Kovariante Ableitung

Datei:KovarianteAbleitung.png

T_{rs...;l}^{ik...}  = T_{rs...,l}^{ik...}&+ \Gamma _{ml}^{i}T_{rs...}^{mk...}\quad&&+\Gamma _{ml}^{i}T_{is...}^{rm...}... &\quad&\text{f }\!\!\ddot{\mathrm{u}}\!\!\text{ r jeden oberen Index} \\
 & +\Gamma _{rl}^{m}T_{ms...}^{ik...}\quad&&+\Gamma _{rl}^{m}T_{rm...}^{ik...}...&&\text{f }\!\!\ddot{\mathrm{u}}\!\!\text{ r jeden unteren Index} \\

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

Eigenschaften

• erhält Tensoreingeschaft <-> Unterschied zur gewöhnlichen Ableitung