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卷绕: 拓扑、几何和分析中的卷绕数（影印版）

样章
作者：
John Roe
定价：
135.00元
ISBN：
978-7-04-059314-3
版面字数：
488.000千字
开本：
16开
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
出版时间：
2023-03-15
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
拓扑学
暂无
  • 前辅文
  • Chapter 1. Prelude: Love, Hate, and Exponentials
    • §1.1. Two sets of travelers
    • §1.2. Winding around
    • §1.3. The most important function in mathematics
    • §1.4. Exercises
  • Chapter 2. Paths and Homotopies
    • §2.1. Path connectedness
    • §2.2. Homotopy
    • §2.3. Homotopies and simple-connectivity
    • §2.4. Exercises
  • Chapter 3. The Winding Number
    • §3.1. Maps to the punctured plane
    • §3.2. The winding number
    • §3.3. Computing winding numbers
    • §3.4. Smooth paths and loops
    • §3.5. Counting roots via winding numbers
    • §3.6. Exercises
  • Chapter 4. Topology of the Plane
    • §4.1. Some classic theorems
    • §4.2. The Jordan curve theorem I
    • §4.3. The Jordan curve theorem II
    • §4.4. Inside the Jordan curve
    • §4.5. Exercises
  • Chapter 5. Integrals and the Winding Number
    • §5.1. Differential forms and integration
    • §5.2. Closed and exact forms
    • §5.3. The winding number via integration
    • §5.4. Homology
    • §5.5. Cauchy’s theorem
    • §5.6. A glimpse at higher dimensions
    • §5.7. Exercises
  • Chapter 6. Vector Fields and the Rotation Number
    • §6.1. The rotation number
    • §6.2. Curvature and the rotation number
    • §6.3. Vector fields and singularities
    • §6.4. Vector fields and surfaces
    • §6.5. Exercises
  • Chapter 7. The Winding Number in Functional Analysis
    • §7.1. The Fredholm index
    • §7.2. Atkinson’s theorem
    • §7.3. Toeplitz operators
    • §7.4. The Toeplitz index theorem
    • §7.5. Exercises
  • Chapter 8. Coverings and the Fundamental Group
    • §8.1. The fundamental group
    • §8.2. Covering and lifting
    • §8.3. Group actions
    • §8.4. Examples
    • §8.5. The Nielsen-Schreier theorem
    • §8.6. An application to nonassociative algebra
    • §8.7. Exercises
  • Chapter 9. Coda: The Bott Periodicity Theorem
    • §9.1. Homotopy groups
    • §9.2. The topology of the general linear group
  • Appendix A. Linear Algebra
    • §A.1. Vector spaces
    • §A.2. Basis and dimension
    • §A.3. Linear transformations
    • §A.4. Duality
    • §A.5. Norms and inner products
    • §A.6. Matrices and determinants
  • Appendix B. Metric Spaces
    • §B.1. Metric spaces
    • §B.2. Continuous functions
    • §B.3. Compact spaces
    • §B.4. Function spaces
  • Appendix C. Extension and Approximation Theorems
    • §C.1. The Stone-Weierstrass theorem
    • §C.2. The Tietze extension theorem
  • Appendix D. Measure Zero
    • §D.1. Measure zero subsets of R and of S1
  • Appendix E. Calculus on Normed Spaces
    • §E.1. Normed vector spaces
    • §E.2. The derivative
    • §E.3. Properties of the derivative
    • §E.4. The inverse function theorem
  • Appendix F. Hilbert Space
    • §F.1. Definition and examples
    • §F.2. Orthogonality
    • §F.3. Operators
  • Appendix G. Groups and Graphs
    • §G.1. Equivalence relations
    • §G.2. Groups
    • §G.3. Homomorphisms
    • §G.4. Graphs
  • Bibliography
  • Index

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