The sphere is presumably closed, so we can use the divergence theorem.
[tex]\nabla\cdot\mathbf f=\dfrac{\partial(xy)}{\partial x}+\dfrac{\partial(yz)}{\partial y}+\dfrac{\partial(xz)}{\partial z}[/tex]
[tex]\nabla\cdot\mathbf f=x+y+z[/tex]
Now convert to spherical coordinates:
[tex]x=\rho\cos\theta\sin\varphi[/tex]
[tex]y=\rho\sin\theta\sin\varphi[/tex]
[tex]z=\rho\cos\varphi[/tex]
The flux is then given by the volume integral,
[tex]\displaystyle\iint_{x^2+y^2+z^2=9}\mathbf f\cdot\mathrm d\mathbf S=\iiint_{x^2+y^2+z^2\le9}(x+y+z)\,\mathrm dV[/tex]
[tex]=\displaystyle\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=3}(\rho\cos\theta\sin\varphi+\rho\sin\theta\sin\varphi+\rho\cos\varphi)\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=0[/tex]
