 Conservative vector field

A vector field $X$ is conservative when $X=\nabla \Phi$ for a potential of a conservative vector field $\Phi$. On a simply connected open subset of ${\mathbb{R}}^3$, every continuously differentiable curl-free vector field is conservative. Locally this follows from the Poincaré lemma applied to the differential one-form $X_{i}d{x}^i$; 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.

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.