## Conservative vector field conditions

My calculus book states that a vector field is conservative if and only if the curl of the vector field is the zero vector. And, as far as I can tell a conservative vector field is the same as a path-independent vector field.

The thing is, I came across this: http://www.math.umn.edu/~nykamp/m2374/readings/pathindex/

The site shows a vector field where the curl is equal to the zero vector, yet the vector field is not conservative.

As far as I can tell, saying "F is conservative iff Curl(F) = 0" contradicts the claims of the site I posted.

What conditions must be met for a vector field to be conservative?

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 Like the site says, the curl should be zero and the domain should be simply connected. Your calculus book probably implies that you work in R^n or some other simply connected space. A conservative field is usually defined as one that is a gradient of some scalar field. Curl of gradient is automatically zero. On the other hand, if you have a field whose curl is zero and its domain is simply connected, you can smoothly deform any closed path into a point, and then it's possible to prove that the field is path-independent, therefore you can construct the scalar field out of the vector field.

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