spherical, sinh-radial, anholonomic, orthonormal

chart coordinates: $x^\tilde{\mu} = \{\rho, \theta, \phi\}$
chart coordinate basis: $e_\tilde{\mu} = \{e_{\tilde{\rho}}, e_{\tilde{\theta}}, e_{\tilde{\phi}}\}$
embedding coordinates: $u^I = \{x, y, z\}$
embedding basis $e_I = \{e_{x}, e_{y}, e_{z}\}$
transform from basis to coordinate:
${{{ \tilde{e}}_{\rho}}^{\rho}} = {\frac{{{A}} {{cosh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{{{w}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}}$; ${{{ \tilde{e}}_{\theta}}^{\theta}} = {\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}$; ${{{ \tilde{e}}_{\phi}}^{\phi}} = {\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}$

transform from coorinate to basis:
${{{ \tilde{e}}^{\rho}}_{\rho}} = {\frac{{{w}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}{{{A}} {{cosh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{ \tilde{e}}^{\theta}}_{\theta}} = {\frac{sinh\left( {{\frac{1}{w}}{({1})}}\right)}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{ \tilde{e}}^{\phi}}_{\phi}} = {\frac{sinh\left( {{\frac{1}{w}}{({1})}}\right)}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}}$

tensor index associated with coordinate $\rho$ is index $\hat{\rho}$ with operator $e_{\hat{\rho}}(\zeta) = $$\frac{{{w}} {{\frac{\partial \zeta}{\partial \rho}}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}{{{A}} {{cosh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}$
tensor index associated with coordinate $\theta$ is index $\hat{\theta}$ with operator $e_{\hat{\theta}}(\zeta) = $$\frac{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{\frac{\partial \zeta}{\partial \theta}}}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}$
tensor index associated with coordinate $\phi$ is index $\hat{\phi}$ with operator $e_{\hat{\phi}}(\zeta) = $$\frac{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{\frac{\partial \zeta}{\partial \phi}}}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}$

flat metric: ${{{ \eta}_x}_x} = {1}$; ${{{ \eta}_y}_y} = {1}$; ${{{ \eta}_z}_z} = {1}$

chart in embedded coordinates:
${{ u}^x} = {{{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}} {{sin\left( \theta\right)}} {{cos\left( \phi\right)}}}$; ${{ u}^y} = {{{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}} {{sin\left( \theta\right)}} {{sin\left( \phi\right)}}}$; ${{ u}^z} = {{{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}} {{cos\left( \theta\right)}}}$

basis operators applied to chart:
${{{ e}_u}^I} = {{{ u}^I}_{,u}}$
${{{ e}_{\hat{\rho}}}^x} = {{{cos\left( \phi\right)}} {{sin\left( \theta\right)}}}$; ${{{ e}_{\hat{\rho}}}^y} = {{{sin\left( \phi\right)}} {{sin\left( \theta\right)}}}$; ${{{ e}_{\hat{\rho}}}^z} = {cos\left( \theta\right)}$; ${{{ e}_{\hat{\theta}}}^x} = {{{cos\left( \theta\right)}} {{cos\left( \phi\right)}}}$; ${{{ e}_{\hat{\theta}}}^y} = {{{cos\left( \theta\right)}} {{sin\left( \phi\right)}}}$; ${{{ e}_{\hat{\theta}}}^z} = {-{sin\left( \theta\right)}}$; ${{{ e}_{\hat{\phi}}}^x} = {-{sin\left( \phi\right)}}$; ${{{ e}_{\hat{\phi}}}^y} = {cos\left( \phi\right)}$
${{{ e}^{\hat{\rho}}}_x} = {{{cos\left( \phi\right)}} {{sin\left( \theta\right)}}}$; ${{{ e}^{\hat{\rho}}}_y} = {{{sin\left( \phi\right)}} {{sin\left( \theta\right)}}}$; ${{{ e}^{\hat{\rho}}}_z} = {cos\left( \theta\right)}$; ${{{ e}^{\hat{\theta}}}_x} = {{{cos\left( \phi\right)}} {{cos\left( \theta\right)}}}$; ${{{ e}^{\hat{\theta}}}_y} = {{{sin\left( \phi\right)}} {{cos\left( \theta\right)}}}$; ${{{ e}^{\hat{\theta}}}_z} = {-{sin\left( \theta\right)}}$; ${{{ e}^{\hat{\phi}}}_x} = {-{sin\left( \phi\right)}}$; ${{{ e}^{\hat{\phi}}}_y} = {cos\left( \phi\right)}$
${{{{{ e}_u}^I}} {{{{ e}^v}_I}}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]}}$
${{{{{ e}_u}^I}} {{{{ e}^u}_J}}} = {\overset{I\downarrow J\rightarrow}{\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]}}$
basis determinant: ${det(e)} = {1}$
${{{{ c}_{\hat{\rho}}}_{\hat{\theta}}}^{\hat{\theta}}} = {\frac{-{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ c}_{\hat{\rho}}}_{\hat{\phi}}}^{\hat{\phi}}} = {\frac{-{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ c}_{\hat{\theta}}}_{\hat{\rho}}}^{\hat{\theta}}} = {\frac{sinh\left( {{\frac{1}{w}}{({1})}}\right)}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ c}_{\hat{\theta}}}_{\hat{\phi}}}^{\hat{\phi}}} = {\frac{-{{{cos\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}}$; ${{{{ c}_{\hat{\phi}}}_{\hat{\rho}}}^{\hat{\phi}}} = {\frac{sinh\left( {{\frac{1}{w}}{({1})}}\right)}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ c}_{\hat{\phi}}}_{\hat{\theta}}}^{\hat{\phi}}} = {\frac{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{cos\left( \theta\right)}}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}}$
${{{ g}_u}_v} = {{{{{ e}_u}^I}} {{{{ e}_v}^J}} {{{{ \eta}_I}_J}}}$
${{{ g}_{\hat{\rho}}}_{\hat{\rho}}} = {1}$; ${{{ g}_{\hat{\theta}}}_{\hat{\theta}}} = {1}$; ${{{ g}_{\hat{\phi}}}_{\hat{\phi}}} = {1}$
${{{ g}_u}_v} = {{{{{ e}_u}^I}} {{{{ e}_v}^J}} {{{{ \eta}_I}_J}}}$
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: ${{{{ c}_{\hat{\rho}}}_{\hat{\theta}}}^{\hat{\theta}}} = {\frac{-{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ c}_{\hat{\rho}}}_{\hat{\phi}}}^{\hat{\phi}}} = {\frac{-{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ c}_{\hat{\theta}}}_{\hat{\rho}}}^{\hat{\theta}}} = {\frac{sinh\left( {{\frac{1}{w}}{({1})}}\right)}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ c}_{\hat{\theta}}}_{\hat{\phi}}}^{\hat{\phi}}} = {\frac{-{{{cos\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}}$; ${{{{ c}_{\hat{\phi}}}_{\hat{\rho}}}^{\hat{\phi}}} = {\frac{sinh\left( {{\frac{1}{w}}{({1})}}\right)}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ c}_{\hat{\phi}}}_{\hat{\theta}}}^{\hat{\phi}}} = {\frac{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{cos\left( \theta\right)}}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}}$
metric: ${{{ g}_{\hat{\rho}}}_{\hat{\rho}}} = {1}$; ${{{ g}_{\hat{\theta}}}_{\hat{\theta}}} = {1}$; ${{{ g}_{\hat{\phi}}}_{\hat{\phi}}} = {1}$
metric inverse: ${{{ g}^{\hat{\rho}}}^{\hat{\rho}}} = {1}$; ${{{ g}^{\hat{\theta}}}^{\hat{\theta}}} = {1}$; ${{{ g}^{\hat{\phi}}}^{\hat{\phi}}} = {1}$
metric derivative: ${{{{ {\partial g}}_a}_b}_c} = {0}$
1st kind Christoffel: ${{{{ \Gamma}_{\hat{\rho}}}_{\hat{\theta}}}_{\hat{\theta}}} = {\frac{-{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ \Gamma}_{\hat{\rho}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ \Gamma}_{\hat{\theta}}}_{\hat{\theta}}}_{\hat{\rho}}} = {\frac{sinh\left( {{\frac{1}{w}}{({1})}}\right)}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ \Gamma}_{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{cos\left( \theta\right)}}}}{{{A}} {{sin\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ \Gamma}_{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\rho}}} = {\frac{sinh\left( {{\frac{1}{w}}{({1})}}\right)}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ \Gamma}_{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}} = {\frac{{{cos\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}{{{A}} {{sin\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$
connection coefficients / 2nd kind Christoffel: ${{{{ \Gamma}^{\hat{\rho}}}_{\hat{\theta}}}_{\hat{\theta}}} = {\frac{-{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ \Gamma}^{\hat{\rho}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ \Gamma}^{\hat{\theta}}}_{\hat{\theta}}}_{\hat{\rho}}} = {\frac{sinh\left( {{\frac{1}{w}}{({1})}}\right)}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ \Gamma}^{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{{{cos\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}}{{{A}} {{sin\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ \Gamma}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\rho}}} = {\frac{sinh\left( {{\frac{1}{w}}{({1})}}\right)}{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$; ${{{{ \Gamma}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}} = {\frac{{{cos\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}{{{A}} {{sin\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}}$
connection coefficients derivative: ${{{{{ {\partial \Gamma}}^{\hat{\rho}}}_{\hat{\theta}}}_{\hat{\theta}}}_{\hat{\rho}}} = {\frac{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}}}}$; ${{{{{ {\partial \Gamma}}^{\hat{\rho}}}_{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\rho}}} = {\frac{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}}}}$; ${{{{{ {\partial \Gamma}}^{\hat{\theta}}}_{\hat{\theta}}}_{\hat{\rho}}}_{\hat{\rho}}} = {\frac{-{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}}}}$; ${{{{{ {\partial \Gamma}}^{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\rho}}} = {\frac{{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}} {{cos\left( \theta\right)}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{sin\left( \theta\right)}}}}$; ${{{{{ {\partial \Gamma}}^{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}} = {\frac{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}{-{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{({{-{1}} + {{cos\left( \theta\right)}^{2}}})}}}}}$; ${{{{{ {\partial \Gamma}}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\rho}}}_{\hat{\rho}}} = {\frac{-{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}}}}$; ${{{{{ {\partial \Gamma}}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}}_{\hat{\rho}}} = {\frac{-{{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}} {{cos\left( \theta\right)}}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{sin\left( \theta\right)}}}}$; ${{{{{ {\partial \Gamma}}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}}_{\hat{\theta}}} = {\frac{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{({{-{1}} + {{cos\left( \theta\right)}^{2}}})}}}}$
connection coefficients squared: ${{{{{ {(\Gamma^2)}}^{\hat{\rho}}}_{\hat{\rho}}}_{\hat{\theta}}}_{\hat{\theta}}} = {\frac{-{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}}}}$; ${{{{{ {(\Gamma^2)}}^{\hat{\rho}}}_{\hat{\rho}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}}}}$; ${{{{{ {(\Gamma^2)}}^{\hat{\rho}}}_{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}} {{cos\left( \theta\right)}}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{sin\left( \theta\right)}}}}$; ${{{{{ {(\Gamma^2)}}^{\hat{\rho}}}_{\hat{\phi}}}_{\hat{\theta}}}_{\hat{\phi}}} = {\frac{{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}} {{cos\left( \theta\right)}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{sin\left( \theta\right)}}}}$; ${{{{{ {(\Gamma^2)}}^{\hat{\theta}}}_{\hat{\rho}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}} {{cos\left( \theta\right)}}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{sin\left( \theta\right)}}}}$; ${{{{{ {(\Gamma^2)}}^{\hat{\theta}}}_{\hat{\theta}}}_{\hat{\theta}}}_{\hat{\theta}}} = {\frac{-{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}}}}$; ${{{{{ {(\Gamma^2)}}^{\hat{\theta}}}_{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}}$; ${{{{{ {(\Gamma^2)}}^{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\theta}}}_{\hat{\phi}}} = {\frac{-{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}}}}$; ${{{{{ {(\Gamma^2)}}^{\hat{\phi}}}_{\hat{\rho}}}_{\hat{\phi}}}_{\hat{\theta}}} = {\frac{{{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}} {{cos\left( \theta\right)}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{sin\left( \theta\right)}}}}$; ${{{{{ {(\Gamma^2)}}^{\hat{\phi}}}_{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\theta}}} = {\frac{-{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}}}}$; ${{{{{ {(\Gamma^2)}}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{({{-{1}} + {{cos\left( \theta\right)}^{2}}})}}}}$
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^{\hat{\rho}}}}} \cdot {{sin\left( \theta\right)}} {{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}} {{\frac{1}{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}} {{\frac{1}{sin\left( \theta\right)}}}} + {{{{A^{\hat{\theta}}}}} \cdot {{cos\left( \theta\right)}} {{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}} {{\frac{1}{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}} {{\frac{1}{sin\left( \theta\right)}}}} + {{{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}} {{\frac{\partial {A^{\hat{\phi}}}}{\partial \phi}}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}} {{\frac{1}{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}} {{\frac{1}{sin\left( \theta\right)}}}} + {{{sin\left( \theta\right)}} {{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}} {{\frac{\partial {A^{\hat{\theta}}}}{\partial \theta}}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}} {{\frac{1}{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}} {{\frac{1}{sin\left( \theta\right)}}}} + {{{w}} {{sin\left( \theta\right)}} {{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}} {{\frac{\partial {A^{\hat{\rho}}}}{\partial \rho}}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}} {{\frac{1}{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}} {{\frac{1}{sin\left( \theta\right)}}}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{\hat{\rho}} \\ \ddot{\hat{\theta}} \\ \ddot{\hat{\phi}}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} {{{{\dot{\hat{\phi}}}^{2}}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}}} + {{{{\dot{\hat{\theta}}}^{2}}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}}} \\ {{{{\dot{\hat{\phi}}}^{2}}} {{cos\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sin\left( \theta\right)}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}}} + {{{-1}} {{\dot{\hat{\rho}}}} \cdot {{\dot{\hat{\theta}}}} \cdot {{sin\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sin\left( \theta\right)}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}}} \\ {{{-1}} {{\dot{\hat{\phi}}}} \cdot {{\dot{\hat{\rho}}}} \cdot {{sin\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sin\left( \theta\right)}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}}} + {{{-1}} {{\dot{\hat{\phi}}}} \cdot {{\dot{\hat{\theta}}}} \cdot {{cos\left( \theta\right)}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}} {{{\frac{1}{A}}{({1})}}} {{\frac{1}{sin\left( \theta\right)}}} {{\frac{1}{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}}}}\end{matrix}\right]}}$

parallel propagators:

${{[\Gamma_\rho]}} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$

$\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left( \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)$ = $\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]$

${ P}_{\rho}$ = ${ⅇ}^{( -{({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left( \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)})})}$ = $\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]$

${{ P}_{\rho}}^{-1}$ = ${ⅇ}^{({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left( \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)})}$ = $\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]$

${{[\Gamma_\theta]}} = {\left[\begin{matrix} 0 & -{1} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$

$\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left( \left[\begin{matrix} 0 & -{1} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)$ = $\left[\begin{matrix} 0 & {-{{\theta_R}}} + {{\theta_L}} & 0 \\ -{({{{\theta_L}} - {{\theta_R}}})} & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]$

${ P}_{\theta}$ = ${ⅇ}^{( -{({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left( \left[\begin{matrix} 0 & -{1} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)})})}$ = $\left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]$

${{ P}_{\theta}}^{-1}$ = ${ⅇ}^{({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left( \left[\begin{matrix} 0 & -{1} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]\right)})}$ = $\left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]$

${{[\Gamma_\phi]}} = {\left[\begin{matrix} 0 & 0 & -{sin\left( \theta\right)} \\ 0 & 0 & -{cos\left( \theta\right)} \\ sin\left( \theta\right) & cos\left( \theta\right) & 0\end{matrix}\right]}$

$\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \phi\left( \left[\begin{matrix} 0 & 0 & -{sin\left( \theta\right)} \\ 0 & 0 & -{cos\left( \theta\right)} \\ sin\left( \theta\right) & cos\left( \theta\right) & 0\end{matrix}\right]\right)$ = $\left[\begin{matrix} 0 & 0 & {{sin\left( \theta\right)}} {{({{{\phi_L}} - {{\phi_R}}})}} \\ 0 & 0 & {{cos\left( \theta\right)}} {{({{{\phi_L}} - {{\phi_R}}})}} \\ -{{{sin\left( \theta\right)}} {{({{{\phi_L}} - {{\phi_R}}})}}} & -{{{cos\left( \theta\right)}} {{({{{\phi_L}} - {{\phi_R}}})}}} & 0\end{matrix}\right]$

${ P}_{\phi}$ = ${ⅇ}^{( -{({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \phi\left( \left[\begin{matrix} 0 & 0 & -{sin\left( \theta\right)} \\ 0 & 0 & -{cos\left( \theta\right)} \\ sin\left( \theta\right) & cos\left( \theta\right) & 0\end{matrix}\right]\right)})})}$ = ${\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( \theta\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( \theta\right)}} {{cos\left( \theta\right)}}}})} & {\frac{1}{6}}{({{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( \theta\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}$

${{ P}_{\phi}}^{-1}$ = ${ⅇ}^{({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \phi\left( \left[\begin{matrix} 0 & 0 & -{sin\left( \theta\right)} \\ 0 & 0 & -{cos\left( \theta\right)} \\ sin\left( \theta\right) & cos\left( \theta\right) & 0\end{matrix}\right]\right)})}$ = ${\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( \theta\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( \theta\right)}} {{cos\left( \theta\right)}}}})} & {\frac{1}{6}}{({-{{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( \theta\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({-{{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} \\ {\frac{1}{6}}{({{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} & {\frac{1}{6}}{({{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}$

propagator commutation:

[ ${ P}_{\rho}$ , ${ P}_{\theta}$ ] = ${ {\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]} {\left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]}} - { {\left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]} {\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]}}$ = $\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]$
[ ${ P}_{\rho}$ , ${ P}_{\phi}$ ] = ${ {\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
${{\frac{\partial}{\partial \rho}}\left( \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \theta}}\left( \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \phi}}\left( \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \rho}}\left( \left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \theta}}\left( \left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \phi}}\left( \left[\begin{matrix} cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & -{sin\left( {{{\theta_L}} - {{\theta_R}}}\right)} & 0 \\ sin\left( {{{\theta_L}} - {{\theta_R}}}\right) & cos\left( {{{\theta_L}} - {{\theta_R}}}\right) & 0 \\ 0 & 0 & 1\end{matrix}\right]\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\frac{\partial}{\partial \rho}}\left({{\left[\begin{matrix} {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{cos\left( \theta\right)}^{2}}}} - {{{{{\phi_L}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}} + {{{{\phi_R}}^{2}} - {{{{{\phi_R}}^{2}}} {{{cos\left( \theta\right)}^{2}}}}}})}})} & {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{sin\left( \theta\right)}} {{cos\left( \theta\right)}}}})} & {\frac{1}{6}}{({{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{2}}{({-{{{{({{{\phi_L}} - {{\phi_R}}})}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})} & {\frac{1}{2}}{({{{{cos\left( \theta\right)}^{2}}} {{({{{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}} - {{{\phi_L}}^{2}}} - {{{\phi_R}}^{2}}})}}})} & {\frac{1}{6}}{({{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}})} \\ {\frac{1}{6}}{({-{{{sin\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{6}}{({-{{{cos\left( \theta\right)}} {{({{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}}} + {{{{{\phi_L}}^{3}} - {{{6}} {{{\phi_L}}}}} - {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}}}} + {{{{6}} {{{\phi_R}}}} - {{{\phi_R}}^{3}}}})}}}})} & {\frac{1}{2}}{({-{({{{{{\phi_L}}^{2}} - {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}} + {{{\phi_R}}^{2}}})}})}\end{matrix}\right]} + {{\mathcal{O}(A^4)}}}\right)} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right]}$
${{\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: ${{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}}$
volume integral: ${{({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({\frac{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{sin\left( \theta\right)}}}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}\right)}\right)})}} {{\Delta \phi}}$
finite volume (0,0)-form:
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{({{{{{\frac{1}{{\mathcal{V}(x_C)}}}{({1})}}} {{({{{-{({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \hat{\phi}\left({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \hat{\theta}\left({{{{{J(\rho_R)}}} \cdot {{{{e_{\hat{\rho}}}^{\bar{\hat{\rho}}}(\rho_R)}}} \cdot {{{F^{\rho}(\rho_R)}}}} - {{{{J(\rho_L)}}} \cdot {{{{e_{\hat{\rho}}}^{\bar{\hat{\rho}}}(\rho_L)}}} \cdot {{{F^{\rho}(\rho_L)}}}}}\right)}\right)})}} - {({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \hat{\phi}\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \hat{\rho}\left({{{{{J(\theta_R)}}} \cdot {{{{e_{\hat{\theta}}}^{\bar{\hat{\theta}}}(\theta_R)}}} \cdot {{{F^{\theta}(\theta_R)}}}} - {{{{J(\theta_L)}}} \cdot {{{{e_{\hat{\theta}}}^{\bar{\hat{\theta}}}(\theta_L)}}} \cdot {{{F^{\theta}(\theta_L)}}}}}\right)}\right)})}} - {({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \hat{\theta}\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \hat{\rho}\left({{{{{J(\phi_R)}}} \cdot {{{{e_{\hat{\phi}}}^{\bar{\hat{\phi}}}(\phi_R)}}} \cdot {{{F^{\phi}(\phi_R)}}}} - {{{{J(\phi_L)}}} \cdot {{{{e_{\hat{\phi}}}^{\bar{\hat{\phi}}}(\phi_L)}}} \cdot {{{F^{\phi}(\phi_L)}}}}}\right)}\right)})}})}}} + {{S(x_C)}}})}}}}$

${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{({{{{\frac{1}{{{({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({\frac{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{sin\left( \theta\right)}}}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}\right)}\right)})}} {{\Delta \phi}}}}} {{({{{-{({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \hat{\phi}\left({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \hat{\theta}\left({{{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{cosh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{{{w}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}}} {{{F^{\rho}(\rho_R)}}}} - {{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{cosh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{{{w}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}}} {{{F^{\rho}(\rho_L)}}}}}\right)}\right)})}} - {({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \hat{\phi}\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \hat{\rho}\left({{{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}} {{{F^{\theta}(\theta_R)}}}} - {{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}} {{{F^{\theta}(\theta_L)}}}}}\right)}\right)})}} - {({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \hat{\theta}\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \hat{\rho}\left({{{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}} {{{F^{\phi}(\phi_R)}}}} - {{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\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}{{{({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({\frac{{{{A}^{2}}} {{{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}^{2}}} {{sin\left( \theta\right)}}}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}\right)}\right)})}} {{\Delta \phi}}}}} {{({{{-{{{({{{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{cosh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{{{w}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}}} {{{F^{\rho}(\rho_R)}}}} - {{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{cosh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{{{w}} {{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}}}} {{{F^{\rho}(\rho_L)}}}}})}} {{\Delta \theta}} \cdot {{\Delta \phi}}}} - {{{({{{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}} {{{F^{\theta}(\theta_R)}}}} - {{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}} {{{F^{\theta}(\theta_L)}}}}})}} {{\Delta \rho}} \cdot {{\Delta \phi}}}} - {{{({{{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}} {{{F^{\phi}(\phi_R)}}}} - {{{{({\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}})}^{2}}} {{sin\left( \theta\right)}} {{\frac{{{A}} {{sinh\left( {{\frac{1}{w}}{({\rho})}}\right)}} {{sin\left( \theta\right)}}}{sinh\left( {{\frac{1}{w}}{({1})}}\right)}}} {{{F^{\phi}(\phi_L)}}}}})}} {{\Delta \rho}} \cdot {{\Delta \theta}}}})}}} + {{S(x_C)}}})}}}}$

${{u(x_C, t_R)}} = {{{{{F^{\phi}(\phi_L)}}} \cdot {{\Delta \rho}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{w}} {{{A}^{3}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{3}}} {{{\frac{1}{\Delta \phi}}{({1})}}} {{{\frac{1}{w}}{({1})}}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}}} {{\frac{1}{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)}}}} + {{{-1}} {{{F^{\phi}(\phi_L)}}} \cdot {{\Delta \rho}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{w}} {{{A}^{3}}} {{{cos\left( \theta\right)}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{3}}} {{{\frac{1}{\Delta \phi}}{({1})}}} {{{\frac{1}{w}}{({1})}}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}}} {{\frac{1}{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)}}}} + {{{-1}} {{{F^{\phi}(\phi_R)}}} \cdot {{\Delta \rho}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{w}} {{{A}^{3}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{3}}} {{{\frac{1}{\Delta \phi}}{({1})}}} {{{\frac{1}{w}}{({1})}}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}}} {{\frac{1}{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)}}}} + {{{{F^{\phi}(\phi_R)}}} \cdot {{\Delta \rho}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{w}} {{{A}^{3}}} {{{cos\left( \theta\right)}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{3}}} {{{\frac{1}{\Delta \phi}}{({1})}}} {{{\frac{1}{w}}{({1})}}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}}} {{\frac{1}{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)}}}} + {{{{F^{\rho}(\rho_L)}}} \cdot {{\Delta \phi}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{A}^{3}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}} {{sin\left( \theta\right)}} {{{\frac{1}{\Delta \phi}}{({1})}}} {{{\frac{1}{w}}{({1})}}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}}} {{\frac{1}{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)}}}} + {{{-1}} {{{F^{\rho}(\rho_R)}}} \cdot {{\Delta \phi}} \cdot {{\Delta \theta}} \cdot {{\Delta t}} \cdot {{{A}^{3}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{cosh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}} {{{\frac{1}{\Delta \phi}}{({1})}}} {{{\frac{1}{w}}{({1})}}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}}} {{\frac{1}{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)}}}} + {{{{F^{\theta}(\theta_L)}}} \cdot {{\Delta \phi}} \cdot {{\Delta \rho}} \cdot {{\Delta t}} \cdot {{w}} {{{A}^{3}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{3}}} {{sin\left( \theta\right)}} {{{\frac{1}{\Delta \phi}}{({1})}}} {{{\frac{1}{w}}{({1})}}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}}} {{\frac{1}{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)}}}} + {{{-1}} {{{F^{\theta}(\theta_R)}}} \cdot {{\Delta \phi}} \cdot {{\Delta \rho}} \cdot {{\Delta t}} \cdot {{w}} {{{A}^{3}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{3}}} {{sin\left( \theta\right)}} {{{\frac{1}{\Delta \phi}}{({1})}}} {{{\frac{1}{w}}{({1})}}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}}} {{\frac{1}{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)}}}} + {{{{S(x_C)}}} \cdot {{\Delta \phi}} \cdot {{\Delta t}} \cdot {{w}} {{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}} {{({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)})}} {{{\frac{1}{\Delta \phi}}{({1})}}} {{{\frac{1}{w}}{({1})}}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}}} {{\frac{1}{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)}}}} + {{{\Delta \phi}} \cdot {{{u(x_C, t_L)}}} \cdot {{w}} {{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}} {{({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)})}} {{{\frac{1}{\Delta \phi}}{({1})}}} {{{\frac{1}{w}}{({1})}}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{3}}}} {{\frac{1}{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}d \theta\left({\int\limits_{{{\rho_L}}}^{{{\rho_R}}}d \rho\left({{{{A}^{2}}} {{{sinh\left( {{{\rho}} \cdot {{{\frac{1}{w}}{({1})}}}}\right)}^{2}}} {{sin\left( \theta\right)}} {{\frac{1}{{sinh\left( {{\frac{1}{w}}{({1})}}\right)}^{2}}}}}\right)}\right)}}}}}$