Problem 1: In class we showed that curl F =0 - F = Vf. In this problem you will show that Hamiltonian vector fields satisfy an analogous relationship but with divergence instead of the curl. First assume F is Hamiltonian and show that the divergence is zero and the flux is path independent. (i) Show that if F is a Hamiltonian vector field, then div(F) = 0. Suggested strategy: direct calculation; compute div(F). (ii) Show that if T'(t) = (x(t), y(t)) is a parametrization of the curve C from 7' (0) = P to '(1) = Q, then the flux of F through C is path independent. Suggested strategy: try to repeat the steps in the proof (in the book or my class notes) of the fundamental theorem for line integrals, but replace the line integrals of F with flux integrals, and (obviously) use that F is Hamiltonian instead of conservative. Now show the reverse implication: assume div( F) = 0 and show that F is Hamiltonian. (ii) Show that if div(F) = 0 then F is a Hamiltonian vector field. Suggested strategy: first show that if div(F) = 0 then the flux integral of F is path independent (use the Divergence Theorem). Then use path independence (of the flux) to reconstruct the Hamiltonian function H (see the proof that path independence of the line integral implies F is conservative in the textbook).Problem 2: In class we learned that conservative fields are called conservative because mechanical energy E is conserved when an object moves subject to a conservative force: we used chain rule to find # E, and then used ? = F (Newton's Second law) to simplify this expression yielding zero. (i) Use a similar strategy to show that H is conserved along solutions to the differential equation 7"(t) = F, where F is Hamiltonian. (ii) Give a geometric interpretation of this fact