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Integration on manifolds

  • Wendell Fleming
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

The topic of this chapter is integration over subsets of an r-manifold ME 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 AM is defined in Section 8.3.

Keywords

Open Subset Implicit Function Theorem Divergence Theorem Unit Tangent Vector Klein Bottle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag, New York Inc. 1977

Authors and Affiliations

  • Wendell Fleming
    • 1
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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