Given a 2-d conservative vector field $F(x,y)=(f_1(x,y),f_2(x,y))$, I would like to perturb it with another unit vector field $G(x,y)$ so as to make the curl of the new vector field $F(x,y)+G(x,y)$ maximized. Is there any approach to assign $G(x,y)$ to achieve this aim?
My naive idea is to assign $G(x,y)$ as the unit tangent of $F(x,y)$, i.e., $G(x,y)=\frac{1}{r}(-f_2(x,y),f_1(x,y))$, where $r=\sqrt{f_2(x,y)^2+f_1(x,y)^2}$. But I can not prove if this idea is correct. Is there any reference for such a problem?
Thanks.