Solution
= Solution
Use the outward <unit normal> on the unit <sphere> and the convention that $\theta$ is measured from the positive vertical axis. Then $\mathbf F\cdot\mathbf n=\sin\theta$, while the <spherical coordinates> <area element> is $dS=\sin\theta\,d\theta\,d\phi$. The <flux> is therefore
$$
\boxed{\int_S\mathbf F\cdot d\mathbf S=\int_0^{2\pi}\int_0^\pi\sin^2\theta\,d\theta\,d\phi=\pi^2.}
$$