  This set of four lectures reviews various aspects of the theory of a dilute
low-temperature trapped Bose gas, starting with (I) a review of the Bogoliubov
description of the elementary excitations in a uniform system. The treatment is
then generalized (II) to include the new physical effects of a confining
harmonic trap potential on the condensate and its normal modes. An equivalent
hydrodynamic description (III) focuses directly on the density and velocity
fluctuations. The physics of vortices (IV) in an incompressible fluid is
summarized and extended to the case of a trapped Bose condensate.
