(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If [tex]\epsilon _{ijjk}[/tex] is the Levi-Civita symbol:

1)Demonstrate that [tex]\sum _{i} \epsilon _{ijk} \epsilon _{ilm}=\delta _{jl} \delta _{km} -\delta _{jm} \delta _{kl}[/tex].

2)Calculate [tex]\sum _{ij} \epsilon _{ijk} \epsilon _{ijl}[/tex].

3)Given the matrix M, calculate [tex]\sum _{ijk} \sum _{lmn} \epsilon _{ijk} \epsilon _{lmn} M_{il} M_{jm} M_{kn}[/tex].

2. Relevant equations

Maybe some properties on tensors but I'm not sure.

3. The attempt at a solution

I'm trying to start with 1) first. I'm so new with tensors that I don't understand well what I have to do.

I know that Levi-Civita symbol is either worth -1, 0 or 1 though I didn't understand what is an even and odd permutation of ijk. And I know Kronecker's delta which is worth either 0 or 1, depending if i=j or not.

So starting with [tex]\sum _{i} \epsilon _{ijk} \epsilon _{ilm}[/tex], can I assume i to go from 1 to 3? And what about j and k? Fixed constants which are either 1, 2 or 3?

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# Homework Help: Tensors, Levi-Civita symbol

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