  We derive a general and exact equation of motion for a quantised vortex in an
inhomogeneous two-dimensional Bose-Einstein condensate. This equation expresses
the velocity of a vortex as a sum of local ambient density and phase gradients
in the vicinity of the vortex. We perform Gross-Pitaevskii simulations of
single vortex dynamics in both harmonic and hard-walled disk-shaped traps, and
find excellent agreement in both cases with our analytical prediction. The
simulations reveal that, in a harmonic trap, the main contribution to the
vortex velocity is an induced ambient phase gradient, a finding that
contradicts the commonly quoted result that the local density gradient is the
only relevant effect in this scenario. We use our analytical vortex velocity
formula to derive a point-vortex model that accounts for both density and phase
contributions to the vortex velocity, suitable for use in inhomogeneous
condensates. Although good agreement is obtained between Gross-Pitaevskii and
point-vortex simulations for specific few-vortex configurations, the effects of
nonuniform condensate density are in general highly nontrivial, and are thus
difficult to efficiently and accurately model using a simplified point-vortex
description.
