Conservative field defined over a not simply-connected region

Question

I am wondering if a field can be conservative if the region where it is defined is not simply-connected. By definition $F$ is conservative if there exists a differentiable function which satisfies $F=grad(u)$ If I find such function which is defined in the not simply connected region, am I OK?

Answer 1

I am not sure what is your concern. It is perfectly valid to ask for a vector field to be equal to a gradient of some function. Let me explain briefly why simple connectedness comes up in the study of conservative vector fields. Whenever you have a vector field which is a gradient, it has vanishing curl. In simply-connected regions the converse is true: every vector field whose curl vanishes is a gradient. This implication in general fails to be true in domains which are not simply connected.

Answer 2

Yes, for example $X=\frac{\partial}{\partial x}$ is $\mathrm{grad}\,u$ on $\mathbb{R}^2-\{(0,0)\}$, where $u(x,y)=x$.