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Vector Analysis Identity simplification/manipulation

  • Thread starter jspectral
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  • #1
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Homework Statement



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]


Homework Equations



The '14 basic vector identities'

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.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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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:

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