Let S be the hemisphere x^2 + y^2 + z^2 = a^2, z \geq 0, oriented with unit normal pointing upward. Let C be the boundary curve, x^2 + y^2 = a^2, z= 0, oriented counterclockwise. Calculate

(a) \int_{S}{dx \wedge dy + 2zdz \wedge dx}
(b) \int_{C}{xdy + z^{2}dx}

Compare your answers and explain.

Solution:

Using spherical coordinates, I got \pi a^2 for both parts. The only part I'm not sure about is the explanation. Is this an application of Green's Theorem?