More abstractly, in the presence of a Riemannian metric, vector fields correspond to differential . The conservative vector fields correspond to the exact , that is, to the forms which are the exterior derivative formula_66 of a function (scalar field) formula_67 on formula_3. The irrotational vector fields correspond to the closed , that is, to the formula_69 such that formula_70. As any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed are exact if formula_3 is simply connected.