Appendix 2 â Afleiding van de Afgeleide van de ChristoffelâSymbolen in Algemene Vorm
Er wordt aangetoond hoe het Christoffelâsymbool uitsluitend afhangt van de elementen van de metrische tensor en diens afgeleiden. Dit is bijzonder handig bij implementatie in spreadsheets of computerprogrammaâs.
Christoffelâsymbool
\begin{equation}
\begin{aligned}
\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)
\end{aligned}
\end{equation}
Afgeleide van het Christoffelâsymbool
\begin{equation}
\begin{aligned}
\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)
\\
&\quad
+
\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)
\end{aligned}
\end{equation}
Afgeleide van de inverse metriek
\begin{equation}
\begin{aligned}
\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}}
\end{aligned}
\end{equation}
Volledige afgeleide van het Christoffelâsymbool
\begin{equation}
\begin{aligned}
\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)
\\
&\quad
+
\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)
\end{aligned}
\end{equation}
Compacte vorm
\begin{equation}
\begin{aligned}
\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)
\end{aligned}
\end{equation}