Relation between Levi-civita and Kronecker- delta symbol

The conversation is about the definition of the Levi-Civita symbol, defined as εijk=+1 if ijk = (123, 231, 312), εijk = −1if ijk = (213, 321, 132), and εijk= 0 otherwise. It is also mentioned that εijk is nonzero only when all three indices are different. The conversation then moves on to using this definition to show that εijkεinm= δjnδkm − δjmδkn , where i,j,k,n,and m are free indices.
  • #1
Pushoam
962
51

Homework Statement


definition of εijk
εijk=+1 if ijk = (123, 231, 312)
εijk = −1if ijk = (213, 321, 132) , (1.1.1)
εijk= 0,otherwise .
That is,εijk is nonzero only when all three indices are different.

From the definition in Eq. (1.1.1), show that
εijkεinm= δjnδkm − δjmδkn , (1.1.4)
where of course there is an implied sum over the i index in Eq. (1.1.4), but the indices j, k, n,and m are free.

Homework Equations

The Attempt at a Solution


By using definition of εijk; I can show that under the values which i,j,k,n,m can take, L.H.S.=R.H.S.

But is there any other better way to show it?
Is there a way such that one will start from L.H.S. and reach to R.H.S.?
 
Last edited:
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  • #2
We do not have access to your book so we cannot tell how Eq. (1.1.1) defines the Levi-Civita symbol.
 
  • #3
Orodruin said:
We do not have access to your book so we cannot tell how Eq. (1.1.1) defines the Levi-Civita symbol.
I have written the definition.
 

1. What is the Levi-Civita symbol and how is it related to the Kronecker delta symbol?

The Levi-Civita symbol, also known as the permutation symbol, is a mathematical symbol used to represent the sign of a permutation of a set of numbers. The Kronecker delta symbol, on the other hand, is a mathematical symbol used to represent the identity matrix. These two symbols are related because they both involve the concept of permutation and are often used in tensor calculus.

2. How are the Levi-Civita and Kronecker delta symbols defined?

The Levi-Civita symbol is defined as 1 if the permutation of the indices is even, -1 if the permutation is odd, and 0 if any two indices are equal. The Kronecker delta symbol is defined as 1 if the two indices are equal, and 0 otherwise.

3. What are the properties of the Levi-Civita and Kronecker delta symbols?

The Levi-Civita symbol is antisymmetric, meaning that it changes sign when the order of its indices is switched. It is also invariant under rotations and reflections. The Kronecker delta symbol is symmetric, meaning that it remains unchanged when the order of its indices is switched. It is also diagonal, meaning that all its non-zero entries occur on the main diagonal.

4. How are the Levi-Civita and Kronecker delta symbols used in physics?

In physics, the Levi-Civita and Kronecker delta symbols are commonly used in tensor calculus to represent vector and tensor operations. They are also used in the study of electromagnetism, quantum mechanics, and general relativity.

5. Can the Levi-Civita and Kronecker delta symbols be generalized to higher dimensions?

Yes, the Levi-Civita and Kronecker delta symbols can be generalized to higher dimensions. In three dimensions, the Levi-Civita symbol has three indices, while in n-dimensional space, it would have n indices. Similarly, the Kronecker delta symbol can also be generalized to higher dimensions with n indices.

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