A vector field $F=yi+2j+k$ exists over a surface S defined by $x^2+y^2+z^2=9$ bounded by $x=0$, $y=0$ and $z=0$ in first octant. Find flux of $F$ over the surface indicated (Ans:$9(1+3\pi/4)$)
calculating the divergence using gauss theorem, we see that $$\iint_{S}\mathbf F\cdot dS=\iiint_V\operatorname{div}(\mathbf F)\,dV$$ but $\operatorname{div}(\mathbf F)=0,$ so the flux over the surface is $0$.
Now calculating flux over the three surfaces of sphere in xy, xz and yz respectively. $S_{xy}=9π/4$, $S_{xz}=9π/2$ and $S_{yz}=0$
Putting these all together, we have $$ \iint_S \mathbf{F} \cdot d\mathbf{S} = 27π/4 $$
What am I doing wrong?