Kovariante Ableitung

Datei:KovarianteAbleitung.png

\begin{align}
T_{{\color{Orange}rs...}; \color{Red}l}^{\color{Blue}ik...}
 =
T_{{\color{Orange}rs...}, \color{Red}l}^{\color{Blue} ik...}&
 +
\Gamma _{ m\color{Red}l}^{\color{Blue}i}T_{\color{Orange}rs...}^{m\color{Blue}k...}\quad&&
 +
\Gamma _{m\color{Red}l}^{\color{Blue}i}T_{\color{Orange}rs...}^{{\color{Blue}r}m\color{Blue}...}... &\quad&
{\color{Blue}\text{f }\!\!\ddot{\mathrm{u}}\!\!\text{ r jeden oberen Index}} \\
 & +
\Gamma _{\color{Orange}r\color{Red}l}^{m}T_{m\color{Orange}s...}^{ \color{Blue}ik...}\quad&&
+
\Gamma _{\color{Orange}r\color{Red}l}^{m}T_{{\color{Orange}r}m\color{Orange}...}^{ \color{Blue}ik...}...&&
\color{Orange}{\text{f }\!\!\ddot{\mathrm{u}}\!\!\text{ r jeden unteren Index}} \\
\end{align}

$\begin{array}{rlrlrl}{T}_{rs...;l}^{ik...}=T_{rs...,l}^{ik...}& +{\mathrm{\Gamma }}_{ml}^i{T}_{rs...}^{mk...}& & +{\mathrm{\Gamma }}_{ml}^i{T}_{rs...}^{rm...}...& & \text{f }\ddot{\mathrm{u}}\text{ r jeden oberen Index}\\ & +{\mathrm{\Gamma }}_{rl}^m{T}_{ms...}^{ik...}& & +{\mathrm{\Gamma }}_{rl}^m{T}_{rm...}^{ik...}...& & \text{f }\ddot{\mathrm{u}}\text{ r jeden unteren Index}\\ \end{array}$ ${A^{\lambda }}_{\mu ;\nu }=\frac{\partial {A^{\lambda }}_{\mu }}{\partial x^{\nu }}+{\mathrm{\Gamma }}_{\rho \nu }^{\lambda }{A^{\rho }}_{\mu }-{\mathrm{\Gamma }}_{\mu \nu }^{\rho }{A^{\lambda }}_{\rho }$

Eigenschaften

• erhält Tensoreingeschaft <-> Unterschied zur gewöhnlichen Ableitung