  The instability condition of the non-vortex state toward vortex formation is
exa mined within the Bogoliubov theory when a Bose-Einstein condensate is under
exte rnally forced rotation. The obtained critical angular velocity combined
with the previous stability cond itions for a votex yields a detailed phase
diagram in the critical velocity vs t he system parameter. This facilitates
vortex formation experiments for alkali atom gases confined in a harmonic
potential.
