Only conservative vector fields are path independent?

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Only conservative vector fields are path independent, meaning the integral of a vector field along a path depends solely on the endpoints, not the specific path taken. This property is closely tied to the concept of line integrals, where if a vector field is conservative, the integral around any closed path equals zero. The discussion raises questions about the precise definitions and contexts, such as whether the focus is on manifolds or subsets of Euclidean spaces. Clarification on these contexts is essential to fully understand the implications of path independence. Understanding these principles is crucial for exploring the fundamental theorem of line integrals.
princejan7
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does anyone have a proof of this?
 
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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##?
 

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