Understanding the Levi-Civita Identity: Simplifying the Notation

[cozmo]

Can somebody show me how

\epsilon_{mni}a_{n}(\epsilon_{ijk}b_j c_{k})

Turns in to

\epsilon_{imn}\epsilon_{ijk}a_{n}b_j c_{k}


Something about the first \epsilon I'm not seeing here when the terms are moved around.

 

[vela]

The tensor changes sign when you transpose two indices, right? A cyclic permutation is an even number of transpositions, so \epsilon_{mni} = \epsilon_{nim} = \epsilon_{imn}.

 

[cozmo]

When \epsilon_{ijk} and a_{n} change places the \epsilon_{mni} changes to a cyclic permutation that is still positive and \epsilon_{mni} =\epsilon_{imn}=\epsilon_{nim} but each one of these will give a different final answer.

I don't see how \epsilon_{mni} turns to \epsilon_{imn} when \epsilon_{ijk} doesn't change.

This is from a problem proving the A X (B X C) = (A*C)B-(A*B)C identity.

 

[vela]

You can swap a_n and \epsilon_{ijk} because real numbers commute. Swapping them has nothing to do with reordering the indices of \epsilon_{mni}.

\epsilon_{mni} = \epsilon_{imn} for all i, m, and n, so you can simply replace \epsilon_{mni} with \epsilon_{imn} in the summation. There's no relabeling of indices going on if that's what you think is happening.