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Proving properties of the Levi-Civita tensor

  • Thread starter Dixanadu
  • Start date
  • #1
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.
 

Answers and Replies

  • #2
56
2
You are absolutely correct! The definition of the Levi-Civita (i.e. swapping (non-cyclical) => minus sign).
 
  • #3
Dick
Science Advisor
Homework Helper
26,258
618

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.
 
  • #4
254
2
Okay, thanks a bunch guys! yea it makes sense now :)
 

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