  The Ginzburg-Landau theory of a trapped Fermi gas with a BEC-BCS crossover is
derived by the path-integral method. In addition to the standard
Ginzburg-Landau equation, a second equation describing the total atom density
is obtained. These two coupled equations are necessary to describe both
homogeneous and inhomogeneous systems. The Ginzburg-Landau theory is valid near
the transition temperature $T_c$ on both sides of the crossover. In the
weakly-interacting BEC region, it is also accurate at zero temperature where
the Ginzburg-Landau equation can be mapped onto the Gross-Pitaevskii (GP)
equation. The applicability of GP equation at finite temperature is discussed.
On the BEC side, the fluctuation of the order parameter is studied and the
renormalization to the molecule coupling constant is obtained.
