# Thread: Is this tensor equation correct?

1. ## Is this tensor equation correct?

I am to prove $\displaystyle \ det[g_{ij}]=\ V^2$ where $\displaystyle \ g_{ij}$ is metric tensor and $\displaystyle \ V=\epsilon_i\cdot\epsilon_j\times\epsilon_k$
Now,I was wondering if the identity
$\displaystyle \epsilon_{ijk}\epsilon_{lmn}= \begin{vmatrix} \ g_{11}&\ g_{21}&\ g_{31}\\ g_{12}&\ g_{22}&\ g_{32}\\ g_{13}&\ g_{23}&\ g_{33} \end{vmatrix}$
is correct---in which case the problem is solved easily.Note that the antisymmetric tensors are both co-variant.
Surely,the best way to know whether it is correct or not is to prove it...but somehow I am not getting it.If it is correct can anyone give me some hint?

2. Possibly I have done wrong in posting this topic in this subforum.I hereby ask the moderator to close this thread.

3. Originally Posted by kolahalb
Possibly I have done wrong in posting this topic in this subforum.I hereby ask the moderator to close this thread.
it was suggested by one of the Helpers that this thread should be moved to this subforum, as it is more a differential geometry problem than a linear algebra problem