  We address the importance of symmetry and symmetry breaking on linear
response theories of fermionic BCS superfluids. The linear theory of a
noninteracting Fermi gas is reviewed and several consistency constraints are
verified. The challenge to formulate linear response theories of BCS
superfluids consistent with density and spin conservation laws comes from the
presence of a broken U(1)$_{\textrm{EM}}$ symmetry associated with
electromagnetism (EM) and we discuss two routes for circumventing this. The
first route follows Nambu's integral-equation approach for the EM vertex
function, but this method is not specific for BCS superfluids. We focus on the
second route based on a consistent-fluctuation-of-the order-parameter (CFOP)
approach where the gauge transformation and the fluctuations of the order
parameter are treated on equal footing. The CFOP approach allows one to
explicitly verify several important constraints: The EM vertex satisfies not
only a Ward identity which guarantees charge conservation but also a $Q$-limit
Ward identity associated with the compressibility sum rule. In contrast, the
spin degrees of freedom associated with another U(1)$_z$ symmetry are not
affected by the Cooper-pair condensation that breaks only the
U(1)$_{\textrm{EM}}$ symmetry. As a consequence the collective modes from the
fluctuations of the order parameter only couple to the density response
function but decouple from the spin response function, which reflects the
different fates of the two U(1) symmetries in the superfluid phase. Our
formulation lays the ground work for application to more general theories of
BCS-Bose Einstein Condensation crossover both above and below $T_c$.

Superconductivity;Quantum Gases