Only conservative vector fields are path independent?

[princejan7]

does anyone have a proof of this?

 

[micromass]

Could you elaborate on your statement? What does it mean for a vector field to be path independent? Could you give the precise statement (which I guess have to do something with integrals over a path). Furthermore, in what generality are you interested? Manifolds, surfaces in $\mathbb{R}^n$, open subsets of $\mathbb{R}^2$?

 

[mathwonk]

A=B implies A-B = 0. so independence of path for integrals between two points implies the integral around the closed path they form equals zero.
http://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx