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
The statement is false.
Consider D the unit circle, $\textbf{v}=(y,-x)$, $\textbf{y}=(0,x)$, and $f=y^2$.
We have \begin{align*}\int_D \nabla f \cdot(\textbf{v}\cdot\nabla\textbf{y})\,dA &= \int_D (0,2y)\cdot \left[\begin{array}{cc}0 & 0 \\1 & 0\end{array}\right]\left[\begin{array}{c}y\\-x\end{array}\right]\,dA\\ &= \int_D 2y^2\,dA \end{align*} which clearly can't vanish.