Editing Conservative vector field (section)

{{short description|Vector field that is the gradient of some function}}
{{More footnotes|date=May 2009}}
In [[vector calculus]], a '''conservative vector field''' is a [[vector field]] that is the [[gradient]] of some [[function (mathematics)|function]].<ref>{{cite book|title = Vector calculus|first1 = Jerrold|last1 = Marsden|author-link1 = Jerrold Marsden |first2 = Anthony|last2 = Tromba|publisher = W.H.Freedman and Company|edition = Fifth|year = 2003|pages = 550&ndash;561}}</ref> A conservative vector field has the property that its [[line integral]] is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field under the line integral being conservative. A conservative vector field is also '''irrotational'''; in three dimensions, this means that it has vanishing [[curl (mathematics)|curl]]. An irrotational vector field is necessarily conservative provided that the domain is [[simply connected]].

Conservative vector fields appear naturally in [[mechanics]]: They are vector fields representing [[force]]s of [[physical system]]s in which [[energy]] is [[conservation of energy|conserved]].<ref>George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'', 6th edition, Elsevier Academic Press (2005)</ref> For a conservative system, the [[work (physics)|work]] done in moving along a path in a configuration space depends on only the endpoints of the path, so it is possible to define [[potential energy]] that is independent of the actual path taken.