  A method is presented which allows the exact construction of conserved (i.e.
divergence-free) current vectors from appropriate sets of multipole moments.
Physically, such objects may be taken to represent the flux of particles or
electric charge inside some classical extended body. Several applications are
discussed. In particular, it is shown how to easily write down the class of all
smooth and spatially-bounded currents with a given total charge. This
implicitly provides restrictions on the moments arising from the smoothness of
physically reasonable vector fields. We also show that requiring all of the
moments to be constant in an appropriate sense is often impossible; likely
limiting the applicability of the Ehlers-Rudolph-Dixon notion of quasirigid
motion. A simple condition is also derived that allows currents to exist in two
different spacetimes with identical sets of multipole moments (in a natural
sense).
