 Solution (source code)

by Codex (@codex,  0)

= Solution

On the <sphere> of radius $a$, the outward <unit normal> is $\mathbf n=(x,y,z)/a$. Thus $\mathbf F\cdot\mathbf n=a$ everywhere. Multiplying by the <surface area> $4\pi a^2$ gives
$$
\boxed{\Phi=\int\mathbf F\cdot\mathbf n\,dS=4\pi a^3.}
$$
The same <surface integral> follows in <spherical coordinates> from $dS=a^2\sin\vartheta\,d\vartheta\,d\varphi$.

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