Levi-Civita Identity Proof Help (εijk εijl = 2δkl)

In summary, the conversation discusses a proof that involves making a substitution and using summation convention. The speaker initially thought it would be straightforward, but ended up with an incorrect result. The other person reminds them to take into account the summation convention and the fact that δjj is equal to 3.
  • #1
John Delaney
3
1
Homework Statement
Prove εijk εijl = 2δkl
Relevant Equations
εijk εilm = δjl δkm - δjm δkl
I assumed that this would be a straightforward proof, as I could just make the substitution l=j and m=l, but upon doing this, I end up with:

δjj δkl - δjl δkj

= δkl - δlk

Clearly I did not take the right approach in this proof and have no clue as to how to proceed.
 
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  • #2
Well you are right that you can do it just by making a 'substitution'. However, bear in mind that summation convention is in use here. You have to sum over the indices that are repeating. With that, you should be able to get the right result.
 
  • #3
Antarres said:
Well you are right that you can do it just by making a 'substitution'. However, bear in mind that summation convention is in use here. You have to sum over the indices that are repeating. With that, you should be able to get the right result.

Thank you for the reminder, completely forgot that δjj = 3 rather than 1.
 

Related to Levi-Civita Identity Proof Help (εijk εijl = 2δkl)

What is the Levi-Civita identity?

The Levi-Civita identity is a mathematical identity that relates the permutation symbol (εijk) and the Kronecker delta symbol (δkl). It is commonly used in vector calculus and tensor analysis.

What is the purpose of the Levi-Civita identity?

The Levi-Civita identity is used to simplify calculations involving vector or tensor operations. It allows for the manipulation of equations involving the permutation symbol and Kronecker delta symbol, making it easier to solve problems in vector calculus and tensor analysis.

How is the Levi-Civita identity derived?

The Levi-Civita identity can be derived from the properties of the permutation symbol and Kronecker delta symbol. It involves using the properties of these symbols and manipulating them to arrive at the final identity, εijk εijl = 2δkl. The proof is commonly taught in advanced mathematics courses.

Why is the Levi-Civita identity important in physics?

The Levi-Civita identity is important in physics because it is used to simplify calculations in vector calculus and tensor analysis, which are essential tools in many areas of physics. It is also used in the theory of relativity and electromagnetism, making it a crucial concept in these fields.

Can the Levi-Civita identity be extended to higher dimensions?

Yes, the Levi-Civita identity can be extended to higher dimensions. In three dimensions, the identity can be written as εijk εijl = 2δkl. In four dimensions, it becomes εαβγδ εαβγδ = 24, where α, β, γ, and δ represent indices in four-dimensional space. This extension is useful in solving problems in higher dimensions, such as in the theory of relativity.

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