= Christoffel symbols of a radially conformal polar metric
{c}
{title2=$ds^2=e^{2\chi(r)}(dr^2+r^2d\theta^2)$}
For the displayed <metric tensor>, its inverse has components $g^{rr}=e^{-2\chi}$ and $g^{\theta\theta}=e^{-2\chi}/r^2$. The nonzero second-kind <Christoffel symbols> are $\Gamma^r{}_{rr}=\chi'$, $\Gamma^r{}_{\theta\theta}=-r-r^2\chi'$ and $\Gamma^\theta{}_{r\theta}=\Gamma^\theta{}_{\theta r}=1/r+\chi'$. They follow by differentiating the two diagonal metric entries and raising the first index of the first-kind connection coefficients. <Polar coordinates> require treating $r=0$ separately.
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