  The low-lying hydrodynamic normal modes of a dilute Bose-Einstein gas in an
isotropic harmonic trap determine the corresponding Bogoliubov amplitudes. In
the Thomas-Fermi limit, these modes have large low-temperature occupation
numbers, and they permit an explicit construction of the dynamic structure
function $S(q,\omega)$. The total noncondensate number $N'(0)$ at zero
temperature increases like $R^6$, where $R$ is the condensate radius measured
in units of the oscillator length. The lowest dipole modes are constructed
explicitly in the Bogoliubov approximation.
