  We consider the evolution of perturbed cosmological spacetime with multiple
scalar fields in Einstein gravity. A complete set of scalar-type perturbation
equations is presented in a gauge-ready form, and we derived the closed set of
second-order differential equations in several useful forms. Conserved
behaviors of the perturbed three-space curvature in the comoving gauge,
$\phi_v$, under several conditions are clarified. Under the slow-roll
conditions, the adiabatic and isocurvature modes decouple from each other, and
in the large-scale limit we have (i) the adiabatic mode is generally conserved,
(ii) for a couple of special potential the isocurvature modes decouple from
each other and are described by conserved quantities, (iii) in the two field
system, the isocurvature mode is described by a conserved quantity for the
general potential.
