An inverse-square law is a dependence 
on the distance 
from a source, where 
is independent of 
. For a radial vector field
in three-dimensional Euclidean space, suppose
its radial component has spherical symmetry
and its outward flux integral 
is constant through concentric spheres. Since a sphere of radius 
has surface area 
, the spherical symmetry
gives
 |
where 
is the outward radial unit vector (Staelin 2011, p. 18).
The divergence of this vector field is zero away from the source. For nonzero 
, the flux
integral through an enclosing sphere is consistent
with the divergence theorem because the vector field has a singularity
at the source. Thus the theorem cannot be applied there without accounting for the
singularity.
More generally, in 
-dimensional
Euclidean space with 
, the corresponding radial law is
 |
where 
is the hyper-surface area of the unit hypersphere,
and 
is the gamma function. When 
, the inverse-square case is therefore specific to three
dimensions. The gravitational inverse-square law is the hypothesis underlying Newton's shell theorem.
