  We investigate the thermodynamic behaviour of a Bose gas interacting with
repulsive forces and confined in a harmonic anisotropic trap. We develop the
formalism of mean field theory for non uniform systems at finite temperature,
based on the generalization of Bogoliubov theory for uniform gases. By
employing the WKB semiclassical approximation for the excited states we derive
systematic results for the temperature dependence of various thermodynamic
quantities: condensate fraction, density profiles, thermal energy, specific
heat and moment of inertia. Our analysis points out important differences with
respect to the thermodynamic behaviour of uniform Bose gases. This is mainly
the consequence of a major role played by single particle states at the
boundary of the condensate. We find that the thermal depletion of the
condensate is strongly enhanced by the presence of repulsive interactions and
that the critical temperature is decreased with respect to the predictions of
the non-interacting model. Our work points out an important scaling behaviour
exhinited by the system in large $N$ limit. Scaling permits to express all the
relevant thermodynamic quantities in terms of only two parameters: the reduced
temperature $t=T/T_c^0$ and the ratio between the $T=0$ value of the chemical
potential and the critical temperature $T_c^0$ for Bose-Einstein condensation.
Comparisons with first experimental results and ab-initio calculations are
presented.
