  Through the use of a lattice U(1) Ward-Takahashi identity, one can find a
precise definition of flux and electric four-current that does not rely on the
continuum limit. The magnetic four-current defined for example by the
DeGrand-Toussaint construction introduces order a^2 errors in the field
distributions. We advocate using a single definition of flux in order to be
consistent with both the electric and magnetic Maxwell's equations at any
lattice spacing. In a U(1) theory the monopoles are slightly smeared by this
choice, i.e. are no longer associated with a single lattice cube. In Abelian
projected SU(2) the consistent definition suggests further modifications. For
simulations in the scaling window, we do not foresee large changes in the
standard analysis of the dual Abrikosov vortex in the maximal Abelian gauge
because the order a^2 corrections have small fluctuations and tend to cancel
out. However in other gauges, the consequences of our definitions could lead to
large effects which may help in understanding the choice of gauge. We also
examine the effect of truncating all monopoles except for the dominant cluster
on the profile of the dual Abrikosov vortex.
