What is a conservative vector field on ?
State Green's theorem in the plane .
(a) Consider a smooth vector field defined on all of which satisfies
or otherwise, show that is conservative.
(b) Now let . Show that there exists a smooth function such that .
Calculate , where is a smooth curve running from to . Deduce that there does not exist a smooth function which satisfies and which is, in addition, periodic with period 1 in each coordinate direction, i.e. .