Proving properties of the Levi-Civita tensor

[Dixanadu]

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.

 

[Bryson]

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

 

[Dick]

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.

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.

 

[Dixanadu]

Okay, thanks a bunch guys! yea it makes sense now :)

 

[paranormal]

I would like to first commend you for taking on the task of proving properties of the Levi-Civita tensor. This is an important concept in mathematics and physics, and understanding its properties can greatly aid in solving complex problems.

To address your question, yes, you are on the right track. The sign of the Levi-Civita tensor does indeed reverse when two indices are swapped. This is known as the antisymmetry property of the tensor. This can be seen in your attempt at a solution, where swapping the last two indices resulted in a change of sign.

To prove question 2, you can use this antisymmetry property and the given equations \epsilon_{ijk}=-\epsilon_{jik}=-\epsilon_{ikj}. By swapping the last two indices in the equation \epsilon_{ijk}=-\epsilon_{jik}, we get -\epsilon_{jki}=-(-\epsilon_{ijk})=\epsilon_{ijk}. This shows that \epsilon_{ijk}=\epsilon_{jki}, and by using the same logic, we can also prove that \epsilon_{jki}=\epsilon_{kij}.

I hope this helps in your understanding of the Levi-Civita tensor and its properties. Keep up the good work!