Divergence Theorem - why can't I use it here to immediately evaluate this surface flux?

Question

For this question I found in a video:
Find the surface flux through the piece of cylinder with equation $x^2 + y^2 = a^2$ for $0 \leq z \leq h$ in the first octant, with vector field $F = (z,x,y)$.
The answer says that it is $\frac{1}{2} ( ah^2 + a^2 h)$.
However, why can't I use the divergence theorem, with $\mathrm{div}F = 0 + 0 + 0 = 0$, so the triple integral $\iiint \mathrm{div}F dV = 0$?

Answer 1

Because the surface at hand is not the boundary of a domain in $\mathbb{R}^3$.