Rather a silly problem! Yes, the flow F(x,y,z)= (x, y, z), which is a flow directly away from the origin is "curl free" or "irrotational": . You don't need Gauss's theorem, the flux integral is 0. Are you sure you have the right F and are you sure you want rather than just ? If I remember correctly, the latter is the "flux" of F through the surface.