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Question on Elementary Index Notation

  1. Sep 8, 2009 #1
    I have a question regarding the attached file. How do you get those indicies when you multiply the kronecker deltas with A, B, and C? For instance, C - subscript m remains the same on the left side of the expression, but then becomes C subscript i on the right side.

    How does this logically work out? What are the rules for these operations?

    Thanks.
     

    Attached Files:

  2. jcsd
  3. Sep 8, 2009 #2

    Astronuc

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    Re: Intro to elementary index notation

    [tex]\varepsilon_{kij}\varepsilon_{klm}A_jB_lC_m\,=\,(\delta_{il}\delta_{jm}\,-\,\delta_{im}\delta_{jl})(A_jB_lC_m)\\
    =\,A_mB_iC_m\,-\,A_lB_lC_i\,=\,B_iA_mC_m\,-\,C_iA_lB_l\,=\,B_i(\bold{A}\cdot{\bold{C}})\,-\,C_i(\bold{A}\cdot{\bold{B}})[/tex]

    The key is [tex]\varepsilon_{kij}\varepsilon_{klm}\,=\,(\delta_{il}\delta_{jm}\,-\,\delta_{im}\delta_{jl})[/tex]

    Is that clear?
     
    Last edited: Sep 8, 2009
  4. Sep 9, 2009 #3

    HallsofIvy

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    Do you understand that they are using the "summation convention"? That, since j, l, and m are repeated, there is an implied sum as j, l, and m take on values 1, 2, and 3. The final result cannot depend on j, l, or m.
     
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