Consider a vector field $F: \mathbb{R}^3 \to \mathbb{R}^3$ that has, among the components, an absolute value containing the variables, and suppose that I need to find the potential of that vector field. An example could be $$F=(|f_1(x,y,z)|,f_2(x,y,z),f_3(x,y,z))$$
Suppose also that I manage to prove that $F$ is irrotational in all $\mathbb{R}^3$.
The question is: where do I look for the potential?
Should I try to find two different potentials in the two separate regions? That is one potential $U_1$ in $A_1$ where $f_1(x,y,z)>0$ considering $F=(f_1(x,y,z),f_2(x,y,z),f_3(x,y,z))$ and then another potential $U_2$ in $A_2$ where $f_1(x,y,z)<0$ and $F=(-f_1(x,y,z),f_2(x,y,z),f_3(x,y,z))$?
Or should I find one potential directly in all $\mathbb{R}^3$? (In that case the potential would contain an absolute value)