I currently learning vector calculus and am having difficulty finding any auxiliary resources about the geometric interpretation of conservative vector fields. When learning the definition, and even more so when learning the properties, I feel like there should be some geometric reasoning behind both the properties of conservative vector fields, as well as the resulting path independence as related to the potential function.
If anyone any good resources that address this question or anything related to it, I would greatly apreciate it. Again, not looking for an algebraic proof, rather some intuition behind the interplay between the potential function and the properties of its gradient vector field. Thanks.