A conservative vector field is a type of vector field in which the total work done by the field along a path depends only on the initial and final positions (the endpoints of the path) and not on the specific path taken. In other words, if you move from point A to point B in a conservative vector field, the work done is the same regardless of the trajectory taken between these two points.
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A vector field is conservative when for a potential of a conservative vector field . On a simply connected open subset of , every continuously differentiable curl-free vector field is conservative. Locally this follows from the Poincare lemma applied to the differential one-form ; globally, simple connectivity makes its line integral independent of the path. The fundamental theorem for line integrals expresses that path independence as the difference of the potential at the endpoints.