1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Index Notation Indentity
  1. Index Notation (Replies: 9)

  2. Index Notation Help! (Replies: 2)

  3. Tensor index notation (Replies: 4)

Loading...