Proofs of Equations Involving Metric Tensors and Christoffel Symbols

[Jinroh]

Hi I'm new here and I hope that you will be able to give me a lot of help. My english is far to be perfect but sufficiant to asks you a lot of questions... (i hope so ).

First question :

I'm looking for a complete proof (with all steps) of :

$$\partial _h g = gg_{ij} \partial _h g_{ij} \Rightarrow \partial _h g_{ij} = \frac{{\partial _h g}}<br /> {{gg_{ij} }}$$

Second question :

I'm looking for a complete proof (with all steps) of :

$$\nabla _k \left( {\nabla _j v_i } \right) = \partial _{kj} v_i - \left( {\partial _k \Gamma _{ji}^l } \right)v_l - \Gamma _{ji}^l \partial _k v_l - \Gamma _{ik}^r \partial _j v_r + \Gamma _{ik}^r \Gamma _{jr}^l v_l + \Gamma _{jk}^r \partial _r v_i + \Gamma _{jk}^r \Gamma _{ri}^l v_l$$


Thanks a lot if you can help me. In all physics-mathematics french forums nobody seems to be able to do that or to have the time to copy a part of their courses.

 

[Hurkyl]

If you replace the $..$ with [ itex ] .. [ /itex ], (without the spaces) it will show up as an image.

 

[Jinroh]

humm...

thanks for the information...

Another question. Who can say me which script is used to post Latex in a forum and where it can be found ?

 

[Hurkyl]

Oh, and using [ tex ] .. [ /tex ] will draw it bigger; the former is better suited for putting text like $a = b$ in a paragraph.

chroot wrote the script for our forum.