  We consider possible generation of singularities of a vector field
transported by diffeomorphisms with derivatives of uniformly bounded
determinants. A particular case of volume preserving diffeomrphism is the most
important, since it has direct applications to the incompressible, inviscid
hydrodynamics. We find relations between the directions of the vector field and
the eigenvectors of the derivative of the back-to-label map near the
singularity. We also find an invariant when we follow the motion of the
integral curves of the vector field. For the 3D Euler equations these results
have immediate implications about the directions of the vortex stretching and
the material stretching near the possible singularities. We also have similar
applications to the other inviscid, incompressible fluid equations such as the
2D quasi-geostrophic equation and the 3D magnetohydrodynamics equations.
