  In this paper, we discuss the evolution operator and the transition
probabilities expressed as functions of the vacuum mass squared differences,
the vacuum mixing angles, and the matter density parameter for three flavor
neutrino oscillations in matter of varying density in the plane wave
approximation. The applications of this to neutrino oscillations in a model of
the Earth's matter density profile, step function matter density profiles,
constant matter density profiles, linear matter density profiles, and finally
in a model of the Sun's matter density profile are discussed. We show that for
matter density profiles, which do not fluctuate too much, the total evolution
operator consisting of $n$ operators can be replaced by one single evolution
operator in the semi-classical approximation.
