In a proof from Boas mathematical methods, the solution has (where $\vec{C}$ is an arbitrary vector)
$$\iiint_\tau \nabla\cdot(\vec{V} \times \vec{C}) \, d\tau = \iint_S (\vec{V}\times \vec{C}) \cdot \hat{n} \, d\sigma$$
which then turns into
$$\vec{C}\cdot \iiint_\tau (\nabla\times \vec{V}) \, d\tau = \vec{C}\cdot \iint_S (\hat{n}\times\vec{V}) \, d\sigma$$
which seems to be missing terms as $\nabla\cdot(\vec{V} \times \vec{C}) = \vec{C}\cdot(\nabla \times \vec{V}) - \vec{V}\cdot(\nabla\times \vec{C})$. There are similar missing terms on the right hand side. Can anyone help me understand why these terms vanish?
$\vec{C}$ is an arbitrary vector, not a vector field, so it is constant. Therefore any derivatives of it, including the curl, are zero.