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Understanding the Kronecker Delta function

  1. Sep 20, 2007 #1
    1. The problem statement, all variables and given/known data
    I'm having some trouble understanding the Kronecker Delta function and how it is used. I understand the basics of it, if i=j, delta=1, if not, delta=0. However, I don't understand why:

    [tex]\delta_{ii}=3[/tex]
    and
    [tex]\delta_{ij}\delta_{ij}=3[/tex]


    2. Relevant equations

    [tex] \delta_{ij}= \left\{\begin{array}{cc}1,&\mbox{ if }i=j,\\0, & \mbox{ if } i\neq k\end{array}\right.[/tex]

    3. The attempt at a solution

    I have not been able to find any proofs using the Kronecker delta online anywhere so I'm not exactly sure of its function. My best guess of why [tex]\delta_{ii}=3[/tex] would be that you take the determinant of the identity function which would give you three. However, I'm not exactly sure what to do for the second one.

    Also could someone explain why [tex]\delta_{ii}=\delta_{11}+\delta_{22}+\delta_{33}=1+1+1=3[/tex]? That's what was in my book but they didn't have any explanation of why.
     
  2. jcsd
  3. Sep 20, 2007 #2
    Where did you find these equalities? They are not correct. I think that they were meant to be these:

    [tex]\sum_{i = 1}^{3} \delta_{ii}=3[/tex]
    and
    [tex]\sum_{i = 1}^{3}\sum_{j = 1}^{3} \delta_{ij}=3[/tex]
     
  4. Sep 20, 2007 #3

    arildno

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    The sigma summation sign is generally omitted in the Einstein summation convention, antonantal.

    The reason is that the sign is entirely superfluous.
     
  5. Sep 20, 2007 #4

    malawi_glenn

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  6. Sep 20, 2007 #5
    Thanks guys, that makes perfect sense now. I had forgotten that they could have been in Einstein notation.
     
  7. Sep 20, 2007 #6
    I knew about the Einstein notation but I didn't think it was used in the equalities above because the index variable appears only in the subscript position.

     
  8. Sep 20, 2007 #7

    arildno

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    That depends.

    In expressions where there are no possibilities of misunderstanding, why bother make the distinction between subscripts and superscripts?

    Just throw that notational element out as well.
     
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