  After deriving the isothermal Hellmann-Feynman theorem (IHFT) that is
suitable for mixed states in a canonical ensemble, we use this theorem to
obtain the isothermal magnetic-field sweep theorems for the free, average and
trapping energies, and for the entropy, specific heat, pressure and atomic
compressibility of strongly-correlated ultra-cold quantum gases. In particular,
we apply the sweep theorems to two-component Fermi gases in the
weakly-interacting BCS and BEC limits, showing that the temperature dependence
of the contact parameter can be determined by the variation of either the
entropy or specific heat with respect to the scattering length. We also use the
IHFT to obtain the Virial theorem in a canonical ensemble, and discuss its
implications for quantum gases.
