Hi there,

I have across an identity connecting the Levi Civita and the Kronecker delta symbols:

Dimitri,
This is quite a new thing which I am come across.Can you give me some background on it.

love arun

arun,

I will get back to you in a few hours...im in a net cafe in france and cannot use this azerty keyboard....sorry

see ya

dimitri

okay,

very briefly

The other symbol called levi cevita symbol is defined according to the position of the index values... for instance $\epsilon (123)=1=-\epsilon (321)=\epsilon (132)$... in other words its value remains the same if all index values are permuted by the sign changes if only 2 index values change position ... there are other properties which you can easily find on the net
see ya

dimitri

(ps. im still in this net cafe and still cant use this keyboard....)

My rough estimation of what you have said is,

Kronecker d symbol has the properties of a 2x2 matrix(identity matrix)

levicevita symbol has the properties of a 3x3 determinant with det value 1.
Am i right in saying that??

love arun
P.S-> Don't they have a qwerty keyboard??

The levi-civita symbol is actually a special case (the 3 dimensional case) of an object called the totally antisymmetric unit pseudotensor (what a name!).

In n dimensions the symbol has n indices. The reason for this has to do with its (pseudo)tensorial nature - see below.
It is defined as follows:

e_1234...n =1
The remaining components are defined by the rule that: e_abcd...n =(-1)^x where x is the number of pairs of indices that have to be swapped to rearrange abcd...n to 1234...n

So for example in the 4 dimensional case:
e₁₂₃₄ =1
e₁₂₄₃ =-1
e₁₄₂₃ =1
and so on.

In 3 dimensions this reduces to the permutation rule.

It follows from this that if any two of the indices are the same then that component is zero, e.g. e₁₂₃₂ =0

This pseudotensor has the very nice and rare property that its components are the same (up to a sign) in all coordinate systems. The reason that it is defined with the same number of indices as dimensions is that if the number of dimensions was greater than the number of indices, then it wouldn't have the above property. If on the other hand there were more indices that dimensions then all the components would be zero.

It is: $e_{\mathrm{ijk}}{e}_{\mathrm{imn}}={\delta }_{\mathrm{jm}}{\delta }_{\mathrm{kn}}-{\delta }_{\mathrm{jn}}{\delta }_{\mathrm{km}}$
1. The LHS is a sum of 3 terms, but because of the pair swapping rule, for given j,k,m,n only 1 of the terms can be nonzero. If j=k or k=m then all the terms are zero - which agrees with the RHS
2. Since only one of the LHS terms can contribute, we can consider just the case i=1 without losing any generality
3. In this case {j,k}={2,3} or {3,2} and {m,n}={2,3} or {3,2} as well
4. So either: j=m and k=n or
j=n and k=m
5. Now looking at e_1jk e_1mn , if j=m and k=n then it will be 1, since the two factors will be equal. This agrees with the RHS where the negative term cancels
6. On the other hand, if j=n and k=m then e_1jk e_1mn =-1 since one factor is 1 and the other is just the same with one pair of indices swapped and therefore must be -1. This also agrees with the RHS, where the positive term cancels
7. But that's it, we have examined all the cases. So the result is shown.

Simon

By the way, there are some interesting identities derivable from the one above:

$e_{\mathrm{ijk}}{e}_{\mathrm{ijk}}=6$ There is also a formula for $e_{\mathrm{ijk}}{e}_{\mathrm{pqr}}$ in terms of Kronecker $\delta$s but it is long and not thatuseful.
It is amazing how easily vector identities come out when using this component notation. About other applications: since e_ijk is a rather important example of a pseudotensor, it plays a big part in general relativity, where such objects are ubiquitous.
It makes it possible to extend concepts like the curl of a vector, and also integral theorems like those of Gauss and Stokes, to more than 3 dimensions. Since relativity requires us to deal with a 4 dimensional spacetime, the Levi-Civita symbol is vital.

Simon