So I am working with some exams and stumbled upon this true/false problem:
The vectorfield $G=[\frac{y}{x^2+y^2}, \frac{-x}{x^2+y^2}]$ can be written as $G=\nabla \phi$ on an area $I$ that is simply connected in $R^2$ that does not contain $(0,0)$.
So I am supposed to find out if this statement is true or false. The solution says true, and I don't understand why. If G is the gradient of a vectorfield $\phi$, must it not be conservative? And conservative vectorfields must be defined on a region that is simply connected. How can the problem say that it is defined on a simply connected domain when there is a hole in the domain? I know that in $R^3$ a simple connected domain can contain holes like this, but I not in $R^2$? So my question is then, must the domain be simply connected for a vectorfield to be conservative?