Let $\mathbf{F}=(\mathbf{u}\cdot\mathbf{x})\mathbf{v}$ where $\mathbf{u},\mathbf{v}$ are fixed. I am trying to find explicit expressions for $\nabla\cdot\mathbf{F}$ and $\nabla\times\mathbf{F}$. Let's start with the divergence:
$$\begin{aligned}\nabla\cdot\mathbf{F} &=(\nabla (\mathbf{u}\cdot \mathbf{x}))\cdot \mathbf{v}+(\mathbf{u}\cdot \mathbf{x})\underbrace{\nabla\cdot \mathbf{v}}_{=\,0}\\&=\Big(\mathbf{u}\times(\nabla\times \mathbf{x})+\mathbf{x}\times(\underbrace{\nabla\times \mathbf{u}}_{=\,0})+(\mathbf{u}\cdot \nabla)\mathbf{x}+(\mathbf{x}\cdot\nabla)\mathbf{u}\Big)\cdot \mathbf{v}\\&=\mathbf{v}\cdot(\mathbf{u}\times(\nabla\times \mathbf{x}))+(\mathbf{x}\cdot\nabla)(\mathbf{u}\cdot\mathbf{v})+(\mathbf{u}\cdot\nabla)(\mathbf{x}\cdot\mathbf{v})\end{aligned}$$
Is this correct? Can I simplify it further? Thank you for any help.
$\nabla(\mathbf u\cdot\mathbf x)=\mathbf u$, so $\text{div}\,\mathbf F=\mathbf u\cdot\mathbf v$. You can check that $\text{curl}\,\mathbf F=\mathbf u\times\mathbf v$.