Consider the following change of variables in phase space f: maps the reals and is smooth and invertable change of coordinates Q=f(q), q = f-1(Q). Given f, define a change of variables on phase space (q,p) -> (Q,P) by the pair of relations Pj = (∇f-1)Tjk(f(q))pk q runs from 1 to n show that its canonical. I know that for this to be canonical (dQi/dqj)(q,p) = (dpj/dPi)(Q,P) (dQi/dpj)(q,p) = -(dqj/dPi)(Q,P) i'm having a couple problems, is f the generating function that i have to find explicitly? Can i use the sympletic method such that MJM^T = J what is the point of the transpose for the (∇f-1)Tjk part for? i thought f had to be symmetric.