  By applying a position-dependent detuning to a spin-orbit-coupled Hamiltonian
with equal Rashba and Dresselhaus coupling, we exploit the behavior of the
angular momentum of a harmonically trapped Bose-Einstein condensed atomic gas
and discuss the distinctive role of its canonical and spin components. By
developing the formalism of spinor hydrodynamics we predict the precession of
the dipole oscillation caused by the synthetic rotational field, in analogy
with the precession of the Foucault pendulum, the excitation of the scissors
mode, following the sudden switching off of the detuning, and the occurrence of
Hall-like effects. When the detuning exceeds a critical value we observe a
transition from a vortex free, rigidly rotating quantum gas to a gas containing
vortices with negative circulation which results in a significant reduction of
the total angular momentum.
