  In the first part of this chapter we analyze the contact intensity $C$, which
has been introduced by Tan [Ann. Phys. 323, 2952 (2008)] and appears in several
physical observables of the strongly correlated two-component Fermi gas. We
calculate the contact $C$ in the full BCS-BEC crossover for a uniform
superfluid Fermi gas by using an efficient parametetrization of the
ground-state energy. In the case of harmonic confinement, within the
Thomas-Fermi approximation, we derive analytical formulas of $C$ in the three
relevant limits of the crossover. In the second part of this chapter we discuss
the extended superfluid hydrodynamics we have recently proposed to describe
static and dynamical collective properties of the Fermi gas in the BCS-BEC
crossover. In particular we show the relation with the effective theory for the
Goldstone field derived by Son and Wingate [Ann. Phys. 321, 197 (2006)] on the
basis of conformal invariance. By using our equations of extended hydrodynamics
we determine nonlinear sound waves, static response function and structure
factor of a generic superfluid at zero temperature.
