  We extend the fully non-linear and exact cosmological perturbation equations
in a Friedmann background universe to include the background curvature. The
perturbation equations are presented in a gauge ready form, so any temporal
gauge condition can be adopted freely depending on the problem to be solved.
%The background curvature term explicitly appears only in the energy and
momentum constraint equations. We consider the scalar, and vector perturbations
without anisotropic stress. As an application, we analyze the equations in the
special case of irrotational zero-pressure fluid in the comoving gauge
condition. We also present the fully nonlinear formulation for a minimally
coupled scalar field.
