Introduction to Calculus and Analysis: Volume II
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer New York
1989
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | 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 |
| Beschreibung: | 1 Online-Ressource (XXV, 954p. 120 illus) |
| ISBN: | 9781461389583 9781461389606 |
| DOI: | 10.1007/978-1-4613-8958-3 |
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| 500 | |a 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 | ||
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| indexdate | 2024-07-10T01:21:07Z |
| institution | BVB |
| isbn | 9781461389583 9781461389606 |
| language | English |
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| spelling | Courant, Richard Verfasser aut Introduction to Calculus and Analysis Volume II by Richard Courant, Fritz John New York, NY Springer New York 1989 1 Online-Ressource (XXV, 954p. 120 illus) txt rdacontent c rdamedia cr rdacarrier 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 Mathematics Global analysis (Mathematics) Analysis Mathematik John, Fritz Sonstige oth https://doi.org/10.1007/978-1-4613-8958-3 Verlag Volltext |
| spellingShingle | Courant, Richard Introduction to Calculus and Analysis Volume II Mathematics Global analysis (Mathematics) Analysis Mathematik |
| title | Introduction to Calculus and Analysis Volume II |
| title_auth | Introduction to Calculus and Analysis Volume II |
| title_exact_search | Introduction to Calculus and Analysis Volume II |
| title_full | Introduction to Calculus and Analysis Volume II by Richard Courant, Fritz John |
| title_fullStr | Introduction to Calculus and Analysis Volume II by Richard Courant, Fritz John |
| title_full_unstemmed | Introduction to Calculus and Analysis Volume II by Richard Courant, Fritz John |
| title_short | Introduction to Calculus and Analysis |
| title_sort | introduction to calculus and analysis volume ii |
| title_sub | Volume II |
| topic | Mathematics Global analysis (Mathematics) Analysis Mathematik |
| topic_facet | Mathematics Global analysis (Mathematics) Analysis Mathematik |
| url | https://doi.org/10.1007/978-1-4613-8958-3 |
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