Does anybody happen to know the integration by parts formula for $\iint(\varphi\cdot \operatorname{curl}(u) dV)$, where both $\varphi$ and $u$ are 3D vectors? Is there a good reference for similar formulae?
The "intuitive" solution would be
$$\iint\varphi\cdot \operatorname{curl}(u) dV = \oint (\varphi \times u) \cdot dS - \iint \operatorname{curl}(\varphi) \cdot u dV$$
but this doesn't seem to quite work.
The vector analysis identity for the divergence of a cross product is:
$$\nabla\cdot(\phi\times u)=u\cdot(\nabla\times\phi)-\phi\cdot(\nabla\times u).$$
Taking the volume integral of both sides and applying the Divergence Theorem to the LHS,
$$\oint(\phi\times u)\cdot dS=\iiint u\cdot(\nabla\times\phi)dV-\iiint\phi\cdot(\nabla\times u)dV.$$
Rearranging,
$$\iiint\phi\cdot(\nabla\times u)dV=\iiint u\cdot(\nabla\times\phi)dV-\oint(\phi\times u)\cdot dS.$$
So your intuitive guess of the formula was of by an overall sign error, which you can blame on the antisymmetry of the cross product.
