spiral, 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 $e_I = \{e_{x}, e_{y}\}$
transform from basis to coordinate:
${{{ \tilde{e}}_r}^r} = {1}$;
${{{ \tilde{e}}_{\phi}}^{\phi}} = {1}$
transform from coorinate to basis:
${{{ \tilde{e}}^r}_r} = {1}$;
${{{ \tilde{e}}^{\phi}}_{\phi}} = {1}$
tensor index associated with coordinate $r$ is index $r$ with operator $e_{r}(\zeta) = $$\frac{\partial \zeta}{\partial r}$
tensor index associated with coordinate $\phi$ is index $\phi$ with operator $e_{\phi}(\zeta) = $$\frac{\partial \zeta}{\partial \phi}$
flat metric:
${{{ \eta}_x}_x} = {1}$;
${{{ \eta}_y}_y} = {1}$
chart in embedded coordinates:
${{ u}^x} = {{{r}} {{cos\left( {{\phi} + {log\left( r\right)}}\right)}}}$;
${{ u}^y} = {{{r}} {{sin\left( {{\phi} + {log\left( r\right)}}\right)}}}$
basis operators applied to chart:
${{{ e}_u}^I} = {{{ u}^I}_{,u}}$
${{{ e}_r}^x} = {{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}}$;
${{{ e}_r}^y} = {{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}}$;
${{{ e}_{\phi}}^x} = {-{{{r}} {{sin\left( {{\phi} + {log\left( r\right)}}\right)}}}}$;
${{{ e}_{\phi}}^y} = {{{r}} {{cos\left( {{\phi} + {log\left( r\right)}}\right)}}}$
${{{ e}^r}_x} = {cos\left( {{\phi} + {log\left( r\right)}}\right)}$;
${{{ e}^r}_y} = {sin\left( {{\phi} + {log\left( r\right)}}\right)}$;
${{{ e}^{\phi}}_x} = {{\frac{1}{r}}{({-{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} + {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}})}}$;
${{{ e}^{\phi}}_y} = {{\frac{1}{r}}{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}}$
${{{{{ e}_u}^I}} {{{{ e}^v}_I}}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{matrix} 1 & 0 \\ 0 & 1\end{matrix}\right]}}$
${{{{{ e}_u}^I}} {{{{ e}^u}_J}}} = {\overset{I\downarrow J\rightarrow}{\left[\begin{matrix} 1 & 0 \\ 0 & 1\end{matrix}\right]}}$
basis determinant: ${det(e)} = {r}$
${{{{ c}_a}_b}^c} = {0}$
${{{ g}_u}_v} = {{{{{ e}_u}^I}} {{{{ e}_v}^J}} {{{{ \eta}_I}_J}}}$
${{{ g}_r}_r} = {{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}}$;
${{{ g}_r}_{\phi}} = {r}$;
${{{ g}_{\phi}}_r} = {r}$;
${{{ g}_{\phi}}_{\phi}} = {{r}^{2}}$
${{{ g}_u}_v} = {{{{{ e}_u}^I}} {{{{ e}_v}^J}} {{{{ \eta}_I}_J}}}$
metric determinant: ${det(g)} = {{r}^{2}}$
${{{{ \Gamma}_a}_b}_c} = {{{{\frac{1}{2}}{({1})}}} {{({{{{{{{{ g}_a}_b}_{,c}} + {{{{ g}_a}_c}_{,b}}} - {{{{ g}_b}_c}_{,a}}} + {{{{ c}_a}_b}_c} + {{{{ c}_a}_c}_b}} - {{{{ c}_c}_b}_a}})}}}$
commutation coefficients:
${{{{ c}_a}_b}^c} = {0}$
metric:
${{{ g}_r}_r} = {{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}}$;
${{{ g}_r}_{\phi}} = {r}$;
${{{ g}_{\phi}}_r} = {r}$;
${{{ g}_{\phi}}_{\phi}} = {{r}^{2}}$
metric inverse:
${{{ g}^r}^r} = {1}$;
${{{ g}^r}^{\phi}} = {-{{\frac{1}{r}}{({1})}}}$;
${{{ g}^{\phi}}^r} = {-{{\frac{1}{r}}{({1})}}}$;
${{{ g}^{\phi}}^{\phi}} = {\frac{{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}}{{r}^{2}}}$
metric derivative:
${{{{ {\partial g}}_r}_{\phi}}_r} = {1}$;
${{{{ {\partial g}}_{\phi}}_r}_r} = {1}$;
${{{{ {\partial g}}_{\phi}}_{\phi}}_r} = {{{2}} {{r}}}$
1st kind Christoffel:
${{{{ \Gamma}_r}_{\phi}}_{\phi}} = {-{r}}$;
${{{{ \Gamma}_{\phi}}_r}_r} = {1}$;
${{{{ \Gamma}_{\phi}}_r}_{\phi}} = {r}$;
${{{{ \Gamma}_{\phi}}_{\phi}}_r} = {r}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma}^r}_r}_r} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{ \Gamma}^r}_r}_{\phi}} = {-{1}}$;
${{{{ \Gamma}^r}_{\phi}}_r} = {-{1}}$;
${{{{ \Gamma}^r}_{\phi}}_{\phi}} = {-{r}}$;
${{{{ \Gamma}^{\phi}}_r}_r} = {\frac{{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}}{{r}^{2}}}$;
${{{{ \Gamma}^{\phi}}_r}_{\phi}} = {{\frac{1}{r}}{({{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}})}}$;
${{{{ \Gamma}^{\phi}}_{\phi}}_r} = {{\frac{1}{r}}{({{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}})}}$;
${{{{ \Gamma}^{\phi}}_{\phi}}_{\phi}} = {1}$
connection coefficients derivative:
${{{{{ {\partial \Gamma}}^r}_r}_r}_r} = {\frac{1}{{r}^{2}}}$;
${{{{{ {\partial \Gamma}}^r}_{\phi}}_{\phi}}_r} = {-{1}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_r}_r}_r} = {-{\frac{4}{{r}^{3}}}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_r}_{\phi}}_r} = {-{\frac{2}{{r}^{2}}}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_{\phi}}_r}_r} = {-{\frac{2}{{r}^{2}}}}$
connection coefficients squared:
${{{{{ {(\Gamma^2)}}^r}_r}_r}_r} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {(\Gamma^2)}}^r}_r}_r}_{\phi}} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{{ {(\Gamma^2)}}^r}_r}_{\phi}}_r} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{{ {(\Gamma^2)}}^r}_r}_{\phi}}_{\phi}} = {-{1}}$;
${{{{{ {(\Gamma^2)}}^r}_{\phi}}_r}_r} = {-{{\frac{1}{r}}{({1})}}}$;
${{{{{ {(\Gamma^2)}}^r}_{\phi}}_{\phi}}_r} = {-{1}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_r}_r}_r} = {\frac{{{2}} {{({{{3} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{{2}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}}})}}}{{r}^{3}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_r}_r}_{\phi}} = {\frac{{{2}} {{({{{3} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{{2}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}}})}}}{{r}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_r}_r} = {\frac{{{2}} {{({{{3} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{{2}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}}})}}}{{r}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_{\phi}}_{\phi}} = {-{1}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R}^{\phi}}_r}_r}_{\phi}} = {\frac{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}{{r}^{2}}}$;
${{{{{ R}^{\phi}}_r}_{\phi}}_r} = {\frac{-{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}}{{r}^{2}}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R}^{\phi}}^r}_r}_{\phi}} = {\frac{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}{{r}^{2}}}$;
${{{{{ R}^{\phi}}^r}_{\phi}}_r} = {\frac{-{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}}{{r}^{2}}}$;
${{{{{ R}^{\phi}}^{\phi}}_r}_{\phi}} = {\frac{-{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}}{{r}^{3}}}$;
${{{{{ R}^{\phi}}^{\phi}}_{\phi}}_r} = {\frac{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}{{r}^{3}}}$
Ricci curvature, $\sharp\flat$:
${{{ R}^r}_r} = {\frac{-{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}}{{r}^{2}}}$;
${{{ R}^{\phi}}_r} = {\frac{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}{{r}^{3}}}$
Gaussian curvature:
$\frac{-{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}}{{r}^{2}}$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}}^r}_r} = {\frac{-{{{2}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}}{{r}^{2}}}$;
${{{ {(R^{TF})}}^{\phi}}_r} = {\frac{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}{{r}^{3}}}$;
${{{ {(R^{TF})}}^{\phi}}_{\phi}} = {\frac{{{2}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}{{r}^{2}}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G}^{\phi}}_r} = {\frac{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}{{r}^{3}}}$;
${{{ G}^{\phi}}_{\phi}} = {\frac{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}{{r}^{2}}}$
Schouten, $\sharp\flat$:
${{{ P}^a}_b} = {0}$
Weyl, $\sharp\sharp\flat\flat$:
${{{{{ C}^{\phi}}^r}_r}_{\phi}} = {\frac{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}{{r}^{2}}}$;
${{{{{ C}^{\phi}}^r}_{\phi}}_r} = {\frac{-{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}}{{r}^{2}}}$;
${{{{{ C}^{\phi}}^{\phi}}_r}_{\phi}} = {\frac{-{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}}{{r}^{3}}}$;
${{{{{ C}^{\phi}}^{\phi}}_{\phi}}_r} = {\frac{{{4}} {{({{{1} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}} + {{{{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}})}}}{{r}^{3}}}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C}_r}_r}_r}_{\phi}} = {{\frac{1}{r}}{({-{{{4}} {{({{{121} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{{125}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{244}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{{{{127}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} - {{{123}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{{{2}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{4}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}}})}}}})}}$;
${{{{{ C}_r}_r}_{\phi}}_r} = {{\frac{1}{r}}{({{{4}} {{({{{121} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{{125}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{244}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{{{{127}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} - {{{123}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} + {{{{2}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{4}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}}})}}})}}$;
${{{{{ C}_{\phi}}_r}_r}_{\phi}} = {-{{{4}} {{({{{{121} - {{{244}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{{125}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} + {{{127}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} + {{{{{2}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{123}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{4}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}}})}}}}$;
${{{{{ C}_{\phi}}_r}_{\phi}}_r} = {{{4}} {{({{{{121} - {{{244}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{{125}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{2}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} + {{{127}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} + {{{{{2}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{123}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{4}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}}})}}}$
Plebanski, $\sharp\sharp\flat\flat$:
${{{{{ P}^r}^{\phi}}_r}_{\phi}} = {\frac{-{{{2}} {{({{911965} + {{{801}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{8}}}} + {{{{{{{993746}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{1864424}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{42084}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{10}}}}} - {{{23778}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} + {{{{998634}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{957}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{{148}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}} - {{{934654}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{144}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{22833}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{{{1610}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}} - {{{12}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} - {{{65578}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} - {{{12}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{8}}}}}})}}}}{{{3}} {{{r}^{4}}}}}$;
${{{{{ P}^r}^{\phi}}_{\phi}}_r} = {\frac{{{2}} {{({{911965} + {{{801}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{8}}}} + {{{{{{{993746}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{1864424}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{42084}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{10}}}}} - {{{23778}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} + {{{{998634}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{957}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{{148}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}} - {{{934654}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{144}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{22833}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{{{1610}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}} - {{{12}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} - {{{65578}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} - {{{12}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{8}}}}}})}}}{{{3}} {{{r}^{4}}}}}$;
${{{{{ P}^{\phi}}^r}_r}_{\phi}} = {\frac{{{2}} {{({{911965} + {{{801}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{8}}}} + {{{{{{{993746}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{1864424}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{42084}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{10}}}}} - {{{23778}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} + {{{{998634}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{957}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{{148}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}} - {{{934654}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{144}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{22833}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{{{1610}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}} - {{{12}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} - {{{65578}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} - {{{12}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{8}}}}}})}}}{{{3}} {{{r}^{4}}}}}$;
${{{{{ P}^{\phi}}^r}_{\phi}}_r} = {\frac{-{{{2}} {{({{911965} + {{{801}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{8}}}} + {{{{{{{993746}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} - {{{1864424}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{42084}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{10}}}}} - {{{23778}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} + {{{{998634}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}} - {{{4}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{957}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{{148}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}} - {{{934654}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}}}} - {{{144}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} + {{{22833}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}} + {{{{{{1610}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}} - {{{12}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{6}}}}} - {{{65578}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{4}}}}} - {{{12}} {{{sin\left( {{\phi} + {log\left( r\right)}}\right)}^{2}}} {{{cos\left( {{\phi} + {log\left( r\right)}}\right)}^{8}}}}}})}}}}{{{3}} {{{r}^{4}}}}}$
divergence: ${{{{ A}^i}_{,i}} + {{{{{{ \Gamma}^i}_i}_j}} {{{ A}^j}}}} = {{\frac{\partial {A^{r}}}{\partial r}} + {\frac{\partial {A^{\phi}}}{\partial \phi}} + {{{{A^{r}}}} \cdot {{{\frac{1}{r}}{({1})}}}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{r} \\ \ddot{\phi}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} {{{2}} {{\dot{\phi}}} \cdot {{\dot{r}}}} + {{{r}} {{{\dot{\phi}}^{2}}}} + {{{{\dot{r}}^{2}}} {{{\frac{1}{r}}{({1})}}}} \\ {{{-4}} {{\dot{\phi}}} \cdot {{\dot{r}}} \cdot {{r}} {{\frac{1}{{r}^{2}}}}} + {{{-1}} {{{\dot{\phi}}^{2}}} {{{r}^{2}}} {{\frac{1}{{r}^{2}}}}} + {{{-2}} {{{\dot{r}}^{2}}} {{\frac{1}{{r}^{2}}}}}\end{matrix}\right]}}$
parallel propagators:
${{[\Gamma_r]}} = {\left[\begin{matrix} -{{\frac{1}{r}}{({1})}} & -{1} \\ \frac{{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}}{{r}^{2}} & {\frac{1}{r}}{({{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}})}\end{matrix}\right]}$
$\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} -{{\frac{1}{r}}{({1})}} & -{1} \\ \frac{{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}}{{r}^{2}} & {\frac{1}{r}}{({{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}})}\end{matrix}\right]\right)$
= $\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} -{{\frac{1}{r}}{({1})}} & -{1} \\ \frac{{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}}{{r}^{2}} & {\frac{1}{r}}{({{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}})}\end{matrix}\right]\right)$
${ P}_r$ = ${ⅇ}^{( -{({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} -{{\frac{1}{r}}{({1})}} & -{1} \\ \frac{{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}}{{r}^{2}} & {\frac{1}{r}}{({{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}})}\end{matrix}\right]\right)})})}$
negIntConn $-{({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} -{{\frac{1}{r}}{({1})}} & -{1} \\ \frac{{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}}{{r}^{2}} & {\frac{1}{r}}{({{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}})}\end{matrix}\right]\right)})}$
negIntConn $-{({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} -{{\frac{1}{r}}{({1})}} & -{1} \\ \frac{{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}}{{r}^{2}} & {\frac{1}{r}}{({{{({{cos\left( {{\phi} + {log\left( r\right)}}\right)} - {sin\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}} + {{({{sin\left( {{\phi} + {log\left( r\right)}}\right)} + {cos\left( {{\phi} + {log\left( r\right)}}\right)}})}^{2}}})}\end{matrix}\right]\right)})}$