  Vector calculus in three dimensions with a Euclidian metric is the lingua
franca of classical physics, including classical electrodynamics. This article
corrects some long-standing imprecision in a fundamental result.
  Some textbooks assert that a vector function defined in the whole of a three
dimensional space is uniquely determined by its divergence, its curl, and the
condition that the function goes to zero as the radius (distance from an
origin) goes to infinity. This article suggests that this condition is not
sufficient for uniqueness. A proof is given that a sufficient condition for
uniqueness is for the vector function to approach zero more rapidly than radius
to the minus 3/2 power as the radius goes to infinity.
  The issue is important because the same uniqueness condition also determines
the uniqueness of the decomposition of a vector field into a transverse field
plus a solenoidal field, as is done in the Coulomb gauge of electrodynamics.
