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}

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

${A^{\color {Violet}\lambda }}_{{\color {Orange}\mu };{\color {Red}\nu }}={\frac {\partial {A^{\color {Violet}\lambda }}_{\color {Orange}\mu }}{\partial {x^{\color {Red}\nu }}}}+\Gamma _{\rho {\color {Red}\nu }}^{\color {Violet}\lambda }{A^{\rho }}_{\color {Orange}\mu }-\Gamma _{{\color {Orange}\mu }{\color {Red}\nu }}^{\rho }{A^{\color {Violet}\lambda }}_{\rho }$

Eigenschaften

• erhält Tensoreingeschaft <-> Unterschied zur gewöhnlichen Ableitung