Stoke' Theorem problem surface integral

Question

Suppose $F = <-y ,x ,z>$ and $S$ is the part of the sphere $x^2 + y^2 + z^2 = 25$ below the plane $z=4$, oriented with the outward-pointing normal (so that the normal at $(5,0,0)$ is i). Compute the flux integral //curl F.dS using Stoke’s theorem?

Answer 1

Hint: the surface $S$ can be easily parametrized using spherical coordinates: $$ x = 5\cos\theta\sin\phi,\qquad y = 5\sin\theta\sin\phi,\qquad z = 5\cos\phi,$$ $$\theta\in[0,2\pi],\qquad\phi\in[\arctan(3/4),\pi].$$

Answer 2

Denote the flux with $\Phi_F$. Notice that $div \bar F = 0$. Using Ostrogradsky theorem, we find:

If the plane is part of the surface, the entire surface is closed, so:

$$\Phi_F = \iint_S \bar F.\bar n dO = \iiint_V div \bar FdV = 0$$