Christoffel symbols

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