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