Relationship between Kronecker Delta and the Alternating Tensor

**jameselmore91** · February 1st 2011, 08:34 AM

In my Vector Calculus class the book discusses the relationship between Kronecker Delta ( $\delta_{ij}$ ) and the alternating tensor ( $\epsilon_{ijk}$ ) as the following:

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

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?

Thanks in Advanced,

James Elmore

P.S.
Is this the right board for this topic?

**Ackbeet** · February 1st 2011, 08:37 AM

Are you asking how the relationship was dreamed up in the first place? Or are you asking how do you prove the relationship?

**jameselmore91** · February 1st 2011, 09:05 AM

I'm more interested in the proof of the relationship

**Ackbeet** · February 1st 2011, 09:23 AM

Take a look here.

**jameselmore91** · February 1st 2011, 01:27 PM

Originally Posted by Ackbeet

Take a look here.

That link leaves me in a similar place.

Where did they get the first equation of:

$\epsilon_{ijk}\epsilon_{lmn} = \mbox{det}\left(\begin{array}{ccc}\delta_{il}&\del ta_{im}&\delta_{in}\\\delta_{jl}&\delta_{jm}&\delt a_{jn}\\\delta_{kl}&\delta_{km}&\delta_{kn}\end{ar ray}\right)$

**Ackbeet** · February 1st 2011, 01:28 PM

Scroll down a little farther.

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