Vector Analysis Identity simplification/manipulation

[jspectral]

Homework Statement

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

$$\nabla\cdot(g\nabla \times (g\mathbf{G}))$$

Homework Equations

The '14 basic vector identities'

The Attempt at a Solution

I tried using the identity $$\nabla\cdot(\mathbf{F} \times \mathbf{G}) = \mathbf{G}\cdot(\nabla\times\mathbf{F}) - \mathbf{F}\cdot(\nabla\times\mathbf{G})$$

But I'm not sure if i can treat $$g\nabla$$ as a vector?

Really I'm quite clueless.

 

[Dick]

No, I wouldn't treat $$g\nabla$$ 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)=$$\nabla\times\mathbf{X}$$. div(X)=$$\nabla\cdot\mathbf{X}$$.