  Various widely-used mean-field type theories for a dilute Bose gas are
critically examined in the light of the recent discovery of Bose-Einstein
condensation of atomic gases in a confined geometry. By numerically solving the
mean-field equations within the framework of the Bogoliubov approximation both
stationary non-uniform case and the vortex case under rotation in a
cylindrically symmetric vessel are investigated. We obtain spatial structures
of condensate, non-condensate, anomalous correlation. The low lying excitation
spectra, the local density of states and the circulating current density in a
vortex corresponding to various levels of mean-field theories are predicted.
