  Morphodynamic descriptions of fluid deformable surfaces are relevant for a
range of biological and soft matter phenomena, spanning materials that can be
passive or active, as well as ordered or topological. However, a principled,
geometric formulation of the correct hydrodynamic equations has remained
opaque, with objective rates proving a central, contentious issue. We argue
that this is due to a conflation of several important notions that must be
disambiguated when describing fluid deformable surfaces. These are the Eulerian
and Lagrangian perspectives on fluid motion, and three different types of gauge
freedom: in the ambient space; in the parameterisation of the surface, and; in
the choice of frame field on the surface. We clarify these ideas, and show that
objective rates in fluid deformable surfaces are time derivatives that are
invariant under the first of these gauge freedoms, and which also preserve the
structure of the ambient metric. The latter condition reduces a potentially
infinite number of possible objective rates to only two: the material
derivative and the Jaumann rate. The material derivative is invariant under the
Galilean group, and therefore applies to velocities, whose rate captures the
conservation of momentum. The Jaumann derivative is invariant under all
time-dependent isometries, and therefore applies to local order parameters, or
symmetry-broken variables, such as the nematic $Q$-tensor. We provide examples
of material and Jaumann rates in two different frame fields that are pertinent
to the current applications of the fluid mechanics of deformable surfaces.
