  We present a covariant formalism for studying nonlinear perturbations of
scalar fields. In particular, we consider the case of two scalar fields and
introduce the notion of adiabatic and isocurvature covectors. We obtain
differential equations governing the evolution of these two covectors, as well
as the evolution equation for the covector associated with the curvature
perturbation. The form of these equations is very close to the analogous
equations obtained in the linear theory, but our equations are fully nonlinear
and exact. As an application of our formalism, we expand these equations at
second order in the perturbations. On large scales, we obtain a closed system
of coupled scalar equations giving the evolution of the second-order adiabatic
and entropy perturbations in terms of the first-order perturbations. These
equations in general contain a nonlocal term which, however, rapidly decays in
an expanding universe.
