Christoffel symbols

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The Christoffel symbols are related to the metric tensor by

\Gamma ^{\lambda}_{\mu \nu}=\frac{1}{2}g^{\lambda \rho}\left(g_{\mu \rho},_{\nu}+g_{\rho \nu},_{\mu}-g_{\mu \nu},_{\rho}\right)

where the comma is a partial derivative. For example

g_{\mu \nu},_{\rho}=\frac{\partial g_{\mu \nu}}{\partial x^{\rho}}

The Christoffel symbols are part of a covariant derivative opperation, represented by a semicolin or capitalized D ,mapping tensor elements to tensor elements. For example

T^\lambda ;_{\rho }=\frac{DT^\lambda }{\partial x^{\rho }} = \frac{\partial T^\lambda }{\partial x^{\rho }}+\Gamma ^{\lambda }_{\mu \rho}T^{\mu}

Also for example

T_{\lambda};_{\rho }=\frac{DT_{\lambda}}{\partial x^{\rho }} = \frac{\partial T_{\lambda}}{\partial x^{\rho }}-\Gamma ^{\mu }_{\lambda \rho}T_{\mu}

And differentiating with respect to an invariant example

\frac{DT^\lambda }{d\tau } = \frac{d T^\lambda }{d\tau}+\Gamma ^{\lambda }_{\mu \nu }T^{\mu}\frac{\partial x^{\nu }}{d\tau }

References

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