sphere surface, anholonomic, orthonormal

chart coordinates: $x^\tilde{\mu} = \{\theta, \phi\}$
chart coordinate basis: $e_\tilde{\mu} = \{e_{\tilde{\theta}}, e_{\tilde{\phi}}\}$
embedding coordinates: $u^I = \{x, y, z\}$
embedding basis $e_I = \{e_{x}, e_{y}, e_{z}\}$
flat metric: ${{{ \eta} _I} _J} = {\overset{I\downarrow J\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \right]}}$

transform from basis to coordinate:
${{{ \tilde{e}} _A} ^a} = {\overset{a\downarrow A\rightarrow}{\left[ \begin{matrix} r & 0 \\ 0 & {{r}} {{\sin\left( \theta\right)}}\end{matrix} \right]}}$

transform from coorinate to basis:
${{{ \tilde{e}} ^a} _A} = {\overset{a\downarrow A\rightarrow}{\left[ \begin{matrix} \frac{1}{r} & 0 \\ 0 & \frac{1}{{{r}} {{\sin\left( \theta\right)}}}\end{matrix} \right]}}$

tensor index associated with coordinate $\theta$ has operator $e_{\theta}(\zeta) = $${\frac{1}{r}} {\frac{\partial \zeta}{\partial \theta}}$
tensor index associated with coordinate $\phi$ has operator $e_{\phi}(\zeta) = $$\frac{\frac{\partial \zeta}{\partial \phi}}{{{r}} {{\sin\left( \theta\right)}}}$

chart in embedded coordinates:
${u} = {\overset{I\downarrow}{\left[ \begin{matrix} {{r}} {{\sin\left( \theta\right)}} {{\cos\left( \phi\right)}} \\ {{r}} {{\sin\left( \theta\right)}} {{\sin\left( \phi\right)}} \\ {{r}} {{\cos\left( \theta\right)}}\end{matrix} \right]}}$

basis operators applied to chart:
${{{ e} _u} ^I} = {{{ u} ^I} _{,u}}$
${{{ e} _u} ^I} = {\overset{u\downarrow I\rightarrow}{\left[ \begin{matrix} {{\cos\left( \phi\right)}} {{\cos\left( \theta\right)}} & {{\sin\left( \phi\right)}} {{\cos\left( \theta\right)}} & -{\sin\left( \theta\right)} \\ -{\sin\left( \phi\right)} & \cos\left( \phi\right) & 0\end{matrix} \right]}}$

${{{ e} ^u} _I} = {\overset{a\downarrow I\rightarrow}{\left[ \begin{matrix} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}} & {{\cos\left( \theta\right)}} {{\sin\left( \phi\right)}} & -{\sin\left( \theta\right)} \\ -{\sin\left( \phi\right)} & \cos\left( \phi\right) & 0\end{matrix} \right]}}$

${{{{{ e} _u} ^I}} {{{{ e} ^v} _I}}} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^u} _J}}} = {\overset{I\downarrow J\rightarrow}{\left[ \begin{matrix} {1} + {{{{\cos\left( \theta\right)}^{2}}} {{{\cos\left( \phi\right)}^{2}}}}{-{{\cos\left( \phi\right)}^{2}}} & -{{{\cos\left( \phi\right)}} {{\sin\left( \phi\right)}} {{{\sin\left( \theta\right)}^{2}}}} & -{{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}} {{\cos\left( \phi\right)}}} \\ -{{{\sin\left( \phi\right)}} {{\cos\left( \phi\right)}} {{{\sin\left( \theta\right)}^{2}}}} & {-{{{{\cos\left( \theta\right)}^{2}}} {{{\cos\left( \phi\right)}^{2}}}}} + {{\cos\left( \phi\right)}^{2}} + {{\cos\left( \theta\right)}^{2}} & -{{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}} {{\sin\left( \phi\right)}}} \\ -{{{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} {{\cos\left( \phi\right)}}} & -{{{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \phi\right)}}} & {\sin\left( \theta\right)}^{2}\end{matrix} \right]}}$
${{{{ c} _a} _b} ^c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & -{\frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}} \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$

${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]}}$

${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
metric determinant: ${det(g)} = {1}$
${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{{{{{ g} _a} _b} _{,c}} + {{{{ g} _a} _c} _{,b}}}{-{{{{ g} _b} _c} _{,a}}} + {{{{ c} _a} _b} _c} + {{{{ c} _a} _c} _b}}{-{{{{ c} _c} _b} _a}}}\right)}}}$
commutation coefficients: ${c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & -{\frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}} \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$

metric: ${g} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]}}$

metric inverse: ${g} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]}}$

metric derivative: ${{\partial g}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$

1st kind Christoffel: ${\Gamma} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & -{\frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$

connection coefficients / 2nd kind Christoffel: ${\Gamma} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} 0 & 0 \\ 0 & -{\frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}}\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 \\ \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}} & 0\end{matrix} \right]\end{matrix} \right]}}$

connection coefficients derivative: ${{\partial \Gamma}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ \frac{1}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} -{\frac{1}{{{{r}^{2}}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}}}} & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$

connection coefficients squared: ${{(\Gamma^2)}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & -{\frac{{\cos\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & -{\frac{{\cos\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}}\end{matrix} \right]}\end{matrix} \right]}}$

Riemann curvature, $\sharp\flat\flat\flat$: ${R} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} \\ -{\frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{\frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}} \\ \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$

Riemann curvature, $\sharp\sharp\flat\flat$: ${R} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} \\ -{\frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}} & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & -{\frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}} \\ \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]\end{matrix} \right]}}$

Ricci curvature, $\sharp\flat$: ${R} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} & 0 \\ 0 & \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}\end{matrix} \right]}}$

Gaussian curvature: $\frac{{{2}} {{{\sin\left( \theta\right)}^{2}}}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}$
trace-free Ricci, $\sharp\flat$: ${{(R^{TF})}} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} & 0 \\ 0 & \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}\end{matrix} \right]}}$

Einstein / trace-reversed Ricci curvature, $\sharp\flat$: ${G} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} & 0 \\ 0 & \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}\end{matrix} \right]}}$

Schouten, $\sharp\flat$: ${P} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}}$

Weyl, $\sharp\sharp\flat\flat$: ${C} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} \\ -{\frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{\frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}} \\ \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$

Weyl, $\flat\flat\flat\flat$: ${C} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} \\ -{\frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}} & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & -{\frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}} \\ \frac{{\sin\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}} & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]\end{matrix} \right]}}$

Plebanski, $\sharp\sharp\flat\flat$: ${P} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{1}{{{2}} {{{r}^{4}}}} \\ -{\frac{1}{{{2}} {{{r}^{4}}}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{\frac{1}{{{2}} {{{r}^{4}}}}} \\ \frac{1}{{{2}} {{{r}^{4}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$

divergence: ${{{{ A} ^i} _{,i}} + {{{{{{ \Gamma} ^i} _i} _j}} {{{ A} ^j}}}} = {{{{\frac{\partial {A^{\hat{\phi}}}}{\partial \phi}}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}} + {{{{A^{\hat{\theta}}}}} \cdot {{\cos\left( \theta\right)}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}} + {{{\frac{\partial {A^{\hat{\theta}}}}{\partial \theta}}} {{\frac{1}{r}}}}}$
geodesic:
${\overset{a\downarrow}{\left[ \begin{matrix} \ddot{\hat{\theta}} \\ \ddot{\hat{\phi}}\end{matrix} \right]}} = {\overset{a\downarrow}{\left[ \begin{matrix} {{{\dot{\hat{\phi}}}^{2}}} {{\cos\left( \theta\right)}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}} \\ {{-1}} {{\dot{\hat{\phi}}}} \cdot {{\dot{\hat{\theta}}}} \cdot {{\cos\left( \theta\right)}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}\end{matrix} \right]}}$

parallel propagators:

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

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

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

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

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

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

${ P} _{\phi}$ = $\exp\left( -{\int\limits_{{{\phi_L}}}^{{{\phi_R}}} {\left[ \begin{matrix} 0 & -{\cos\left( \theta\right)} \\ \cos\left( \theta\right) & 0\end{matrix} \right]}d \phi}\right)$ = $\left[ \begin{matrix} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} {{\left({{-{{\phi_L}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}} + {{\phi_R}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{{\left({{{\phi_L}}{-{{\phi_R}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}} \\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}\right)}}}{{{2}} {{\cos\left( \theta\right)}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}\end{matrix} \right]$

${{ P} _{\phi}}^{-1}$ = $\exp\left({\int\limits_{{{\phi_L}}}^{{{\phi_R}}} {\left[ \begin{matrix} 0 & -{\cos\left( \theta\right)} \\ \cos\left( \theta\right) & 0\end{matrix} \right]}d \phi}\right)$ = $\left[ \begin{matrix} \frac{{-{{\phi_L}}} + {{\phi_R}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{{\left({{-{{\phi_L}}} + {{\phi_R}} + {{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}{-{{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}}\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}} \\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}} {{\left({{1}{-{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}\right)}}}{{{2}} {{\cos\left( \theta\right)}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}\end{matrix} \right]$

propagator commutation:

[ ${ P} _{\theta}$ , ${ P} _{\phi}$ ] = ${ {\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]} {\left[ \begin{matrix} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}\right)}}} {{\left({{-{{\phi_L}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}}}} + {{\phi_R}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{{\left({{{\phi_L}}{-{{\phi_R}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}}}}\right)}} {{\cos\left( {\theta_L}\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}} \\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}\right)}}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}}\right)}}}{{{2}} {{\cos\left( {\theta_L}\right)}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}\right)}}}\end{matrix} \right]}}{-{ {\left[ \begin{matrix} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}\right)}}} {{\left({{-{{\phi_L}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}}}} + {{\phi_R}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{{\left({{{\phi_L}}{-{{\phi_R}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}}}}\right)}} {{\cos\left( {\theta_R}\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}} \\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}\right)}}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}}\right)}}}{{{2}} {{\cos\left( {\theta_R}\right)}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}\right)}}}\end{matrix} \right]} {\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]}}}$ = $\left[ \begin{matrix} \frac{{-{{\phi_L}}} + {{\phi_R}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}}}} + {{{{\phi_L}}} \cdot {{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}{-{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}}} + {{{{\phi_L}}} \cdot {{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}} + {\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}\right)}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}}}{-{{{{\phi_R}}} \cdot {{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}{-{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}}}}{-{{{{\phi_R}}} \cdot {{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}} + {\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}\right)}}}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}\right)}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{-{{{{{\phi_L}}^{3}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{\cos\left( {\theta_L}\right)}}}} + {{{{{\phi_R}}^{3}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{\cos\left( {\theta_L}\right)}}}{-{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{\cos\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{\cos\left( {\theta_L}\right)}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}{-{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}} {{\sqrt{{{{{{\phi_L}}^{4}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}} + {{{{{\phi_R}}^{4}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}{-{{{4}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{3}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{4}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{3}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{6}} {{{{\phi_L}}^{2}}} {{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}}} {{\cos\left( {\theta_R}\right)}}}} + {{{{\phi_L}}} \cdot {{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}} + {\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}\right)}} {{\sqrt{{{{{{\phi_L}}^{4}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}} + {{{{{\phi_R}}^{4}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}{-{{{4}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{3}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{4}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{3}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{6}} {{{{\phi_L}}^{2}}} {{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}}} {{\cos\left( {\theta_R}\right)}}} + {{{{\phi_R}}} \cdot {{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}{-{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}} {{\sqrt{{{{{{\phi_L}}^{4}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}} + {{{{{\phi_R}}^{4}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}{-{{{4}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{3}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{4}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{3}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{6}} {{{{\phi_L}}^{2}}} {{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}}} {{\cos\left( {\theta_R}\right)}}}{-{{{{\phi_R}}} \cdot {{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}} + {\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}\right)}} {{\sqrt{{{{{{\phi_L}}^{4}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}} + {{{{{\phi_R}}^{4}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}{-{{{4}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{3}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{4}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{3}}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{6}} {{{{\phi_L}}^{2}}} {{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}}} {{\cos\left( {\theta_R}\right)}}}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{\cos\left( {\theta_L}\right)}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{\cos\left( {\theta_L}\right)}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{\cos\left( {\theta_L}\right)}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}} {{{\cos\left( {\theta_R}\right)}^{2}}} {{\cos\left( {\theta_L}\right)}}}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}} {{\left({{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}\right)}}} \\ \frac{{-{{{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}} {{\cos\left( {\theta_R}\right)}}}} + {{{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}} {{\cos\left( {\theta_R}\right)}}} + {{{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}{-{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}} {{\cos\left( {\theta_L}\right)}}}{-{{{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}} + {\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}} {{\cos\left( {\theta_L}\right)}}}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}\right)}} {{\cos\left( {\theta_R}\right)}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}} {{\cos\left( {\theta_L}\right)}}} & \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}}}\right)}{-{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}{-{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}}\right)}}{-{\exp\left({{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}} + {\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_R}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_R}\right)}^{2}}}}}}}\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( {\theta_L}\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( {\theta_L}\right)}^{2}}}}}}\right)}}}\end{matrix} \right]$

propagator partials
${{\frac{\partial}{\partial \theta}}\left( \left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]\right)} = {\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}$
${{\frac{\partial}{\partial \phi}}\left( \left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]\right)} = {\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}$
${{\frac{\partial}{\partial \theta}}\left( \left[ \begin{matrix} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} {{\left({{-{{\phi_L}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}} + {{\phi_R}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{{\left({{{\phi_L}}{-{{\phi_R}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}} \\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}\right)}}}{{{2}} {{\cos\left( \theta\right)}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}\end{matrix} \right]\right)} = {\left[ \begin{matrix} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}} {{\left({{-{{{\phi_L}}^{4}}}{-{{{\phi_R}}^{4}}} + {{{{{\phi_L}}^{4}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}} + {{{{{\phi_R}}^{4}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}{-{{{6}} {{{{\phi_L}}^{2}}} {{{{\phi_R}}^{2}}}}} + {{{6}} {{{{\phi_L}}^{2}}} {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}} + {{{4}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{3}}}} + {{{4}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{3}}}}{-{{{4}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}{-{{{4}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}}\right)}}}{{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}} {{\left({{{{\phi_L}}^{2}} + {{{\phi_R}}^{2}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}}}\right)}}} & \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} {{\sin\left( \theta\right)}} {{\left({{-{{{{\phi_L}}} \cdot {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}} + {{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}} + {{{{\phi_R}}} \cdot {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}{-{{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}}{-{{{{{\phi_L}}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{{\phi_R}}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{{\phi_L}}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{1}{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}}} + {{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{1}{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}} + {{{{{\phi_R}}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{1}{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}}{-{{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{1}{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}}}{-{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{1}{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}}} + {{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{1}{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{1}{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}}{-{{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{1}{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}}}}\right)}}}{{{2}} {{\left({{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}\right)}}} \\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} {{\sin\left( \theta\right)}} {{\left({{-{{{{{\phi_L}}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_L}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{{\phi_R}}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{{\phi_R}}^{3}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{\phi_L}}} \cdot {{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{\phi_R}}} \cdot {{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}}{{{2}} {{\left({{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} & \frac{{{\left({{-{{{\phi_L}}^{2}}}{-{{{\phi_R}}^{2}}} + {{{{{\phi_L}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}} + {{{{{\phi_R}}^{2}}} {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}}}{-{{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}}\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\end{matrix} \right]}$
${{\frac{\partial}{\partial \phi}}\left( \left[ \begin{matrix} \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} {{\left({{-{{\phi_L}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}} + {{\phi_R}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}\right)}}}{{{2}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{{\left({{{\phi_L}}{-{{\phi_R}}}{-{{{{\phi_L}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}} + {{{{\phi_R}}} \cdot {{\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}}}\right)}} {{\cos\left( \theta\right)}}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}} \\ \frac{{{\frac{1}{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}} {{\left({{-{1}} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}\right)}}}{{{2}} {{\cos\left( \theta\right)}} {{\left({{-{{\phi_L}}} + {{\phi_R}}}\right)}}} & \frac{{1} + {\exp\left({{{2}} {{\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}}}\right)}}{{{2}} {{\exp\left({\sqrt{{-{{{{{\phi_L}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{{\phi_R}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{\phi_L}}} \cdot {{{\phi_R}}} \cdot {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}\end{matrix} \right]\right)} = {\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}$
volume element: ${{{r}^{2}}} {{\sin\left( \theta\right)}}$
volume integral: ${-{{{\Delta (cos(\theta))}} \cdot {{{r}^{2}}}}} {{\Delta \phi}}$
finite volume (0,0)-form:
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{{\mathcal{V}(x_C)}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$

${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{ {-{{{\Delta (cos(\theta))}} \cdot {{{r}^{2}}}}} {{\Delta \phi}}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$

${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{ {-{{{\Delta (cos(\theta))}} \cdot {{{r}^{2}}}}} {{\Delta \phi}}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$

${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{{S(x_C)}}} \cdot {{\Delta t}}}}$