  In the present work we present the full treatment of scalar and vector
cosmological perturbations in a non-singular bouncing universe in the context
of metric $f(R)$ cosmology. Scalar metric perturbations in $f(R)$ cosmology
were previously calculated in the Jordan frame, in the present paper we
successfully use the Einstein frame to calculate the scalar metric
perturbations where the cosmological bounce takes place in the Jordan frame.
The Einstein frame picture presented corrects and completes a previous
calculation of scalar perturbations and adds new information. Behavior of fluid
velocity potential and the pure vector fluid velocity terms are elaborately
calculated for the first time in $f(R)$ cosmology for a bouncing universe in
presence of exponential gravity. It is shown that the vector perturbations can
remain bounded and almost constant during the non-singular bounce in $f(R)$
gravity unlike general relativistic models where we expect the vector
perturbations to be growing during the contracting phase and decaying during
the expanding phase. The paper shows that the Einstein frame can be used for
calculation of scalar and vector metric perturbations in a bouncing universe
for most of the cases except the case of asymmetric non-singular bounces.
