Now I’ll actually give the proof of the Darboux theorem that a symplectic manifold is locally symplectomorphic to with the usual form.

**Proof of the Darboux theorem **

We will prove the equivalent:

Theorem 1Let be a manifold with closed symplectic forms , and with . Then there are neighborhoods of and a diffeomorphism with .

The idea is to consider the continuously varying family of 2-forms

where

We will consider a small neighborhood of and a smooth map such that is a diffeomorphism, , and