Appendix 2 — Derivation of the Derivative of the Christoffel Symbols in General Form

It is demonstrated how the Christoffel symbol depends exclusively on the components of the metric tensor and its derivatives. This is particularly convenient for implementation in spreadsheets or computer programs.

Christoffel symbol

\[ \Gamma^{\rho}{}_{\mu\nu} = \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) \]

Derivative of the Christoffel symbol

\[ \frac{\partial \Gamma^{\rho}{}_{\mu\nu}}{\partial x^{\gamma}} = \frac{1}{2} \frac{\partial g^{\rho\alpha}}{\partial x^{\gamma}} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) + \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\gamma}} + \frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\gamma}} - \frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\gamma}} \right) \]

Derivative of the inverse metric

\[ \frac{\partial g^{\rho\alpha}}{\partial x^{\gamma}} = \frac{\partial \frac{1}{g_{\rho\alpha}}}{\partial x^{\gamma}} = \frac{-1}{g^2_{\rho\alpha}}\cdot\frac{\partial g_{\rho\alpha}}{\partial x^{\lambda}} = - (g^{\rho\alpha})^{2} \frac{\partial g_{\rho\alpha}}{\partial x^{\gamma}} \]

Full derivative of the Christoffel symbol

\[ \frac{\partial \Gamma^{\rho}_{\mu\nu}}{\partial x^{\gamma}} = -\frac{1}{2} (g^{\rho\alpha})^{2} \frac{\partial g_{\rho\alpha}}{\partial x^{\gamma}} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) + \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\gamma}} + \frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\gamma}} - \frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\gamma}} \right) \]

Compact form

\[ \frac{\partial \Gamma^{\rho}_{\mu\nu}}{\partial x^{\gamma}} = - g^{\rho\alpha} \frac{\partial g_{\rho\alpha}}{\partial x^{\gamma}} \Gamma^{\rho}_{\mu\nu} + \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\gamma}} + \frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\gamma}} - \frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\gamma}} \right) \]