Use the divergence theorem to evaluate s f · ds, where f(x, y, z) = z2xi + y3 3 + tan z j + (x2z + y2)k and s is the top half of the sphere x2 + y2 + z2 = 9. (hint: note that s is not a closed surface. first compute integrals over s1 and s2, where s1 is the disk x2 + y2 ≤ 9, oriented downward, and s2 = s1 ∪ s.)

By the divergence theorem, the surface integral over is



where denotes the space bounded by . Assuming the vector field is given to be



then



Converting to spherical coordinates, we take



so that the triple integral becomes





Now the integral over alone will be the difference of the integral over and the integral over , i.e.



We can parameterize the points in by



so that the integral over is






So, the integral over alone evaluates to