Relationship between Kronecker Delta and the Alternating Tensor

• Feb 1st 2011, 08:34 AM
jameselmore91
Relationship between Kronecker Delta and the Alternating Tensor
In my Vector Calculus class the book discusses the relationship between Kronecker Delta ($\displaystyle \delta_{ij}$) and the alternating tensor ($\displaystyle \epsilon_{ijk}$) as the following:

$\displaystyle \epsilon_{ijk}\epsilon_{klm} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}$

The book gives a lengthy explanation of why these two expressions are equivalent, but gives no insight as to how this relation was derived.

Can anyone give any insight into the derivation of this equation?

James Elmore

P.S.
Is this the right board for this topic?
• Feb 1st 2011, 08:37 AM
Ackbeet
Are you asking how the relationship was dreamed up in the first place? Or are you asking how do you prove the relationship?
• Feb 1st 2011, 09:05 AM
jameselmore91
I'm more interested in the proof of the relationship
• Feb 1st 2011, 09:23 AM
Ackbeet
• Feb 1st 2011, 01:27 PM
jameselmore91
Quote:

Originally Posted by Ackbeet

That link leaves me in a similar place.

Where did they get the first equation of:

$\displaystyle \epsilon_{ijk}\epsilon_{lmn} = \mbox{det}\left(\begin{array}{ccc}\delta_{il}&\del ta_{im}&\delta_{in}\\\delta_{jl}&\delta_{jm}&\delt a_{jn}\\\delta_{kl}&\delta_{km}&\delta_{kn}\end{ar ray}\right)$
• Feb 1st 2011, 01:28 PM
Ackbeet