Christoffel symbols in polar coordinates

Differentiating the coordinate map gives vectors $(\cos \theta ,\sin \theta )$ and $(-r\sin \theta ,r\cos \theta )$, whose inner products are $1,0,r^2$. Thus $G=\mathrm{diag}(1,r^2)$, $G^{-1}=\mathrm{diag}(1,r^{-2})$, and the only nonzero metric derivative is ${\mathrm{\partial }}_r{g}_{\theta \theta }=2r$. Formula [F1] gives ${\mathrm{\Gamma }}^r{}_{\theta \theta }=-(2r)\mathrm{/}2=-r$ and ${\mathrm{\Gamma }}^{\theta }{}_{r\theta }={\mathrm{\Gamma }}^{\theta }{}_{\theta r}=(2r)\mathrm{/}(2r^2)=1\mathrm{/}r$.

The remaining entries are ${\mathrm{\Gamma }}^r{}_{rr}={\mathrm{\Gamma }}^r{}_{r\theta }={\mathrm{\Gamma }}^r{}_{\theta r}={\mathrm{\Gamma }}^{\theta }{}_{rr}={\mathrm{\Gamma }}^{\theta }{}_{\theta \theta }=0$. For the first and fourth, every metric derivative is zero. In the middle two the only potentially nonzero term ${\mathrm{\partial }}_r{g}_{\theta \theta }$ is multiplied by $g^{r\theta }=0$; in the last, the term $-{\mathrm{\partial }}_r{g}_{\theta \theta }$ is multiplied by $g^{\theta r}=0$. At $r=1$ the three displayed nonzero entries are $-1,1,1$. None of these formulas applies at $r=0$: there the angular coordinate vector vanishes and the coordinate map is not a chart. There is therefore no singularity of the Euclidean metric asserted at the origin.