{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}})}}})} & {\frac{1}{2}}{({-{{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}}})}}}})} & {\frac{1}{6}}{({-{{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{3}}} {{sin\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_R}\right)}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}}}} + {{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_R}\right)}}}})}}}})} \\ {\frac{1}{2}}{({-{{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}}})}}}})} & {\frac{1}{2}}{({-{{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}})}}}})} & {\frac{1}{6}}{({-{{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}}}} - {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_L}\right)}}}} + {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_L}\right)}}} + {{{{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_R}\right)}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}}} + {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}}} + {{{{{{{\phi_R}}^{3}}} {{cos\left( {\theta_R}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}}}} - {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_L}\right)}}}}})}}}})} \\ {\frac{1}{6}}{({{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_L}\right)}}}} + {{{{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{3}}} {{sin\left( {\theta_R}\right)}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_R}\right)}}} + {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_R}\right)}}} + {{{{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_L}\right)}}}} - {{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}}}}})}}})} & {\frac{1}{6}}{({{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}}}} - {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}}} + {{{{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_L}\right)}}}} - {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_R}\right)}}}})}}})} & 0\end{matrix}\right] & 0 \\ \left[\begin{matrix} {\frac{1}{2}}{({-{{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}})}}}})} & {\frac{1}{2}}{({{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}})}}})} & {\frac{1}{6}}{({{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_L}\right)}}}} + {{{{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{3}}} {{sin\left( {\theta_R}\right)}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_R}\right)}}} + {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_R}\right)}}} + {{{{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_L}\right)}}}} - {{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}}}}})}}})} \\ {\frac{1}{2}}{({{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}})}}})} & {\frac{1}{2}}{({{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}})}}})} & {\frac{1}{6}}{({{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}}}} - {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}}} + {{{{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_L}\right)}}}} - {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_R}\right)}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{3}}} {{sin\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_R}\right)}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}}}} + {{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_R}\right)}}}})}}}})} & {\frac{1}{6}}{({-{{{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}}}} - {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_L}\right)}}}} + {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_L}\right)}}} + {{{{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_R}\right)}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}}} + {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}}} + {{{{{{{\phi_R}}^{3}}} {{cos\left( {\theta_R}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}}}} - {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_L}\right)}}}}})}}}})} & 0\end{matrix}\right] & \left[\begin{matrix} {\frac{1}{2}}{({-{{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}})}}}})} & {\frac{1}{2}}{({{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}})}}})} & {\frac{1}{6}}{({{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_L}\right)}}}} + {{{{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{3}}} {{sin\left( {\theta_R}\right)}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_R}\right)}}} + {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_R}\right)}}} + {{{{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_L}\right)}}}} - {{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}}}}})}}})} \\ {\frac{1}{2}}{({{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}}} + {{{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}})}}})} & {\frac{1}{2}}{({{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}})}}})} & {\frac{1}{6}}{({{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{cos\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_L}\right)}}}} + {{{{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_R}\right)}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}}} + {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_R}\right)}}} + {{{{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_L}\right)}}}} - {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}}}}})}}})} \\ {\frac{1}{6}}{({-{{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{3}}} {{sin\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_R}\right)}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}}}} + {{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_R}\right)}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( {{{\theta_L}} - {{\theta_R}}}\right)}} {{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_R}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{cos\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_R}\right)}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}}}} + {{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_R}\right)}}}})}}}})} & 0\end{matrix}\right] & 0 \\ 0 & 0 & \left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}})}})} & {\frac{1}{2}}{({{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}} - {{{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}}} + {{{{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}}} + {{{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}}})} & {\frac{1}{6}}{({{{-{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{3}}} {{sin\left( {\theta_R}\right)}}}} + {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_R}\right)}}} + {{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_R}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_R}\right)}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}}} + {{{{{{{\phi_L}}^{3}}} {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_L}\right)}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_L}\right)}}}}})} \\ {\frac{1}{2}}{({{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}} {{cos\left( {\theta_R}\right)}}}} + {{{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} + {{{{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}} {{cos\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}} {{sin\left( {\theta_L}\right)}}}}})} & {\frac{1}{2}}{({{{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_L}\right)}^{2}}}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( {\theta_R}\right)}^{2}}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{{\phi_L}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}} - {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_L}\right)}^{2}}}}} + {{{{{\phi_R}}^{2}}} {{{cos\left( {\theta_R}\right)}^{2}}}}})} & {\frac{1}{6}}{({{{-{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}}}} - {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_R}\right)}}}} + {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_R}\right)}}} + {{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_R}\right)}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}}} + {{{{{{{\phi_L}}^{3}}} {{cos\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_L}\right)}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}}} - {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_L}\right)}}}}})} \\ {\frac{1}{6}}{({-{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{sin\left( {\theta_R}\right)}}}} - {{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_L}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{sin\left( {\theta_L}\right)}}} - {{{{{\phi_L}}^{3}}} {{sin\left( {\theta_R}\right)}}}} + {{{{6}} {{{\phi_L}}} \cdot {{sin\left( {\theta_R}\right)}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_L}\right)}}}} + {{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{sin\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{sin\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{sin\left( {\theta_R}\right)}}}})}})} & {\frac{1}{6}}{({-{({{{{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_L}\right)}}} - {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{cos\left( {\theta_R}\right)}}}} - {{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_L}\right)}}}} + {{{{{{\phi_L}}^{3}}} {{cos\left( {\theta_L}\right)}}} - {{{{{\phi_L}}^{3}}} {{cos\left( {\theta_R}\right)}}}} + {{{{6}} {{{\phi_L}}} \cdot {{cos\left( {\theta_R}\right)}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_L}\right)}}}} + {{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{cos\left( {\theta_R}\right)}}} - {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_L}\right)}}}} + {{{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_L}\right)}}} - {{{6}} {{{\phi_R}}} \cdot {{cos\left( {\theta_R}\right)}}}} + {{{{{\phi_R}}^{3}}} {{cos\left( {\theta_R}\right)}}}})}})} & 0\end{matrix}\right]\end{matrix}\right]$
propagator partials
${{\frac{\partial}{\partial r}}\left( \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \theta}}\left( \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \phi}}\left( \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial r}}\left( \left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \theta}}\left( \left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \phi}}\left( \left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial r}}\left({{\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( \theta\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( \theta\right)}} {{cos\left( \theta\right)}}}})} & {\frac{1}{6}}{({{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( \theta\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}}\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \theta}}\left({{\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( \theta\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( \theta\right)}} {{cos\left( \theta\right)}}}})} & {\frac{1}{6}}{({{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( \theta\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}}\right)} = {\left[\begin{matrix} {{sin\left( \theta\right)}} {{cos\left( \theta\right)}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}} & {\frac{1}{2}}{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{4}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( \theta\right)}^{2}}}} - {{{2}} {{{{\phi_L}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{2}} {{{{\phi_R}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}})} & {\frac{1}{6}}{({{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{4}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( \theta\right)}^{2}}}} - {{{2}} {{{{\phi_L}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{2}} {{{{\phi_R}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}})} & -{{{sin\left( \theta\right)}} {{cos\left( \theta\right)}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}} & {\frac{1}{6}}{({-{{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} \\ {\frac{1}{6}}{({-{{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \phi}}\left({{\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( \theta\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( \theta\right)}} {{cos\left( \theta\right)}}}})} & {\frac{1}{6}}{({{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( \theta\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}}\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
volume element: ${{{r}^{2}}} {{sin\left( \theta\right)}}$
volume integral: ${{{\frac{1}{3}}{({-{{{\Delta (cos(\theta))}} \cdot {{{\Delta (r^3)}}}}})}}} {{\Delta \phi}}$
finite volume (0,0)-form:
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{({{{{{\frac{1}{{\mathcal{V}(x_C)}}}{({1})}}} {{({{{-{({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \hat{\phi}\left({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \hat{\theta}\left({{{{{J(r_R)}}} \cdot {{{{e_{r}}^{\bar{r}}(r_R)}}} \cdot {{{F^{r}(r_R)}}}} - {{{{J(r_L)}}} \cdot {{{{e_{r}}^{\bar{r}}(r_L)}}} \cdot {{{F^{r}(r_L)}}}}}\right)}\right)})}} - {({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \hat{\phi}\left({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left({{{{{J(\theta_R)}}} \cdot {{{{e_{\hat{\theta}}}^{\bar{\hat{\theta}}}(\theta_R)}}} \cdot {{{F^{\theta}(\theta_R)}}}} - {{{{J(\theta_L)}}} \cdot {{{{e_{\hat{\theta}}}^{\bar{\hat{\theta}}}(\theta_L)}}} \cdot {{{F^{\theta}(\theta_L)}}}}}\right)}\right)})}} - {({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \hat{\theta}\left({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left({{{{{J(\phi_R)}}} \cdot {{{{e_{\hat{\phi}}}^{\bar{\hat{\phi}}}(\phi_R)}}} \cdot {{{F^{\phi}(\phi_R)}}}} - {{{{J(\phi_L)}}} \cdot {{{{e_{\hat{\phi}}}^{\bar{\hat{\phi}}}(\phi_L)}}} \cdot {{{F^{\phi}(\phi_L)}}}}}\right)}\right)})}})}}} + {{S(x_C)}}})}}}}$
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{({{{{\frac{1}{{{{\frac{1}{3}}{({-{{{\Delta (cos(\theta))}} \cdot {{{\Delta (r^3)}}}}})}}} {{\Delta \phi}}}}} {{({{{-{({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \hat{\phi}\left({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \hat{\theta}\left({{{{{{r_R}}^{2}}} {{sin\left( \theta\right)}} {{{F^{r}(r_R)}}}} - {{{{{r_L}}^{2}}} {{sin\left( \theta\right)}} {{{F^{r}(r_L)}}}}}\right)}\right)})}} - {({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \hat{\phi}\left({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left({{{{{r}^{2}}} {{sin\left( \theta\right)}} {{r}} {{{F^{\theta}(\theta_R)}}}} - {{{{r}^{2}}} {{sin\left( \theta\right)}} {{r}} {{{F^{\theta}(\theta_L)}}}}}\right)}\right)})}} - {({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \hat{\theta}\left({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left({{{{{r}^{2}}} {{sin\left( \theta\right)}} {{r}} {{sin\left( \theta\right)}} {{{F^{\phi}(\phi_R)}}}} - {{{{r}^{2}}} {{sin\left( \theta\right)}} {{r}} {{sin\left( \theta\right)}} {{{F^{\phi}(\phi_L)}}}}}\right)}\right)})}})}}} + {{S(x_C)}}})}}}}$
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{({{{{\frac{1}{{{{\frac{1}{3}}{({-{{{\Delta (cos(\theta))}} \cdot {{{\Delta (r^3)}}}}})}}} {{\Delta \phi}}}}} {{({{{-{{{({{{{{{r_R}}^{2}}} {{sin\left( \theta\right)}} {{{F^{r}(r_R)}}}} - {{{{{r_L}}^{2}}} {{sin\left( \theta\right)}} {{{F^{r}(r_L)}}}}})}} {{\Delta \theta}} \cdot {{\Delta \phi}}}} - {{{{\frac{1}{4}}{({{{sin\left( \theta\right)}} {{({{{{{{{F^{\theta}(\theta_L)}}} \cdot {{{{r_L}}^{4}}}} - {{{{F^{\theta}(\theta_L)}}} \cdot {{{{r_R}}^{4}}}}} - {{{{F^{\theta}(\theta_R)}}} \cdot {{{{r_L}}^{4}}}}} + {{{{F^{\theta}(\theta_R)}}} \cdot {{{{r_R}}^{4}}}}})}}})}}} {{\Delta \phi}}}} - {{{{\frac{1}{4}}{({{{{{{{F^{\phi}(\phi_L)}}} \cdot {{{{r_L}}^{4}}}} - {{{{F^{\phi}(\phi_L)}}} \cdot {{{{r_R}}^{4}}}}} - {{{{F^{\phi}(\phi_R)}}} \cdot {{{{r_L}}^{4}}}}} + {{{{{F^{\phi}(\phi_R)}}} \cdot {{{{r_R}}^{4}}}} - {{{{F^{\phi}(\phi_L)}}} \cdot {{{{r_L}}^{4}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{F^{\phi}(\phi_L)}}} \cdot {{{{r_R}}^{4}}} {{{cos\left( \theta\right)}^{2}}}} + {{{{{F^{\phi}(\phi_R)}}} \cdot {{{{r_L}}^{4}}} {{{cos\left( \theta\right)}^{2}}}} - {{{{F^{\phi}(\phi_R)}}} \cdot {{{{r_R}}^{4}}} {{{cos\left( \theta\right)}^{2}}}}}})}}} {{\Delta \theta}}}})}}} + {{S(x_C)}}})}}}}$
${{u(x_C, t_R)}} = {{{{3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\phi}(\phi_L)}}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{{r_L}}^{4}}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}} {{{\frac{1}{\Delta \phi}}{({1})}}}} + {{{-3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\phi}(\phi_L)}}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{{r_L}}^{4}}} {{{cos\left( \theta\right)}^{2}}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}} {{{\frac{1}{\Delta \phi}}{({1})}}}} + {{{-3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\phi}(\phi_L)}}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{{r_R}}^{4}}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}} {{{\frac{1}{\Delta \phi}}{({1})}}}} + {{{3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\phi}(\phi_L)}}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{{r_R}}^{4}}} {{{cos\left( \theta\right)}^{2}}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}} {{{\frac{1}{\Delta \phi}}{({1})}}}} + {{{-3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\phi}(\phi_R)}}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{{r_L}}^{4}}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}} {{{\frac{1}{\Delta \phi}}{({1})}}}} + {{{3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\phi}(\phi_R)}}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{{r_L}}^{4}}} {{{cos\left( \theta\right)}^{2}}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}} {{{\frac{1}{\Delta \phi}}{({1})}}}} + {{{3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\phi}(\phi_R)}}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{{r_R}}^{4}}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}} {{{\frac{1}{\Delta \phi}}{({1})}}}} + {{{-3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\phi}(\phi_R)}}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{{r_R}}^{4}}} {{{cos\left( \theta\right)}^{2}}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}} {{{\frac{1}{\Delta \phi}}{({1})}}}} + {{{3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\theta}(\theta_L)}}} \cdot {{\Delta t}} \cdot {{{{r_L}}^{4}}} {{sin\left( \theta\right)}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}}} + {{{-3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\theta}(\theta_L)}}} \cdot {{\Delta t}} \cdot {{{{r_R}}^{4}}} {{sin\left( \theta\right)}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}}} + {{{-3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\theta}(\theta_R)}}} \cdot {{\Delta t}} \cdot {{{{r_L}}^{4}}} {{sin\left( \theta\right)}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}}} + {{{3}} \cdot {{{\frac{1}{4}}{({1})}}} {{{F^{\theta}(\theta_R)}}} \cdot {{\Delta t}} \cdot {{{{r_R}}^{4}}} {{sin\left( \theta\right)}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}}} + {{{-3}} {{{F^{r}(r_L)}}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{{r_L}}^{2}}} {{sin\left( \theta\right)}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}}} + {{{3}} {{{F^{r}(r_R)}}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{{r_R}}^{2}}} {{sin\left( \theta\right)}} {{{\frac{1}{\Delta (cos(\theta))}}{({1})}}} {{{\frac{1}{{\Delta (r^3)}}}{({1})}}}} + {{{{S(x_C)}}} \cdot {{\Delta t}}} + {{u(x_C, t_L)}}}$