Understanding the Properties of Levi-Civita Symbol in Tensor Calculus

In summary, the Levi-Civita symbol is either worth -1, 0 or 1, depending on the permutation of ijk. The Kronecker delta is either worth 0 or 1, depending on the permutation of ij. The sum of the ijth epsilon functions is equal to 2 if the permutation of ijk is odd, and 0 if it's even.
  • #1
fluidistic
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Homework Statement


If [tex]\epsilon _{ijjk}[/tex] is the Levi-Civita symbol:
1)Demonstrate that [tex]\sum _{i} \epsilon _{ijk} \epsilon _{ilm}=\delta _{jl} \delta _{km} -\delta _{jm} \delta _{kl}[/tex].
2)Calculate [tex]\sum _{ij} \epsilon _{ijk} \epsilon _{ijl}[/tex].
3)Given the matrix M, calculate [tex]\sum _{ijk} \sum _{lmn} \epsilon _{ijk} \epsilon _{lmn} M_{il} M_{jm} M_{kn}[/tex].

Homework Equations



Maybe some properties on tensors but I'm not sure.

The Attempt at a Solution


I'm trying to start with 1) first. I'm so new with tensors that I don't understand well what I have to do.
I know that Levi-Civita symbol is either worth -1, 0 or 1 though I didn't understand what is an even and odd permutation of ijk. And I know Kronecker's delta which is worth either 0 or 1, depending if i=j or not.
So starting with [tex]\sum _{i} \epsilon _{ijk} \epsilon _{ilm}[/tex], can I assume i to go from 1 to 3? And what about j and k? Fixed constants which are either 1, 2 or 3?
 
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  • #2
fluidistic said:

Homework Statement


If [tex]\epsilon _{ijjk}[/tex] is the Levi-Civita symbol:
1)Demonstrate that [tex]\sum _{i} \epsilon _{ijk} \epsilon _{ilm}=\delta _{jl} \delta _{km} -\delta _{jm} \delta _{kl}[/tex].
2)Calculate [tex]\sum _{ij} \epsilon _{ijk} \epsilon _{ijl}[/tex].
3)Given the matrix M, calculate [tex]\sum _{ijk} \sum _{lmn} \epsilon _{ijk} \epsilon _{lmn} M_{il} M_{jm} M_{kn}[/tex].

Homework Equations



Maybe some properties on tensors but I'm not sure.

The Attempt at a Solution


I'm trying to start with 1) first. I'm so new with tensors that I don't understand well what I have to do.
I know that Levi-Civita symbol is either worth -1, 0 or 1 though I didn't understand what is an even and odd permutation of ijk.
(132) would be an odd permutation of (123) because you can obtain it using an odd number of transpositions (swap of a pair), e.g. swap 2 and 3. (312) on the other hand would be an even permutation of (123) because obtaining it requires an even number of transpositions, e.g. swap 1 and 3 to get (321) and then swap 1 and 2 to get (312).

http://en.wikipedia.org/wiki/Parity_of_a_permutation
And I know Kronecker's delta which is worth either 0 or 1, depending if i=j or not.
So starting with [tex]\sum _{i} \epsilon _{ijk} \epsilon _{ilm}[/tex], can I assume i to go from 1 to 3? And what about j and k? Fixed constants which are either 1, 2 or 3?
Yes, the index i runs from 1 to 3; j, k, l, and m are fixed. There aren't too many combinations of j, k, l, and m to check since you can easily see that most are 0.
 
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  • #3
vela said:
(132) would be an odd permutation of (123) because you can obtain it using an odd number of transpositions (swap of a pair), e.g. swap 2 and 3. (312) on the other hand would be an even permutation of (123) because obtaining it requires an even number of transpositions, e.g. swap 1 and 3 to get (321) and then swap 1 and 2 to get (312).

http://en.wikipedia.org/wiki/Parity_of_a_permutation

Yes, the index i runs from 1 to 3; j, k, l, and m are fixed. There aren't too many combinations of j, k, l, and m to check since you can easily see that most are 0.

Thank you very much. Now I understand and I still find that there are lots of choices (4^4 I think) even though most of them are worth 0, I must show all. Part 1) solved.
By the way I had seen the wikipedia article but didn't understand. Your words were much clearer to me. Now I'll try part 2.Edit: I'm a bit confused on the notation of the sum. There are 9 cases. I take the first as [tex]k=l=1[/tex]. Is [tex]\sum _{ij} \epsilon _{ijk} \epsilon _{ijl}[/tex] worth [tex]\epsilon _{111}\epsilon _{111}+\epsilon _{211}\epsilon _{211}+\epsilon _{311}\epsilon _{311}+\epsilon _{121}\epsilon _{121}+\epsilon _{131}\epsilon _{131}+\epsilon _{231}\epsilon _{231}+\epsilon _{321}\epsilon _{321}=0[/tex]?
 
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  • #4
fluidistic said:
Edit: I'm a bit confused on the notation of the sum. There are 9 cases. I take the first as [tex]k=l=1[/tex]. Is [tex]\sum _{ij} \epsilon _{ijk} \epsilon _{ijl}[/tex] worth [tex]\epsilon _{111}\epsilon _{111}+\epsilon _{211}\epsilon _{211}+\epsilon _{311}\epsilon _{311}+\epsilon _{121}\epsilon _{121}+\epsilon _{131}\epsilon _{131}+\epsilon _{231}\epsilon _{231}+\epsilon _{321}\epsilon _{321}=0[/tex]?
The only non-vanishing terms are the last two, and they're both equal to 1, so the sum is equal to 2.
 
  • #5
Indeed you're right, I made a mistake in the last term of the sum, I considered it as -1 instead of 1. My question was about if my sum was right since I don't recall having used a double subindex in sums before.
Now I'll try to tackle part 3).
Nevermind, I see it's only a huge arithmetic "mess". I don't see any trick but to calculate the whole sum...
Thanks for all, and you've taught me what was an even and odd permutation. :)
 
Last edited:

1. What are tensors?

Tensors are mathematical objects used to describe the relationship between vectors and scalars. They have components that vary in multiple dimensions and follow specific transformation rules under coordinate transformations.

2. What is the Levi-Civita symbol?

The Levi-Civita symbol, also known as the permutation symbol, is a mathematical symbol used to represent an alternating tensor. It is defined as 1 if the indices are in an even permutation, -1 if they are in an odd permutation, and 0 if any indices are repeated.

3. How are tensors and the Levi-Civita symbol related?

The Levi-Civita symbol is used to define the components of a tensor in a coordinate-independent way. It is used as a coefficient in the transformation equation for tensors, allowing them to be expressed in different coordinate systems.

4. What is the significance of the Levi-Civita symbol in physics?

The Levi-Civita symbol is used extensively in physics, particularly in the fields of electromagnetism, general relativity, and quantum mechanics. It is used to define important physical quantities such as the electromagnetic field tensor, the Riemann curvature tensor, and the spinor field.

5. How can I use the Levi-Civita symbol in my research?

The Levi-Civita symbol can be used in a variety of ways in research, depending on the specific field of study. It can be used to define and manipulate tensors, to simplify equations and calculations, and to express physical quantities in a coordinate-independent manner. It is also used in computer programming and coding for scientific simulations and calculations.

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