Christoffelsymbole

${\mathrm{\Gamma }}_{\mu \nu }^{\alpha }=\frac{1}{2}{g}^{\alpha \sigma }(g_{\sigma \mu ,\nu }+g_{\sigma \nu ,\mu }-g_{\mu \nu ,\sigma })$

${\mathrm{\Gamma }}_{\mu \nu }^{\alpha }=\frac{\partial x^{\alpha }}{\partial {\xi }^{\beta }}\frac{{\partial }^2{\xi }^{\beta }}{\partial x^{\mu }\partial x^{\nu }}$

Verhalten bei Koordinatentransformation

$\mathrm{\Gamma }{\text{'}}_{\mu \nu }^{\alpha }=\frac{\partial x{\text{'}}^{\alpha }}{\partial x^{\beta }}\frac{\partial x^{\rho }}{\partial x{\text{'}}^{\mu }}\frac{\partial x^{\sigma }}{\partial x{\text{'}}^{\nu }}{\mathrm{\Gamma }}_{\rho \sigma }^{\beta }+\frac{\partial x{\text{'}}^{\alpha }}{\partial x^{\beta }}\frac{{\partial }^2{x}^{\beta }}{\partial x{\text{'}}^{\mu }\partial x{\text{'}}^{\nu }}$