4. Let F be the vector field F(x, y, z) = (xe$^z$, ?ye$^z$, 2). The vector field G is given in terms of a function g: R³ ? R by G(x, y, z) = (?y, x, g(x, y, z)).
a) Determine a function g such that curl(G) = F.
b) Sketch the surface S that is parametrized by ?(?, ?) = (?cos(?), $rac{1}{3}$?sin(?), (1 ? ?)$^2$), where ? ? [0, 1] and ? ? [0, 2?].
c) Evaluate
$\iint_S$ (F, n)d?,
where n is the upward pointing normal vector.