# Derivation of the Christoffel symbol

by elenuccia
Tags: christoffel, derivation, symbol
 Emeritus Sci Advisor PF Gold P: 9,403 Did you obtain a result like $$g_{\rho\sigma,\mu}=\Gamma^\lambda_{\mu\rho}g_{\lambda\sigma}+\Gamma^\la mbda_{\mu\sigma}g_{\rho\lambda}$$already? In that case, consider the quantity $$g_{\mu\sigma,\rho}+g_{\mu\rho,\sigma}-g_{\rho\sigma,\mu}$$and I think you'll be able to figure out the rest. Don't forget that a Levi-Civita connection is torsion free. This implies that the Christoffel symbol is symmetric in the lower indices.