Levi-Civita Symbol multiplied by itself

[RolloJarvis]

Homework Statement

evaluate $$\epsilon_{ijk}\epsilon_{ijk}$$ where \epsilon is is the antisymetric levi-civita symbol in 3D

Homework Equations

determinant of deltas = product of levi-civita -> would take ages to write out.

The Attempt at a Solution

$$\epsilon_{ijk}\epsilon_{ijk}=\delta_{kk}\delta_{ll}-\delta_{lk}\delta_{kl}=1-2\delta_{lk}$$

But i have a feeling the answer is 3? (because of this)

 

[praharmitra]

Well, firstly clean your notation. It should be
$$\epsilon_{ijk}\epsilon_{ijk}=\delta_{kk}\delta_{jj}-\delta_{jk}\delta_{jk}$$
And there is a summation over j and k.
So that gives
$$\sum\limits_{j,k}\delta_{kk}\delta_{jj}-\delta_{jk}\delta_{jk} = 3.3-3 = 6 = 3!$$
Also 3! = 6, not 3.

 

[RolloJarvis]

Thanks a lot. I was in a rush and I am pretty stressed at the moment as its only a few days before exams, the [tex]\epsilon [\tex]was supposed to read [tex]\epsilon_{jlk}[\tex].

How is it okay to swap the order of indicies on the delta funtion as you have done: i.e. [tex]\delta_{ij} = \delta_{ji}[\tex]

 

[praharmitra]

ya, that's a property of the delta function.

 

[RolloJarvis]

again, thanks a lot