I want to prove the following:
If $\nabla \cdot \mathbf{v}= 0$ and $\mathbf{v}$ is parallel to $\partial D$ then
$\int_D \nabla f \cdot (\mathbf{v} \cdot \nabla \mathbf{y}) dx =0 $
where $f$ is a scalar potential and $\mathbf{v},\ \mathbf{y}$ are vector fields.
It must be something very straightforward like applying the divergence theorem or Stokes's but I cannot see it. I'd appreciate any help.
Thanks