Neutrino oscillations with three flavors in matter of varying density

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.