The Christoffel symbols are related to the metric tensor by

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

where the comma is a partial derivative. For example

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

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

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

Also for example

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

And differentiating with respect to an invariant example

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

References

• Relativity