Editing Conservative vector field (section)

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