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Trying to understand Levi-Civita Symbol and notation

  1. Mar 2, 2010 #1

    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:


    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.

  2. jcsd
  3. Mar 2, 2010 #2


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    Hello Werbel22! :smile:

    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. :wink:
  4. Mar 3, 2010 #3
    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?
  5. Mar 3, 2010 #4


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    ?? :confused: if j and k are given, then you do know their values.
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