- #1

Dixanadu

- 254

- 2

## Homework Statement

Hey everyone,

So I've got to prove a couple of equations to do with the Levi-Civita tensor. So we've been given:

[itex]\epsilon_{ijk}=-\epsilon_{jik}=-\epsilon_{ikj}[/itex]

We need to prove the following:

(1) [itex]\epsilon_{ijk}=-\epsilon_{kji}[/itex]

(2) [itex]\epsilon_{ijk}=\epsilon_{jki}=\epsilon_{kij}[/itex]

## Homework Equations

## 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

[itex]\epsilon_{ijk}[/itex]

Then if I swap two indices, ij -> ji, I get

[itex]-\epsilon_{jik}[/itex]

If I swap the last two indices like so:

[itex]-\epsilon_{jik} → +\epsilon_{jki}[/itex].

Is that true? I think that's the only way to prove question 2.