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Table of contents (8 chapters)
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Front Matter
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Back Matter
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About this book
The new Chapter 1 contains all the fundamental properties of linear differential forms and their integrals. These prepare the reader for the introduction to higher-order exterior differential forms added to Chapter 3. Also found now in Chapter 3 are a new proof of the implicit function theorem by successive approximations and a discus sion of numbers of critical points and of indices of vector fields in two dimensions. Extensive additions were made to the fundamental properties of multiple integrals in Chapters 4 and 5. Here one is faced with a familiar difficulty: integrals over a manifold M, defined easily enough by subdividing M into convenient pieces, must be shown to be inde pendent of the particular subdivision. This is resolved by the sys tematic use of the family of Jordan measurable sets with its finite intersection property and of partitions of unity. In order to minimize topological complications, only manifolds imbedded smoothly into Euclidean space are considered. The notion of "orientation" of a manifold is studied in the detail needed for the discussion of integrals of exterior differential forms and of their additivity properties. On this basis, proofs are given for the divergence theorem and for Stokes's theorem in n dimensions. To the section on Fourier integrals in Chapter 4 there has been added a discussion of Parseval's identity and of multiple Fourier integrals.
Authors and Affiliations
Bibliographic Information
Book Title: Introduction to Calculus and Analysis
Book Subtitle: Volume II
Authors: Richard Courant, Fritz John
DOI: https://doi.org/10.1007/978-1-4613-8958-3
Publisher: Springer New York, NY
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eBook Packages: Springer Book Archive
Copyright Information: Springer-Verlag New York, Inc. 1989
Edition Number: 1
Number of Pages: XXV, 954
Additional Information: Originally published by Interscience Publishers, a division of John Wiley and Sons Inc., New York 1974
Topics: Analysis