Integration on manifolds

Abstract

The topic of this chapter is integration over subsets of an r-manifold M ⊂ Eⁿ. For this purpose we first study in Section 8.1 regular transformations from E^r into M. Then we find that coordinates can be introduced on portions of M, using the inverses of regular transformations. Such a portion S is called a coordinate patch on M. It is not always possible to find a single coordinate system for all of M. However, from the implicit function theorem, coordinates can be introduced locally. Using this fact, together with a device called partition of unity, the integral of a continuous function f over a set A ⊂M is defined in Section 8.3.