/sci/ - Science & Math

Prove that the assignation [math]X \rightarrow ~ i_X \omega[/math] defines a surjective linear map [math]\mathcal{H} _{loc} (W) \rightarrow H^1 (W; \mathbb{R})[/math]. Deduce an exact sequence of vector spaces [eqn] 0 \rightarrow \mathcal{H} (W) \rightarrow \mathcal{H} _{loc} (W) \rightarrow H^1 (W; \mathbb{R} ) \rightarrow 0[/eqn]
Where [math](W, \omega)[/math] is a symplectic manifold, [math]\mathcal{H} (W)[/math] is the vector space of Hamiltonian vector fields, [math]\mathcal{H} _{loc} (W)[/math] are locally Hamiltonian vector fields, and the cohomology is de Rham.
Shouldn't take you more than a minute, it is basically just applying the definitions.

>>11360420
Yeah, of course there is, but you'll need to pass a little test to get in.
Prove that the assignation [math]X \rightarrow ~ i_X \omega[/math] defines a surjective linear map [math]\mathcal{H} _{loc} (W) \rightarrow H^1 (W; \mathbb{R})[/math]. Deduce an exact sequence of vector spaces [eqn] 0 \rightarrow \mathcal{H} (W) \rightarrow \mathcal{H} _{loc} (W) \rightarrow H^1 (W; \mathbb{R} ) \rightarrow 0[/eqn]
Where [math](W, \omega)[/math] is a symplectic manifold, [math]\mathcal{H} (W)[/math] is the vector space of Hamiltonian vector fields, [math]\mathcal{H} _{loc} (W)[/math] are locally Hamiltonian vector fields, and the cohomology is de Rham.
Shouldn't take you more than a minute, it is basically just applying the definitions.