2D integration by parts I have the 2D integral $$\iint_{xy}\psi\textbf{u}_H\cdot\nabla_Hh\,{\rm d}x{\rm d}y,$$ with $\psi = \psi(x, y)$, $\textbf{u}_H = (u(x, y), v(x, y))$, $\nabla_H = (\partial/\partial x, \partial/\partial y)$ and $h = h(x, y)$. Is there a 2D integration by parts equivalent, so that I can remove the differentiation of $h$?

The integral seems to be of the form
$$
\iint_\Omega \mathbf{v}\cdot\nabla f \, dA
$$
The integration by parts equivalent is to use
$$
\nabla \cdot (f \mathbf{v}) = \nabla f \cdot \mathbf{v} + f (\nabla\cdot\mathbf{v})
$$
and then Stokes' theorem:
$$
\iint_\Omega \mathbf{v}\cdot\nabla f \, dA
= \iint_\Omega \left( \nabla\cdot (f\mathbf{v}) - f (\nabla\cdot\mathbf{v}) \right) \, dA
= \iint_\Omega \nabla\cdot (f\mathbf{v}) \, dA - \iint_\Omega f (\nabla\cdot\mathbf{v})  \, dA \\
= \oint_{\partial\Omega} f\mathbf{v} \cdot \mathbf{n} \, ds - \iint_\Omega f (\nabla\cdot\mathbf{v})  \, dA,
$$
where $\mathbf{n}$ is the normal on the boundary and $ds$ is boundary length measure.
