Among all smooth, simple closed curves in the plane, oriented counterclockwise, find the one along which the work done by:
$F = <\frac14x^2y + \frac13y^3, x>$
is greatest. Hint: where is curl $F \cdot k$ positive?
I tried evaluating Green's theorem but get stuck, and am not sure how to find the maximum work.
Take the curl of $F$ and find its $z$-component: $$k\cdot\nabla\times F = k\cdot\langle 0,0,1-\frac{x^2}{4}-y^2\rangle=1-\frac{x^2}{4}-y^2,$$
which is non-negative in the region bounded by the ellipse $\frac{x^2}{4}+y^2=1$. This is the region of integration that maximizes the double integral, and thus by Stokes' its bounding curve (the ellipse) maximizes the work.
