How to find the vector potential of the field $F = (-y,x,0)$?

Question

Given the vector field $F = (-y,x,0)$, how can I find a vector potential of this field? I know that we have to find a vector field $G$ such that $F = \nabla \times G$, but I am struggling with finding a right solution.

Answer 1

Observe that the vector field $G = (xz, yz, 0)$ has the property that $\nabla \times G = F.$

To find such a potential, you need to hind a vector field $G = (G_1, G_2, G_3)$ such that $\nabla \times G = F$. But, observe that you can always choose one of the functions $G_1, G_2$ or $G_3$ to be equal $0$, so this makes computations easier.