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How do I cross Del with (scalar*vector)?

  1. Sep 5, 2016 #1
    1. The problem statement, all variables and given/known data
    Show that for any scalar field α and vector field B:

    ∇ x (αB) = ∇α x B + α∇ x B


    2. Relevant equations

    (∇ x B)i = εijk vk,j
    (∇α)i = αi
    (u x v)i = eijkujvk
    3. The attempt at a solution
    Since α is a scalar i wasn't quite sure how to cross it with ∇

    So on the left side I have:
    εijkBk,j αi

    I'm pretty sure I'm supposed to solve out the left side to get to the right because that what I did for the previous problem.

    The right side all I could get it to simplify to was:
    αi x B + α∇ x B
     
  2. jcsd
  3. Sep 5, 2016 #2

    Simon Bridge

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  4. Sep 5, 2016 #3
    So does that mean that my left side is just εijk Bk,j α ? I feel like it's supposed to be broken up into two parts
     
    Last edited: Sep 5, 2016
  5. Sep 5, 2016 #4

    Simon Bridge

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    It needs to if you are going to get RHS = LHS ... if in doubt go back to the definitions.
    Maybe look at ##(\alpha B)_i## also see what the RHS looks like in components. Play around until you understand it.
     
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