How do I verify divergence theorem for spherical surface?

Question

The surface is bounded by $x^2+y^2+z^2 < 4$, $z > 0$. I can solve the problem using Divergence Theorem, but I want to verify my result evaluating the surface integral given by ∫⋅. Could you help me to discover the dS term?

Answer 1

For some paramaterized surface $\Sigma = \textbf{x}(u,v)$ as $u$ and $v$ range over some $T \subseteq \mathbb{R}^2$, we have

$$\iint_\Sigma \textbf{f}(\textbf{x}) \ \cdot dS = \iint_{T} \textbf{f}(\textbf{x}(u, v)) \cdot \left(\frac{\partial \textbf{x}}{\partial u} \times \frac{\partial\textbf{x}}{\partial v}\right) \ \text{d}u \ \text{d}v $$

In particular, using spherical coordinates you can evaluate the surface of your integral over the hemisphere and using polar coordinates the circle, both of which have easy paramaterizations. From here the bounds are pretty simple and the $dS$ terms just computation.