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Kovariante Ableitung

2009-07-07T00:56:34Z

92.226.33.241:

[[Bild:KovarianteAbleitung.png]]
<source lang="latex">
\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}
</source>

<math>
\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}
</math>

<math>
{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
</math>

== Eigenschaften ==
* erhält Tensoreingeschaft <-> Unterschied zur [[gewöhnlichen Ableitung]]

[[Kategorie:Tensor]]

Kovariante Ableitung

2009-07-07T00:56:13Z

92.226.33.241:

[[Bild:KovarianteAbleitung.png]]
<source lang="latex">
\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}
</source>

<math>
\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}
</math>
<math>
{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
</math>

== Eigenschaften ==
* erhält Tensoreingeschaft <-> Unterschied zur [[gewöhnlichen Ableitung]]

[[Kategorie:Tensor]]

Hauptseite

2009-07-07T00:38:48Z

92.226.33.241:

unter [[Antiproportional]] sieht man mehr
Siehe [[Weihnachtsübung zur Allgemeinen Relativitätstheorie II]]

<math>
f(a,b) = (a+b)^2
= a^2+2ab+b^2 </math>