= Solution
Rotational symmetry of the <vector field> means $\mathbf F(r\mathbf n)=h(r)\mathbf n$: rotations fixing $\mathbf n$ force the field to be parallel to $\mathbf n$, and rotations between directions make its magnitude depend only on $r$. Its <flux integral> over the radius-$r$ <sphere> is $4\pi r^2h(r)$. Applying the flux identity from part (i) to the ball, with the spherically symmetric source, gives
$$
4\pi r^2h(r)=4\pi\int_0^r f(s)s^2\,ds.
$$
Therefore the <origin-regular spherical Poisson flux law> is
$$
\boxed{\mathbf F(\mathbf x)=\frac{\mathbf x}{|\mathbf x|^3}
\int_0^{|\mathbf x|}f(s)s^2\,ds.}
$$
Only source values at $s\le|\mathbf x|$ occur. Changing the exterior source can change the scalar potential by an interior constant, but cannot change this radial <gradient>. Regularity on all of $\mathbb R^3$, inherited from the <Poisson equation> setting, matters: on a punctured domain alone an additional $C\mathbf x/r^3$ would have zero <divergence> away from the origin and would not be fixed by the local source. It is excluded here by the origin regularity needed for the ball flux identity.
Back to article page
