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?

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    $\begingroup$ When you ask "am I OK?" are you asking "is the integral of this field along any closed path zero"? This is a common useful property of conservative fields on simply-connected domains. $\endgroup$ – Eric Towers Jan 31 at 14:55
  • $\begingroup$ Yes I am asking this. And I think that a field is conservative and the integral on every closed path is zero are equivalent statements. $\endgroup$ – spyer Jan 31 at 15:02

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.

  • $\begingroup$ I am asking can a field be potential(conservative) if the region where it is defined is not simply connected? $\endgroup$ – spyer Jan 31 at 15:09
  • $\begingroup$ It can. Why not? In fact restriction of a conservative vector field on $\mathbb R^n$ to any open subset will still be a conservative. $\endgroup$ – Blazej Jan 31 at 15:13
  • $\begingroup$ And if it is is it true that the integral on every closed path in the region is equal to zero? $\endgroup$ – spyer Jan 31 at 15:13
  • $\begingroup$ Yes, this is always true if you have a conservative vector field (but not if you demand only that curl vanishes, which is a weaker condition in non simply-connected regions). $\endgroup$ – Blazej Jan 31 at 15:15

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


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