Loi de transformation des symboles de Christoffel

Avec $\Gamma =0$ en cartésiennes, ${\tilde{\Gamma }}_{bc}^a=\frac{\partial y^a}{\partial x^i}\frac{{\partial }^2{x}^i}{\partial y^b\partial y^c}$. On a $\frac{{\partial }^{2}x}{\partial {\theta }^2}=(-r\cos \theta ,-r\sin \theta )$ et $\frac{{\partial }^{2}x}{\partial r\partial \theta }=(-\sin \theta ,\cos \theta )$ ; les lignes de $J^{-1}$ sont $\frac{\partial r}{\partial x^i}=(\cos \theta ,\sin \theta )$ et $\frac{\partial \theta }{\partial x^i}=(-\frac{\sin \theta }{r},\frac{\cos \theta }{r})$. D'où ${\Gamma }_{\theta \theta }^r=-r{\cos }^2\theta -r{\sin }^2\theta =-r$ et ${\Gamma }_{r\theta }^{\theta }=\frac{{\sin }^2\theta }{r}+\frac{{\cos }^2\theta }{r}=\frac{1}{r}$.