  We quantize the recent hydrodynamic analysis of Stringari for the low-energy
collective modes of a trapped Bose gas at $T=0$. This is based on the
time-dependent Gross-Pitaevskii equation, but omits the kinetic energy of the
density fluctuations. We diagonalize the hydrodynamic Hamiltonian in terms of
the normal modes associated with the amplitude and phase of the inhomogeneous
Bose order parameter. These normal modes provide a convenient basis for
calculating observable quantities. As applications, we calculate the depletion
of the condensate at $T=0$ as well as the inelastic light-scattering cross
section $S({\bf q},\omega)$ from low-energy condensate fluctuations. The latter
involves a sum over all normal modes, with a weight proportional to the square
of the ${\bf q}$ Fourier component of the density fluctuation associated with a
given mode. Finally, we show how the Thomas-Fermi hydrodynamic description can
be derived starting from the coupled Bogoliubov equations.
