  Respecting the conservation laws of momentum and energy in a many body theory
is very important for understanding the transport phenomena. The previous
conserving approximation requires that the self-energy of a single particle can
be written as a functional derivative of a full dressed Green's function. This
condition can not be satisfied in the $G_0G$ t-matrix or pair fluctuation
theory which emphasizes the fermion pairing with a stronger than the
Bardeen-Cooper-Schrieffer (BCS) attraction. In the previous
work\cite{stressWI}, we have shown that when the temperature is above the
superfluid transition temperature $T_c$, the $G_0G$ t-matrix theory can be put
into a form that satisfies the stress tensor Ward identity (WI) or local form
of conservation laws by introducing a new type of vertex correction. In this
paper, we will extend the above conservation approximation to the superfluid
phase in the BCS mean field level. To establish the stress tensor WI, we have
to include the fluctuation of the order parameter or the contribution from the
Goldstone mode. The result will be useful for understanding the transport
properties such as the behavior of the viscosity of Fermionic gases in the
superfluid phases.
