  We study gradient magnetometry with an ensemble of atoms with arbitrary spin.
We calculate precision bounds for estimating the gradient of the magnetic field
based on the quantum Fisher information. For quantum states that are invariant
under homogeneous magnetic fields, we need to measure a single observable to
estimate the gradient. On the other hand, for states that are sensitive to
homogeneous fields, a simultaneous measurement is needed, as the homogeneous
field must also be estimated. We prove that for the cases studied in this
paper, such a measurement is feasible. We present a method to calculate
precision bounds for gradient estimation with a chain of atoms or with two
spatially separated atomic ensembles. We also consider a single atomic ensemble
with an arbitrary density profile, where the atoms cannot be addressed
individually, and which is a very relevant case for experiments. Our model can
take into account even correlations between particle positions. While in most
of the discussion we consider an ensemble of localized particles that are
classical with respect to their spatial degree of freedom, we also discuss the
case of gradient metrology with a single Bose-Einstein condensate.
