Helmotz decomposition theorem says, on one hand, that every vector field $F$ sufficiently smooth can be decomposed into the sum of a solenoidal field $\nabla\times \bf A$ and a gradient field $\nabla \psi$. In other words: $$F = \nabla\times {\bf A} + \nabla \psi.$$ On the other hand, this Wikipedia article lists the following identities:
- $\iiint_{\Omega} \nabla \times {\bf A} dV =- \iint _{\partial \Omega} {\bf A} \times d\bf S$,
- $\iiint_{\Omega} \nabla \psi dV = \iint_{\partial \Omega} \psi d\bf S$.
But that just means that the volume integral of any sufficiently smooth field can be transformed into a surface integral, isn't it?