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Proof using Levi-Civita symbol

  • Thread starter cashmerelc
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  • #1
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Homework Statement


Prove [tex]\sum_{j,k}[/tex] [tex]\epsilon_{ijk}[/tex] [tex]\epsilon_{ljk}[/tex] = 2[tex]\delta_{il}[/tex]



Homework Equations


[tex]\epsilon_{ijk}[/tex] [tex]\epsilon_{ljk} = [tex]\delta_{il}[/tex]([tex]\delta_{jj}[/tex][tex]\delta_{kk}[/tex] - [tex]\delta_{jk}[/tex][tex]\delta_{kj}[/tex]) + [tex]\delta_{ij}[/tex]([tex]\delta_{jk}[/tex][tex]\delta_{kl}[/tex] - [tex]\delta_{jl}[/tex][tex]\delta_{kk}[/tex]) + [tex]\delta_{ik}[/tex]([tex]\delta_{jl}[/tex][tex]\delta_{kk}[/tex] - [tex]\delta_{jj}[/tex][tex]\delta_{kl}[/tex])


The Attempt at a Solution



Okay, in cases where subscripts of the Kronecker delta are equal, then [tex]\delta_{jj}[/tex] = 1.

If the subscripts are not equal, then [tex]\delta_{il}[/tex] = 0.

So plugging those into the parenthesis of the above equation gives me:

[tex]\delta_{il}[/tex]([tex]\delta_{jj}[/tex][tex]\delta_{kk}[/tex]) ?

If that is the case, then how could the two inside the parenthesis equal 2? I know I must be missing something.
 
Last edited:

Answers and Replies

  • #2
learningphysics
Homework Helper
4,099
5
In your formula, replace the [tex]\delta_{jj}[/tex], [tex]\delta_{kk}[/tex] etc... where the variables are the same... with 1.

Also, [tex]\delta_{ij}\delta_{jk} = \delta_{ik}[/tex]
 
  • #3
18
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In your formula, replace the [tex]\delta_{jj}[/tex], [tex]\delta_{kk}[/tex] etc... where the variables are the same... with 1.

Also, [tex]\delta_{ij}\delta_{jk} = \delta_{ik}[/tex]
If [tex]\delta_{ij}\delta_{jk} = \delta_{ik}[/tex] does that mean that [tex]\delta_{lk}\delta_{kj} = \delta_{lj}[/tex] and so on?
 
  • #4
learningphysics
Homework Helper
4,099
5
If [tex]\delta_{ij}\delta_{jk} = \delta_{ik}[/tex] does that mean that [tex]\delta_{lk}\delta_{kj} = \delta_{lj}[/tex] and so on?
Yes, exactly.
 
  • #5
18
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Okay, I think one more question will help me get it.

[tex]\delta_{jk}[/tex][tex]\delta_{kj}[/tex] = ?
 
  • #6
learningphysics
Homework Helper
4,099
5
Okay, I think one more question will help me get it.

[tex]\delta_{jk}[/tex][tex]\delta_{kj}[/tex] = ?
= [tex]\delta_{jj} = 1[/tex]
 

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