Are you sure that your question is related to the Stokes Theorem?
Given a Vector field and a surface we know that
The boundary of this curve is the unit circle in the xy plane.
My question is what is the integral you started with and what are you trying to compute? Are you trying to compute the flux of
though the hemisphere?
Edit: Just typing too slow
Use the divergence theorem as hinted by others. The volume you're dealing with is a hemisphere which is easy to deal with in spherical coordinates.
Spherical coordinate system - Wikipedia, the free encyclopedia
The limits for a hemisphere with radius=1 is: