The solution of $\int_S z dx∧dy$ where $S:x^2+y^2+z^2=1, z≧0.$

surface-integrals divergence-theorem

Divergence theorem is applicable for a closed surface. So if you want to apply divergence theorem, you need to use a trick. Close the surface with a disk at $z = 0$ and then apply divergence theorem. That gives you the surface integral for the closed surface. Finally evaluate the surface integral for the disk and subtract. That would give you surface integral for the spherical surface.

So the answer you obtained is for the surface $\{ S: x^2 + y^2 + z ^2 = 1, z \geq 0 \} \cup \{S_1:x^2 + y^2 \leq 1, z = 0 \}$

Now it is easy to see that the surface integral of the vector field $\vec F = (0, 0, z)$ through the disk $S_1$ in plane $z = 0$ will be zero.

Hence the surface integral you obtained using divergence theorem is all coming from the spherical surface $S$.

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