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,
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,