  A general type of mathematical argument is described, which applies to all
the cases in which dynamo maintenance of a steady magnetic field by motion in a
uniform density is known to be impossible. Previous work has demonstrated that
magnetic field decay is unavoidable under conditions of axisymmetry and in
spherical or planar incompressible flows. These known results are encompassed
by a calculation for flows described in terms of a generalized
poloidal-toroidal representation of the magnetic field with respect to an
arbitrary two dimensional surface. We show that when the velocity field is two
dimensional, the dynamo growth, if any, that results, is linear in one of the
projections of the field while the other projections remain constant. We also
obtain criteria for the existence of and classification into two and three
dimensional velocity results which are satisfied by a restricted set of
geometries. In addition, we discuss the forms of spatial variation of the
density and the resistivity that are allowed so that field decay still occurs
for this set of geometries.
