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Vector Analysis Identity simplification/manipulation

  1. Apr 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex]\mathbf{G}(x,y,z)[/tex] be an irrotational vector field and g(x,y,z) a [tex]C^1[/tex] function. Use vector identities to simplify:

    [tex]\nabla\cdot(g\nabla \times (g\mathbf{G})) [/tex]

    2. Relevant equations

    The '14 basic vector identities'

    3. The attempt at a solution

    I tried using the identity [tex]\nabla\cdot(\mathbf{F} \times \mathbf{G}) = \mathbf{G}\cdot(\nabla\times\mathbf{F}) - \mathbf{F}\cdot(\nabla\times\mathbf{G})[/tex]

    But I'm not sure if i can treat [tex] g\nabla [/tex] as a vector?

    Really I'm quite clueless.
  2. jcsd
  3. Apr 17, 2010 #2


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    Science Advisor
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    No, I wouldn't treat [tex]g\nabla [/tex] as a vector, it's an operator. Start from the inside. You've got curl(gG). Irrotational tells you curl(G)=0. How does that let you simplify curl(gG)? BTW curl(X)=[tex]\nabla\times\mathbf{X}[/tex]. div(X)=[tex]\nabla\cdot\mathbf{X}[/tex].
    Last edited: Apr 17, 2010
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