  We investigate three different local approximations for nonlinear
gravitational instability in the framework of cosmological Lagrangian fluid
dynamics of cold dust. They include the Zel'dovich approximation (ZA), the
``non-magnetic'' approximation of Bertschinger \& Jain (1994, NMA), and a new
``local tidal'' approximation (LTA). The LTA is exact for any perturbations
whose gravitational and velocity equipotentials have the same constant shape
with time, including spherical, cylindrical, and plane-parallel perturbations.
We tested all three local approximations with the collapse of a homogeneous
triaxial ellipsoid, for which an exact solution exists for an ellipsoid
embedded in empty space and an excellent approximation is known in the
cosmological context. We find that the LTA is significantly more accurate in
general than the ZA and the NMA. Like the ZA, but unlike the NMA, the LTA
generically leads to pancake collapse. For a randomly chosen mass element in an
Einstein-de Sitter universe, assuming a Gaussian random field of initial
density fluctuations, the LTA predicts that at least 78\% of initially
underdense regions collapse owing to nonlinear effects of shear and tides.
