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

(a) $\displaystyle \int_{S}{dx \wedge dy + 2zdz \wedge dx}$

(b) $\displaystyle \int_{C}{xdy + z^{2}dx}$

Compare your answers and explain.

Solution:

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