Introduction to global analysis, Volume 91 (Pure and Applied Mathematics)

✍ Scribed by Kahn D.W. (editor)

• Publisher: Academic Press
• Year: 1980
• Tongue: English
• Leaves: 347
• Category: Library

✦ Synopsis

Geared toward advanced undergraduates and graduate students, this text introduces the methods of mathematical analysis as applied to manifolds. In addition to examining the roles of differentiation and integration, it explores infinite-dimensional manifolds, Morse theory, Lie groups, dynamical systems, and the roles of singularities and catastrophes. 1980 edition.

✦ Table of Contents

Introduction to Global Analysis
Copyright Page
Contents
Preface
Introduction
Chapter 1. Manifolds and Their Maps
Differentiable Manifolds and Their Maps
The Case of Euclidean Spaces
Power Series
Functions with Prescribed Properties; Norms
Germs and Jets
Problems and Projects
Chapter 2. Embeddings and Immersions of Manifolds
Some Important Examples
The Tangent Space
Existence of Embeddings and Immersions
Approximation of Smooth Mappings
Problems and Projects
Chapter 3. Critical Values, Sard’s Theorem, and Transversality
Critical Points and Values
Sard’s Theorem; Applications
Thom’s Transversality Lemma
Problems and Projects
Chapter 4. Tangent Bundles, Vector Bundles, and Classification
Groups Acting on Spaces
The Tangent Bundle
Vector Bundles
Constructions with Vector Bundles
The Classification of Vector Bundles
Examples of Classifications
Problems and Projects
Chapter 5. Differentiation and Integration on Manifolds
Integration in Several Variables
Exterior Algebra and Forms
Integration on Manifolds
The Poincarè Lemma
Stokes’ Theorem
Problems and Projects
Chapter 6. Differential Operators on Manifolds
Differential Operators on Smooth Bundles
Riemann Metrics and the Laplacian
Characterization of Linear Differential Operators
The Symbol; Ellipticity
Problems and Projects
Chapter 7. Infinite-Dimensional Manifolds
Topological Vector Spaces
Elements of Infinite-Dimensional Manifolds
Hilbert Manifolds; Partition of Unity
Function Spaces
The Unitary Group
Problems and Projects
Chapter 8. Morse Theory and Its Applications
Nondegenerate Critical Points
Homology and Morse Inequalities
Cell Decompositions from a Morse Function
Applications to Geodesics
Problems and Projects
Chapter 9. Lie Groups
Basic Theory of Lie Groups
The Idea of Lie Algebras
The Exponential Map
Closed Subgroups of Lie Groups
Invariant Forms and Integration
Representations of Lie Groups
Lie Groups Acting on Manifolds
Problems and Projects
Chapter 10. Dynamical Systems
Transformation Groups; Invariant and Minimal Sets
Linear Differential Equations in Euclidean Space
Planar Flows: The Poincaè-Bendixson Theorem
Families of Subspaces; The Frobenius Theorem
Problems and Projects
Chapter 11. A Description of Singularities and Catastrophes
Singularities of Smooth Maps; Stability
Finite Determination and Codimension
Unfoldings of Singularities
Elementary Catastrophes
Bibliography
Index