  We present a standard field theoretical derivation of the dynamic density and
spin linear response functions of a dilute superfluid Fermi gas in the BCS-BEC
crossover in both three and two dimensions. The derivation of the response
functions is based on the elegant functional path integral approach which
allows us to calculate the density-density and spin-spin correlation functions
by introducing the external sources for the density and the spin density. Since
the generating functional cannot be evaluated exactly, we consider two gapless
approximations which ensure a gapless collective mode (Goldstone mode) in the
superfluid state: the BCS-Leggett mean-field theory and the
Gaussian-pair-fluctuation (GPF) theory. In the mean-field theory, our results
of the response functions agree with the known results from the random phase
approximation. We further consider the pair fluctuation effects and establish a
theoretical framework for the dynamic responses within the GPF theory. We show
that the GPF response theory naturally recover three kinds of famous
diagrammatic contributions: the Self-Energy contribution, the Aslamazov-Lakin
contribution, and the Maki-Thompson contribution. We also show that unlike the
equilibrium state, in evaluating the response functions, the linear
(first-order) terms in the external sources as well as the induced order
parameter perturbations should be treated carefully. In the superfluid state,
there is an additional order parameter contribution which ensures that in the
static and long wavelength limit, the density response function recovers the
result of the compressibility (compressibility sum rule). We expect that the
$f$-sum rule is manifested by the full number equation which includes the
contribution from the Gaussian pair fluctuations.
