Relation between Levi-civita and Kronecker- delta symbol

1. May 19, 2017

Pushoam

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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: May 19, 2017
2. May 19, 2017

Orodruin

Staff Emeritus
We do not have access to your book so we cannot tell how Eq. (1.1.1) defines the Levi-Civita symbol.

3. May 19, 2017

Pushoam

I have written the definition.