Integration on Manifolds
Integration over n-dimensional manifolds reduces through charts to integration in ℝ n . The objects integrated on oriented manifolds are n-forms, for the following reason. For an ordinary function f : M → ℝ, the contribution of a chart domain U to the integral would clearly depend on the choice of chart h But for an n-form, the integral of its component function pulled down by an orientation-preserving chart is independent of the coordinates, as we see from the changeof-variables formula for multiple integrals in ℝ n . This is the main content of Chapter 5. Section 5.4 contains the technical details and Section 5.3 a summary of necessary background. In the first two sections we give an intuitive view of integration on manifolds.
KeywordsMeasure Zero Measurable Subset Oriented Manifold Intuitive View Arbitrary Measure Space
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