Integration on manifolds
The topic of this chapter is integration over subsets of an r-manifold M ⊂ E n . 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.
KeywordsOpen Subset Implicit Function Theorem Divergence Theorem Unit Tangent Vector Klein Bottle
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