Let $f$ a scalar function and $\vec{v}$ a vector function. We can write a product as:
$f \vec{v} = \vec{F}$
If we differentiate the LHS of above equation by the chain rule we can obtain
\begin{align} \nabla \cdot (f \vec{v}) = f \, \nabla \cdot \vec{v} + \vec{v} \, \nabla f \end{align}
Integrating we have:
\begin{align} \int \nabla \cdot (f \vec{v}) = \int f \, \nabla \cdot \vec{v} + \int \vec{v} \, \nabla f \end{align}
By inner product definition
\begin{align} \int \mathbf{D} (f \vec{v}) = \langle f, \mathbf{D} \vec{v} \rangle + \langle \vec{v} , \mathbf{G} f \rangle \end{align}
where $\mathbf{D}$ and $\mathbf{G}$ are the divergent and gradient operators. That lead us to
\begin{align} \int \mathbf{D} \vec{F} = \langle f, \mathbf{D} \vec{v} \rangle + \langle \vec{v} , \mathbf{G} f \rangle \end{align}
And by Fundamental Theorem of Calculus
\begin{align} \left. \vec{F} \right|_{\partial\Omega} = \langle f, \mathbf{D} \vec{v} \rangle + \langle \vec{v} , \mathbf{G} f \rangle \end{align}
where ${\partial\Omega}$ is the region boundary.
Is that correct?