Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative
And if $D$ is a triply connected set( that is $D$ has two holes)?
I was thinking about a conservative field such as $F(x,y)=(x^2,y^2)$ and just take $D=\mathbb R^2 \setminus B_1(0,0)$ where $B_1(0,0)$ is the ball of radius $1$ center $(0,0)$ but I don´t kow if this is a good example, what do you think?