= Origin-regular spherical Poisson flux law
{title2=$F(x)=\frac{x}{|x|^3}\int_0^{|x|}f(s)s^2\,ds$}
For a spherically symmetric <gradient> field regular at the origin and satisfying <Poisson equation>, the <divergence theorem> on a ball determines its radial magnitude from the enclosed source integral. It depends only on the source at smaller radii. Regularity excludes the source-free but origin-singular addition $Cx/|x|^3$; without that condition, solving the local equation on a punctured domain leaves this extra term undetermined.
Back to article page
