  We study cosmological perturbations generated from quantum fluctuations in
multi-field inflationary scenarios in generalized Einstein theories, taking
both adiabatic and isocurvature modes into account. In the slow-roll
approximation, explicit closed-form long-wave solutions for field and metric
perturbations are obtained by the analysis in the Einstein frame. Since the
evolution of fluctuations depends on specific gravity theories, we make
detailed investigations based on analytic and numerical approaches in four
generalized Einstein theories: the Jordan-Brans-Dicke (JBD) theory, the
Einstein gravity with a non-minimally coupled scalar field, the
higher-dimensional Kaluza-Klein theory, and the $R+R^2$ theory with a
non-minimally coupled scalar field. We find that solutions obtained in the
slow-roll approximation show good agreement with full numerical results except
around the end of inflation. Due to the presence of isocurvature perturbations,
the gravitational potential $\Phi$ and the curvature perturbations ${\cal R}$
and $\zeta$ do not remain constant on super-horizon scales. In particular, we
find that negative non-minimal coupling can lead to strong enhancement of
${\cal R}$ in both the Einstein and higher derivative gravity, in which case it
is difficult to unambiguously decompose scalar perturbations into adiabatic and
isocurvature modes during the whole stage of inflation.
