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Index Notation Indentity

  1. Nov 1, 2012 #1
    1. The problem statement, all variables and given/known data

    Simplify the following, where A and B are arbitrary vector fields:

    f(x) = ∇[itex]\bullet[/itex][A [itex]\times[/itex] (∇ [itex]\times[/itex] B)] - (∇ [itex]\times[/itex] A)[itex]\bullet[/itex](∇ [itex]\times[/itex] B) + (A [itex]\bullet[/itex] ∇)(∇ [itex]\bullet[/itex] B)

    I know that the correct solution is A [itex]\bullet[/itex] ∇2B, according to my professor. However, I can't get that. I think my mistake is in the first couple of lines, but I'll write out my entire solution and hopefully someone can tell me where I messed up. Thanks!

    2. Relevant equations

    3. The attempt at a solution

    f(x) = ∂iεijkAjεkabaBb - εijkjAkεiabaBb + AiijBj

    f(x) = εkijεkabiAjaBb - εijkεiabjAkaBb + AiijBj

    (note that I changed εijk to εkij in the first term)

    f(x) = (δiaδjb - δibδja)∂iAjaBb - (δjaδkb - δjbδka)∂jAkaBb + AiijBj

    f(x) = ∂iAjiBj - ∂iAjjBi - ∂jAkjBk + ∂jAkkBj + AiijBj

    Now the first term cancels with the third term, and the second term cancels with the fourth term, so we are left with:

    f(x) = (A [itex]\bullet[/itex] ∇)(∇ [itex]\bullet[/itex] B)

    But apparently this isn't right.
  2. jcsd
  3. Nov 1, 2012 #2


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    Are you allowed to use the properties of the triple product: a.(bxc) = b.(cxa) etc?
  4. Nov 1, 2012 #3
    We're allowed to use pretty much whatever we want, as long as I understand it and it makes sense.
  5. Nov 1, 2012 #4


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