Q:

Compute flux of vector field curl F through the hemisfere

Could anyone help me with this question?
I need to compute the flux of vector field $\textrm{curl } F$ through the hemisphere
$$x = \sqrt{1 - y^2 - z^2}$$ 
Positively oriented, with the vector field
$$F(x,y,z) = \langle e^{xy}cosz, x^2z,xy \rangle$$
I've already tried to do spherical coordinates but I don't know if it is the way, because the terms became to large and not seems to take me anywhere.

A:

Use Stokes Theorem:
$$ \int_S \nabla \times \vec{F} \; \mathrm{d}\vec{S} = \oint_{\partial S} \vec{F} \; \mathrm{d}\vec{r}$$
where the line integral is along the open boundary of the hemisphere - which for your geometry is just the unit circle in the yz-plane at x = 0; from here the computation should be straightforward. 

