# Proving properties of the Levi-Civita tensor

1. Oct 21, 2013

1. The problem statement, all variables and given/known data
Hey everyone,
So I've got to prove a couple of equations to do with the Levi-Civita tensor. So we've been given:
$\epsilon_{ijk}=-\epsilon_{jik}=-\epsilon_{ikj}$

We need to prove the following:
(1) $\epsilon_{ijk}=-\epsilon_{kji}$
(2) $\epsilon_{ijk}=\epsilon_{jki}=\epsilon_{kij}$

2. Relevant equations

3. The attempt at a solution
So this seems a bit too easy - but my question is this: if I swap two of the indices, the sign reverses. But if I do another swap, (not necessarily the same indices), does the sign reverse again? So basically If I start with this
$\epsilon_{ijk}$
Then if I swap two indices, ij -> ji, I get
$-\epsilon_{jik}$
If I swap the last two indices like so:
$-\epsilon_{jik} → +\epsilon_{jki}$.
Is that true? I think that's the only way to prove question 2.

2. Oct 21, 2013

### Bryson

You are absolutely correct! The definition of the Levi-Civita (i.e. swapping (non-cyclical) => minus sign).

3. Oct 21, 2013

### Dick

Yes, that's exactly the idea. If there are an even number of swaps then the sign doesn't change. If there are an odd number, then it does.

4. Oct 21, 2013