Proving the Divergence Theorem for Bounded Domains and Differentiable Fields

[EngageEngage]

Homework Statement

Let the domain D be bounded by the surface S as in the divergence theorem, and assume that all fields satisfy the appropriate differentiability conditions.
Suppose that:
$$\nabla\cdot\vec{V}=0$$
$$\vec{W}=\nabla\phi with \phi = 0 on S$$
prove:
$$\int\int\int_{D}\vec{V}\cdot\vec{W}dV=0$$

This problem is in the Laplace's, Poisson's and Greens Formulas section. Truthfully I'm not sure where to even get started here. If anyone could give me a push in the right direction I would appreciate it greatly.

 

[EngageEngage]

nevermind; got it!

 

[EngageEngage]

I used a vector identity, but can someone please help me do this one without an identity. This is in a greens identities section, but none of the greens identities look like they would work. Is it right of me to say that
$$\vec{V} = curl \vec{G}$$

where g i ssome vector potential? or would this not help me at all?