Conservative vector field
Abstraction

More abstractly, in the presence of a Riemannian metric, vector fields correspond to differential 1 {\displaystyle 1} -forms. The conservative vector fields correspond to the exact 1 {\displaystyle 1} -forms, that is, to the forms which are the exterior derivative d ϕ {\displaystyle d\phi } of a function (scalar field) ϕ {\displaystyle \phi } on U {\displaystyle U}. The irrotational vector fields correspond to the closed 1 {\displaystyle 1} -forms, that is, to the 1 {\displaystyle 1} -forms ω {\displaystyle \omega } such that d ω = 0 {\displaystyle d\omega =0}. As d 2 = 0 {\displaystyle d^{2}=0}, any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed 1 {\displaystyle 1} -forms are exact if U {\displaystyle U} is simply connected.
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