For what smooth functions $f:\mathbb R^3\to \mathbb R$ is there a smooth vector field $W:\mathbb R^3\to \mathbb R^3$ with $\operatorname{curl}W=V$, where $$V(x,y,z)=(y,x,f(x,y,z)).$$ For $f$ in this class, find such a $W$. Is it unique?
I wrote out $\operatorname{curl}W$ explicitly as $$\operatorname{curl}W=(D_2w_3-D_3w_2,D_3w_1-D_1w_3,D_1w_2-D_2w_1)$$ and the condition in question is $$(D_2w_3-D_3w_2,D_3w_1-D_1w_3,D_1w_2-D_2w_1)=(y,x,f(x,y,z))$$ I guess I have to suppose that everything takes place in the same $\mathbb R^3$ with the same coordinates $x,y,z$. Then $w_i=w_i(x,y,z)$ and $$D_2w_3-D_3w_2=y\\D_3w_1-D_1w_3=x\\D_1w_2-D_2w_1=f(x,y,z)$$
For the first two equations I can choose $w_3=y^2/2-x^2/2$ and $w_1,w_2$ arbitrary. But what to do with the third condition? Should I write $w1,w_2$ as one-variable integrals of $f$ or something? The above was for the "find $W$ part". Also how to find the $f$'s we need?