  A new cohomology, induced by a vector field, is defined on pairs of
differential forms ($1$--differentiable forms) in a manifold. It is proved a
link with the classical de Rham cohomology and an $1$-differentable cohomology
of Lichnerowicz type associated to an one form. Also, the case when the
manifold is complex and the vector field is holomorphic is studied. Finally, an
application of this theory to the harmonicity of $1$-differentiable forms is
studied in a particular case.
