polar, anholonomic, conformal
chart coordinates: $x^\tilde{\mu} = \{r, \theta\}$
chart coordinate basis: $e_\tilde{\mu} = \{e_{\tilde{r}}, e_{\tilde{\theta}}\}$
embedding coordinates: $u^I = \{x, y\}$
embedding basis $e_I = \{e_{x}, e_{y}\}$
flat metric:
${{{ \eta} _x} _x} = {1}$;
${{{ \eta} _y} _y} = {1}$
transform from basis to coordinate:
${{{ \tilde{e}} _r} ^r} = {\sqrt{r}}$;
${{{ \tilde{e}} _{\theta}} ^{\theta}} = {\sqrt{r}}$
transform from coorinate to basis:
${{{ \tilde{e}} ^r} _r} = {\frac{1}{\sqrt{r}}}$;
${{{ \tilde{e}} ^{\theta}} _{\theta}} = {\frac{1}{\sqrt{r}}}$
tensor index associated with coordinate $r$ has operator $e_{r}(\zeta) = $$\frac{\frac{\partial \zeta}{\partial r}}{\sqrt{r}}$
tensor index associated with coordinate $\theta$ has operator $e_{\theta}(\zeta) = $$\frac{\frac{\partial \zeta}{\partial \theta}}{\sqrt{r}}$
chart in embedded coordinates:
${{ u} ^x} = {{{r}} {{\cos\left( \theta\right)}}}$;
${{ u} ^y} = {{{r}} {{\sin\left( \theta\right)}}}$
basis operators applied to chart:
${{{ e} _u} ^I} = {{{ u} ^I} _{,u}}$
${{{ e} _{\hat{r}}} ^x} = {\frac{\cos\left( \theta\right)}{\sqrt{r}}}$;
${{{ e} _{\hat{r}}} ^y} = {\frac{\sin\left( \theta\right)}{\sqrt{r}}}$;
${{{ e} _{\hat{\theta}}} ^x} = {-{{{\sqrt{r}}} {{\sin\left( \theta\right)}}}}$;
${{{ e} _{\hat{\theta}}} ^y} = {{{\sqrt{r}}} {{\cos\left( \theta\right)}}}$
${{{ e} ^{\hat{r}}} _x} = {{{\sqrt{r}}} {{\cos\left( \theta\right)}}}$;
${{{ e} ^{\hat{r}}} _y} = {{{\sqrt{r}}} {{\sin\left( \theta\right)}}}$;
${{{ e} ^{\hat{\theta}}} _x} = {-{\frac{\sin\left( \theta\right)}{\sqrt{r}}}}$;
${{{ e} ^{\hat{\theta}}} _y} = {\frac{\cos\left( \theta\right)}{\sqrt{r}}}$
${{{{{ e} _u} ^I}} {{{{ e} ^v} _I}}} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^u} _J}}} = {\overset{I\downarrow J\rightarrow}{\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]}}$
basis determinant: ${det(e)} = {1}$
${{{{ c} _{\hat{r}}} _{\hat{\theta}}} ^{\hat{\theta}}} = {-{\frac{1}{{{2}} {{{{r}} {{\sqrt{r}}}}}}}}$;
${{{{ c} _{\hat{\theta}}} _{\hat{r}}} ^{\hat{\theta}}} = {\frac{1}{{{2}} {{{{r}} {{\sqrt{r}}}}}}}$
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
${{{ g} _{\hat{r}}} _{\hat{r}}} = {\frac{1}{r}}$;
${{{ g} _{\hat{\theta}}} _{\hat{\theta}}} = {r}$
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
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:
${{{{ c} _{\hat{r}}} _{\hat{\theta}}} ^{\hat{\theta}}} = {-{\frac{1}{{{2}} {{{{r}} {{\sqrt{r}}}}}}}}$;
${{{{ c} _{\hat{\theta}}} _{\hat{r}}} ^{\hat{\theta}}} = {\frac{1}{{{2}} {{{{r}} {{\sqrt{r}}}}}}}$
metric:
${{{ g} _{\hat{r}}} _{\hat{r}}} = {\frac{1}{r}}$;
${{{ g} _{\hat{\theta}}} _{\hat{\theta}}} = {r}$
metric inverse:
${{{ g} ^{\hat{r}}} ^{\hat{r}}} = {r}$;
${{{ g} ^{\hat{\theta}}} ^{\hat{\theta}}} = {\frac{1}{r}}$
metric derivative:
${{{{ {\partial g}} _{\hat{r}}} _{\hat{r}}} _{\hat{r}}} = {-{\frac{1}{{{{r}^{2}}} {{\sqrt{r}}}}}}$;
${{{{ {\partial g}} _{\hat{\theta}}} _{\hat{\theta}}} _{\hat{r}}} = {\frac{1}{\sqrt{r}}}$
1st kind Christoffel:
${{{{ \Gamma} _{\hat{r}}} _{\hat{r}}} _{\hat{r}}} = {-{\frac{1}{{{2}} {{{{{r}^{2}}} {{\sqrt{r}}}}}}}}$;
${{{{ \Gamma} _{\hat{r}}} _{\hat{\theta}}} _{\hat{\theta}}} = {-{\frac{1}{\sqrt{r}}}}$;
${{{{ \Gamma} _{\hat{\theta}}} _{\hat{r}}} _{\hat{\theta}}} = {{\frac{1}{2}} {\frac{1}{\sqrt{r}}}}$;
${{{{ \Gamma} _{\hat{\theta}}} _{\hat{\theta}}} _{\hat{r}}} = {\frac{1}{\sqrt{r}}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma} ^{\hat{r}}} _{\hat{r}}} _{\hat{r}}} = {-{\frac{1}{{{2}} {{{{r}} {{\sqrt{r}}}}}}}}$;
${{{{ \Gamma} ^{\hat{r}}} _{\hat{\theta}}} _{\hat{\theta}}} = {-{\sqrt{r}}}$;
${{{{ \Gamma} ^{\hat{\theta}}} _{\hat{r}}} _{\hat{\theta}}} = {\frac{1}{{{2}} {{{{r}} {{\sqrt{r}}}}}}}$;
${{{{ \Gamma} ^{\hat{\theta}}} _{\hat{\theta}}} _{\hat{r}}} = {\frac{1}{{{r}} {{\sqrt{r}}}}}$
connection coefficients derivative:
${{{{{ {\partial \Gamma}} ^{\hat{r}}} _{\hat{r}}} _{\hat{r}}} _{\hat{r}}} = {\frac{3}{{{4}} {{{r}^{3}}}}}$;
${{{{{ {\partial \Gamma}} ^{\hat{r}}} _{\hat{\theta}}} _{\hat{\theta}}} _{\hat{r}}} = {-{\frac{1}{{{2}} {{r}}}}}$;
${{{{{ {\partial \Gamma}} ^{\hat{\theta}}} _{\hat{r}}} _{\hat{\theta}}} _{\hat{r}}} = {-{\frac{3}{{{4}} {{{r}^{3}}}}}}$;
${{{{{ {\partial \Gamma}} ^{\hat{\theta}}} _{\hat{\theta}}} _{\hat{r}}} _{\hat{r}}} = {-{\frac{3}{{{2}} {{{r}^{3}}}}}}$
connection coefficients squared:
${{{{{ {(\Gamma^2)}} ^{\hat{r}}} _{\hat{r}}} _{\hat{r}}} _{\hat{r}}} = {\frac{1}{{{4}} {{{r}^{3}}}}}$;
${{{{{ {(\Gamma^2)}} ^{\hat{r}}} _{\hat{r}}} _{\hat{\theta}}} _{\hat{\theta}}} = {-{\frac{1}{r}}}$;
${{{{{ {(\Gamma^2)}} ^{\hat{r}}} _{\hat{\theta}}} _{\hat{r}}} _{\hat{\theta}}} = {\frac{1}{{{2}} {{r}}}}$;
${{{{{ {(\Gamma^2)}} ^{\hat{r}}} _{\hat{\theta}}} _{\hat{\theta}}} _{\hat{r}}} = {-{\frac{1}{{{2}} {{r}}}}}$;
${{{{{ {(\Gamma^2)}} ^{\hat{\theta}}} _{\hat{r}}} _{\hat{r}}} _{\hat{\theta}}} = {\frac{1}{{{2}} {{{r}^{3}}}}}$;
${{{{{ {(\Gamma^2)}} ^{\hat{\theta}}} _{\hat{r}}} _{\hat{\theta}}} _{\hat{r}}} = {-{\frac{1}{{{2}} {{{r}^{3}}}}}}$;
${{{{{ {(\Gamma^2)}} ^{\hat{\theta}}} _{\hat{\theta}}} _{\hat{r}}} _{\hat{r}}} = {\frac{1}{{{4}} {{{r}^{3}}}}}$;
${{{{{ {(\Gamma^2)}} ^{\hat{\theta}}} _{\hat{\theta}}} _{\hat{\theta}}} _{\hat{\theta}}} = {-{\frac{1}{r}}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R} ^a} _b} _c} _d} = {0}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R} ^a} ^b} _c} _d} = {0}$
Ricci curvature, $\sharp\flat$:
${{{ R} ^a} _b} = {0}$
Gaussian curvature:
$0$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}} ^a} _b} = {0}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G} ^a} _b} = {0}$
Schouten, $\sharp\flat$:
${{{ P} ^a} _b} = {0}$
Weyl, $\sharp\sharp\flat\flat$:
${{{{{ C} ^a} ^b} _c} _d} = {0}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C} _a} _b} _c} _d} = {0}$
Plebanski, $\sharp\sharp\flat\flat$:
${{{{{ P} ^a} ^b} _c} _d} = {0}$
divergence: ${{{{ A} ^i} _{,i}} + {{{{{{ \Gamma} ^i} _i} _j}} {{{ A} ^j}}}} = {{{{\frac{1}{\sqrt{r}}}} {{\frac{\partial {A^{\hat{\theta}}}}{\partial \theta}}}} + {{{\frac{1}{\sqrt{r}}}} {{\frac{\partial {A^{\hat{r}}}}{\partial r}}}} + {{{\frac{1}{2}}} {{{A^{\hat{r}}}}} \cdot {{\frac{1}{r}}} {{\frac{1}{\sqrt{r}}}}}}$
geodesic:
${\overset{a\downarrow}{\left[ \begin{matrix} \ddot{\hat{r}} \\ \ddot{\hat{\theta}}\end{matrix} \right]}} = {\overset{a\downarrow}{\left[ \begin{matrix} {{{{\dot{\hat{\theta}}}^{2}}} {{\sqrt{r}}}} + {{{\frac{1}{2}}} {{{\dot{\hat{r}}}^{2}}} {{\frac{1}{r}}} {{\frac{1}{\sqrt{r}}}}} \\ {{-3}} \cdot {{\frac{1}{2}}} {{\dot{\hat{\theta}}}} \cdot {{\dot{\hat{r}}}} \cdot {{\frac{1}{{r}^{{{3}} \cdot {{\frac{1}{2}}}}}}}\end{matrix} \right]}}$
parallel propagators:
${{[\Gamma_r]}} = {\left[ \begin{matrix} -{\frac{1}{{{2}} {{r}}}} & 0 \\ 0 & \frac{1}{{{2}} {{r}}}\end{matrix} \right]}$
$\int\limits_{{{r_L}}}^{{{r_R}}} {\left[ \begin{matrix} -{\frac{1}{{{2}} {{r}}}} & 0 \\ 0 & \frac{1}{{{2}} {{r}}}\end{matrix} \right]}d r$
= $\left[ \begin{matrix} \log\left( {\sqrt{{\frac{1}{{r_R}}} {{r_L}}}}\right) & 0 \\ 0 & \log\left( {\sqrt{{\frac{1}{{r_L}}} {{r_R}}}}\right)\end{matrix} \right]$
${ P} _r$ = $\exp\left( -{\int\limits_{{{r_L}}}^{{{r_R}}} {\left[ \begin{matrix} -{\frac{1}{{{2}} {{r}}}} & 0 \\ 0 & \frac{1}{{{2}} {{r}}}\end{matrix} \right]}d r}\right)$