From the introduction: In this paper we have two main goals: the first is to construct the invariant vector fields and one-forms on SU(3) in Euler coordinates that are similar to the Euler angle parameters of SU(2); the second is the application of these structures to the description of the geometric phases of three state systems. The Euler coordinates along with the left and right invariant vector fields and one-forms are constructed here for several reasons. One, the Euler decomposition of a group element is achieved by a method that is very general and could, in principle, be applied to \(\text{SU}(n)\). Two, this particular parametrization is convenient for many calculations, as will be shown. Three, there will be advantages to having a set of invariant vector fields and corresponding one-forms in the same parametrization of the group. Color, flavor, and nuclear models use the group SU(3). However, geometric aspects of the group manifold are not emphasized. That is not to say that the differential geometric structures are not relevant to these models; indeed they are! However, as was shown earlier, the differential geometry of SU(3) is an integral part of the description of three state systems. The Euler coordinates and differential forms provide a complete description of the differential geometry needed to describe the pure state density matrix and geometric phase of three state systems.