Say $F=(F_1(x),F_2(x),F_3(x))$ and $x\in\mathbb{R}^3$.
The standard divergence theorem states: $\int\int F\cdot n dS=\int\int\int\nabla\cdot F dV$.
Then, if $G=(G_1(x),G_2(x),G_3(x))$, what can we say about $\int\int (F\cdot n)GdS$?
Or, if you prefer, you can think of $G$ as a scalar valued function of $x$.
Consider the case of a scalar function $\phi$ (you can apply this to each component of your $G$ if you wish). Then, by the Divergence Theorem, $$\iint (\phi F)\cdot n\,dS = \iiint \nabla\cdot(\phi F)\,dV = \iiint \big(\phi\,\nabla\cdot F+\nabla\phi\cdot F\big)\,dV.$$