# Trying to understand Levi-Civita Symbol and notation

Hello,

I am having a little difficulty understanding what exactly the Levi-Civita symbol is about.

In the past I believed that it was equal to 1, -1 and 0, depending on the number of permutations of i,j,k. I had just accepted that to be the extent of it.

However, now I am seeing things like summations with i,j,k mentioned as WELL as other letters, for example m and n in the link below. I tried reading online about it to understand what exactly it is, especially here:

http://en.wikipedia.org/wiki/Levi-Civita_symbol#Relation_to_Kronecker_delta

How can something that is the value of -1, 1 or 0 be written as a sum, with i=1 up to 3 as written in the link, without any mention of the values of j or k? Whenever I see summation, I think substitute i=1 first, then add with i=2, then add the value with i=3, and that's the answer. But if it's E_ijk E_imn, if I sub into the 'contracted epsilon identity' as shown on wikipedia I get

E_1jk E_1mn + E2jk E_2mn + E3jk E_3mn

How do I know what to do with the levi-civita symbol if I don't know what to with all these other letters?

What does this expression even mean? :S

Sorry for my lack of knowledge here, I haven't had much experience ever using it and I'm worried I'll fall behind in class if I don't get this cleared up.

Thanks!

## Answers and Replies

tiny-tim
Science Advisor
Homework Helper
… However, now I am seeing things like summations with i,j,k mentioned as WELL as other letters, for example m and n in the link below. I tried reading online about it to understand what exactly it is, especially here:

http://en.wikipedia.org/wiki/Levi-Civita_symbol#Relation_to_Kronecker_delta

How can something that is the value of -1, 1 or 0 be written as a sum, with i=1 up to 3 as written in the link, without any mention of the values of j or k? Whenever I see summation, I think substitute i=1 first, then add with i=2, then add the value with i=3, and that's the answer. …

Hello Werbel22! i=1 up to 3 is only for three dimensions.

For the generalisation to n dimensions, i = 1 to n.

The product of two Levi-Civita symbols in n dimensions is given later on that page, as a determinant.

The ∑ with i=1 up to 3 as written in the link is for fixed values of j k m and n (so the RHS is a function of j k m and n) … there's no summing over j and k because they're fixed. So then what is the value of the levi-civita symbol then? I only know it to be -1, 1, 0, and I thought you need to know about j and k as well to find it's value?

tiny-tim
Science Advisor
Homework Helper
?? if j and k are given, then you do know their values.