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Del operator crossed with a scalar times a vector proof

  1. Jan 12, 2013 #1
    "Del" operator crossed with a scalar times a vector proof

    1. The problem statement, all variables and given/known data
    Prove the following identity (we use the summation convention notation)

    [tex]\bigtriangledown\times(\phi\vec{V})=(\phi \bigtriangledown)\times\vec{V}-\vec{V}\times(\bigtriangledown)\phi[/tex]

    2. Relevant equations

    equation for del, the gradient, curl..

    3. The attempt at a solution

    im kind of confused on the first step...I broke it down into the following; however, levi civita symbols aren't my cup of tea and I get pretty confused about it...anyway heres what I did:

    [tex]\bigtriangledown\times(\phi\vec{V})=(\epsilon_{ijk})\partial_i\vec{V}\phi\hat{x}_k[/tex]

    I dont know if this first step is right or if I decomposed the cross product right ?
     
    Last edited by a moderator: Jan 13, 2013
  2. jcsd
  3. Jan 13, 2013 #2
    Re: "Del" operator crossed with a scalar times a vector proof

    You should have the j:th component of V in your expression,
    [tex] \nabla \times (\phi \vec{V}) = \epsilon_{ijk} \partial_i (\phi V_j) \hat{x}_k [/tex]
     
  4. Jan 13, 2013 #3

    dextercioby

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    Re: "Del" operator crossed with a scalar times a vector proof

    The form you've written is completely equal to the one seen in math/physics books

    [tex] \nabla\times \left(\phi\vec{V}\right) = \nabla\phi\times\vec{V} + \phi\nabla\times\vec{V} [/tex]
     
  5. Jan 13, 2013 #4
    Re: "Del" operator crossed with a scalar times a vector proof

    Thanks for the replies, I'm just not sure what to do after what clamtrox said to do, the whole proofing business if pretty new to me :/
     
  6. Jan 13, 2013 #5

    dextercioby

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    Re: "Del" operator crossed with a scalar times a vector proof

    You need two more steps. Apply the product rule for differentiation and then once you obtain a sum of two terms, reconstruct vectors from their components.
     
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