Preliminaries from Manifold Theory

The purpose of this chapter is to summarize the basic facts about manifolds and mappings between them which are most frequently used in the theory of partial differential equations. Manifolds are an abstraction of the idea of a surface in Euclidean space. The virtue of manifold theory is that it provides the geometric insight into the study of partial differential equations, and intrinsic properties of partial differential equations may be revealed. This chapter, included for the sake of completeness, should serve to settle questions of notation and such.