We have the vector field $F(x,y) = (-\frac y{x²+y²}, \frac x{x²+y²})$
We have three sets:
$V = \{(x,y) ∈ ℝ^2 : y > 0\}$
$W = \{(x,y) ∈ ℝ^2 : y < 0\}$
$T = \{(x,y) ∈ ℝ^2 : x > 0, y = 0\}$
We have the set $U = V ∪ W ∪ T$
I need to find the potentials for $ F$ restricted to $V$, $ F$ restricted to $W,$ and $F$ restricted to $U.$ I found the first two:
$F|_V = \arctan(y/x)$
$F|_W = -\arctan(y/x)$
But could anyone help for the last one?