Explaining result: comparing line integral and two-form integral

Dec 2009
30
4
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?