 If the manifold formula_4 is smooth or analytic—that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields.The collection of all smooth vector fields on a smooth manifold formula_4 is often denoted by formula_22 or formula_23 (especially when thinking of vector fields as sections); the collection of all smooth vector fields is also denoted by formula_24 (a fraktur "X").Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del