Using Gauss' Theorem to evaluate a flux integral?

Use Gauss' Theorem to evaluate the flux integral:

SS(curlF).ndS

where SS represents the double integral with respect to the surface S and where

F(x,y,z)=(x,y,z) and surface S is given by:

S={(x,y,z) such that z=1-x^2-y^2 and z>=0}.

I know how Gauss' theorem relates a flux integral to a triple integral of the div of the vector field but when I try to make curlF the vector field so that I can use Gauss' theorem to evaluate the triple integal of div(curlF) over the region bounded by S I can't because I get (curlF)=0 and I don't know where to go from there.Any help would be greatly appreciated.Thanks in advance.