Christoffelsymbole

$\Gamma _{{\color {Red}\mu }{\color {Violet}\nu }}^{\color {Orange}\alpha }={\frac {1}{2}}{{g}^{{\color {Orange}\alpha }\sigma }}\left({{g}_{\sigma {\color {Violet}\mu },{\color {Red}\nu }}}+{{g}_{\sigma {\color {Red}\nu },{\color {Violet}\mu }}}-{{g}_{{\color {Red}\mu }{\color {Violet}\nu },\sigma }}\right)$

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

Verhalten bei Koordinatentransformation

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