Let be a symplectic manifold. It said to be exact if for some one-form on .
(1) If is exact, then there is a canonical isomorphism between the v.f. and 1-forms. In particular, there exists a v.f. such that . Then we have , and , and .
(2) Suppose there exists a vector field on such that its Lie-derivative (notice the difference with ). Then Cartan’s formula says that , where . So is exact, and .