  We derive an expression for the superfluid density of a uniform two-component
Fermi gas through the BCS-BEC crossover in terms of the thermodynamic potential
in the presence of an imposed superfluid flow. Treating the pairing
fluctuations in a Gaussian approximation following the approach of Nozi\`eres
and Schmitt-Rink, we use this definition of $\rho_s$ to obtain an explicit
result which is valid at finite temperatures and over the full BCS-BEC
crossover. It is crucial that the BCS gap $\Delta$, the chemical potential
$\mu$, and $\rho_s$ all include the effect of fluctuations at the same level in
a self-consistent manner. We show that the normal fluid density $\rho_n \equiv
n - \rho_s$ naturally separates into a sum of contributions from Fermi BCS
quasiparticles ($\rho^F_{n}$) and Bose collective modes ($\rho^B_{n}$). The
expression for $\rho^F_{n}$ is just Landau's formula for a BCS Fermi superfluid
but now calculated over the BCS-BEC crossover. The expression for the Bose
contribution $\rho^B_{n}$ is more complicated and only reduces to Landau's
formula for a Bose superfluid in the extreme BEC limit, where all the fermions
have formed stable Bose pairs and the Bogoliubov excitations of the associated
molecular Bose condensate are undamped. In a companion paper, we present
numerical calculations of $\rho_s$ using an expression equivalent to the one
derived in this paper, over the BCS-BEC crossover, including unitarity, and at
finite temperatures.
