Integration on manifolds

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


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.


Open Subset Implicit Function Theorem Divergence Theorem Unit Tangent Vector Klein Bottle 
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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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