Integration on Manifolds

  • Andrew Browder
Part of the Undergraduate Texts in Mathematics book series (UTM)


In this chapter, we define the integral of a k-form over a compact oriented k-manifold, and prove the important generalized Stokes’ theorem, which can be regarded as a far-reaching generalization of the fundamental theorem of calculus. We also define the integral of a function over a (not necessarily oriented) manifold, and describe the integral of a form in terms of the integral of a function. The classical theorems of vector analysis (Green’s theorem, divergence theorem, Stokes’ theorem) appear as special cases of the general Stokes’ theorem. Applications are made to topology (the Brouwer fixed point theorem) and to the study of harmonic functions (the mean value property, the maximum principle, Liouville’s theorem, and the Dirichlet principle).


Harmonic Function Compact Subset Divergence Theorem Borel Function Coordinate Patch 
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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Andrew Browder
    • 1
  1. 1.Mathematics DepartmentBrown UniversityProvidenceUSA

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