Abstract
Integration over n-dimensional manifolds reduces through charts to integration in ℝn. 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 ℝn. 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.
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© 2001 Springer Science+Business Media New York
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Jänich, K. (2001). Integration on Manifolds. In: Vector Analysis. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3478-2_5
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DOI: https://doi.org/10.1007/978-1-4757-3478-2_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-3144-3
Online ISBN: 978-1-4757-3478-2
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