Γ μ ν α = 1 2 g α σ ( g σ μ , ν + g σ ν , μ − g μ ν , σ ) {\displaystyle \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 {Brown}\mu }}}-{{g}_{{\color {Red}\mu }{\color {Violet}\nu },\sigma }}\right)}
Γ μ ν α = ∂ x α ∂ ξ β ∂ 2 ξ β ∂ x μ ∂ x ν {\displaystyle \Gamma _{\mu \nu }^{\alpha }={\frac {\color {Violet}\partial {{x}^{\alpha }}}{\partial {{\xi }^{\beta }}}}{\frac {{{\partial }^{2}}{{\xi }^{\beta }}}{\partial {{x}^{\mu }}\partial {{x}^{\nu }}}}}
Γ ′ μ ν α = ∂ x ′ α ∂ x β ∂ x ρ ∂ x ′ μ ∂ x σ ∂ x ′ ν Γ ρ σ β + ∂ x ′ α ∂ x β ∂ 2 x β ∂ x ′ μ ∂ x ′ ν {\displaystyle \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 {\partial x{{'}^{\alpha }}}{\partial {{x}^{\beta }}}}{\frac {{{\partial }^{2}}{{x}^{\beta }}}{\partial x{{'}^{\mu }}\partial x{{'}^{\nu }}}}}