I might be a n00b in physics but I am quite sure the flux is, the surface integral:

Where is the surface, is the vector field and is normal vector.

The first step is to parametrize the surface. That can be written as for .

Next is you compute the partials,

.

Now compute,

But I could have also computed and got the same vector only with a negative. So which way do I pick it? Remember the problem says the normal vector points downward. So pick a point on the sphere, say, the vector should be for , i.e. a vector pointing some positive scalar multiple in the same direction as its inward normal . Is that true? Let us just check this, on this point , and it is true! So we have the inward normal vector parametrization.

Thus, the surface integral is,

Which simplifies to,