Kovariante Ableitung

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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{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}


{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

Von „http://wiki.physikerwelt.de/wiki/Kovariante_Ableitung“
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