In one of my Calculus III classes the professor presented the following vector field, defined on the set $S$ of all points $(x,y) \neq(0,0)$ : $$ \bbox[6px,border:1px solid black] { \vec{F}=\frac{-y}{x^2+y^2}\vec{i} + \frac{x}{x^2+y^2}\vec{j} } $$
Although $\vec{F}$ is not a gradient on $S$, it is a gradient on the set $\Omega=\mathbb{R}^2-\left\{(x,y)\,|\,y=0, \, x\geq0\right\}$, i.e, all points in the xy-plane except those on the positive x-axis. In the class, my professor wrote a potential function for $\vec{F}$ on $\Omega$ using the following functions:
$$ \bbox[6px,border:1px solid black] { \begin{alignat}{0} \Psi_1(x,y)=-\arctan\left(\frac x y \right), \,y\neq0 &\text{and} &\Psi_2(x,y)=\arctan\left(\frac y x \right), \,x\neq0 \end{alignat} } $$
The first is a potential of $\vec F$ on $\Omega^+=\{(x,y)\in\mathbb{R}^2\,|\,y>0\}$ and on $\Omega^-=\{(x,y)\in\mathbb{R}^2\,|\,y<0\}$.
The last one is a potential of $\vec F$ too, but on $\Omega_+=\{(x,y)\in\mathbb{R}^2\,|\,x>0\}$ and on $\Omega_-=\{(x,y)\in\mathbb{R}^2\,|\,x<0\}$
The potential function wrote by my professor:
$$
\bbox[8px,border:1px solid black] {
\phi (x,y) =
\begin{cases}
\Psi_1(x,y) & \text{if $(x,y) \in R_1$} \\[2ex]
\Psi_2(x,y)+\frac\pi 2 & \text{if $(x,y) \in R_2$} \\[2ex]
\Psi_1(x,y)+2\pi & \text{if $(x,y) \in R_3$} \\
\end{cases}
}
$$
(Note: I remember that my professor started with the argument that since $\nabla\Psi_1=\vec F = \nabla\Psi_2$, then $\Psi_1-\Psi_2=k, \,k \in \mathbb{R}$)
My doubts:
- I've tried a lot, but still have not figured out how to build $\phi (x,y)$ using $\Psi_1$ and $\Psi_2$. So, how to find this specific potential function of $\vec F$ ?
- Why can't I simply write potential functions of $\vec{F}$ in the following ways $$ \phi_A (x,y) = \begin{alignat}{0} \begin{cases} \Psi_1(x,y) & \text{if $(x,y) \in R_1$} \\[2ex] \Psi_2(x,y) & \text{if $(x,y) \in R_2$} \\[2ex] \Psi_1(x,y) & \text{if $(x,y) \in R_3$} \\ \end{cases} &\text{or} &\phi_B(x,y)= \begin{cases} \Psi_1(x,y) & \text{if $y>0$} \\[2ex] 0 & \text{if $y=0$ and $x<0$} \\[2ex] \Psi_1(x,y) & \text{if $y<0$} \\ \end{cases} \end{alignat} $$ since $\nabla\phi_A=\nabla\phi_B=\vec F$ on $\Omega$ ?
