I'm trying to complete an exercise in vectorial calculus, integrating the following vector field in the sense finding (one of) the function it derives from by taking the gradient:
$$ \vec{v} = (\frac{y}{x^2+y^2},\frac{-x}{x^2+y^2},0) $$
Doing it in cylindrical coordinates, I get $f = - \theta $ or $ - \text{arctan}(\frac{y}{x})$
I see that this thing cannot be uniquely defined in the $(x,y)$ plane because it pickups an additive factor of $2\pi$ after a rotation when it should be invariant.
It looks like the functions needing to be treated with a branch cut (I'm thinking complex logarithm). To close this exercise I need to answer why it implies that this function cannot be defined everywhere but I'm not seeing the point they want me to raise; is it linked to what I just commented?