Vector Analysis: $\varphi$, $\mathbf{v}$, and Their Cross Products

[Dustinsfl]

$\newcommand{\unit}[1]{\hat{\mathbf{#1}}}$
Let $\varphi = F_1h_1$ and $\mathbf{v} = \frac{\unit{u}_1}{h_1}$.
Why is $\varphi\nabla\times\mathbf{v} = 0$ but $\nabla\varphi\times\mathbf{v}$ not?

 

[Ackbach]

Of the symbols $F_{1}$, $h_{1}$, and $\hat{\mathbf{u}}_{1}$, which are variable, depending on position, and which are constant?