Divergence Theorem Exercise

Question

The problem is as follows:

Given that

1. $\nabla \cdot \mathbf{F} = 0$ in $V$
2. $\mathbf{F} \cdot \mathbf{n} = 0$ on $\partial V$

Prove that: $\int_V \mathbf{F} \, dV = 0$.

I understand it intuitively, but I can't figure out how to show the result. All I have is below which just leads to $0=0$

$\int_V (\nabla \cdot \mathbf{F}) \, dV = \int_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS$

Answer 1

Perhaps Green's identity? It says $$ \int_V \left(\psi \: \nabla \cdot \mathbf{F} + \mathbf{F} \cdot \nabla \psi \right)dV = \int_{\partial V} \psi \: (\mathbf{F} \cdot \mathbf{n})dS $$ for any differentiable $\psi$. The RHS is zero, the first term of the LHS is zero, and thus it reduces to
$$\int_V \mathbf{F} \cdot \nabla \psi \: dV =0 $$ Taking $\psi$ to be $x_i$ leads to $\int_V F_i dV = 0$.