Tensor Analysis on Manifolds

by Richard L. Bishop, Samuel I. Goldberg

Overview

"This is a first-rate book and deserves to be widely read." — American Mathematical Monthly
Despite its success as a mathematical tool in the general theory of relativity and its adaptability to a wide range of mathematical and physical problems, tensor analysis has always had a rather restricted level of use, with an emphasis on notation and the manipulation of indices. This book is an attempt to broaden this point of view at the stage where the student first encounters the subject. The authors have treated tensor analysis as a continuation of advanced calculus, striking just the right balance between the formal and abstract approaches to the subject.
The material proceeds from the general to the special. An introductory chapter establishes notation and explains various topics in set theory and topology. Chapters 1 and 2 develop tensor analysis in its function-theoretical and algebraic aspects, respectively. The next two chapters take up vector analysis on manifolds and integration theory. In the last two chapters (5 and 6) several important special structures are studied, those in Chapter 6 illustrating how the previous material can be adapted to clarify the ideas of classical mechanics. The text as a whole offers numerous examples and problems.
A student with a background of advanced calculus and elementary differential equation could readily undertake the study of this book. The more mature the reader is in terms of other mathematical knowledge and experience, the more he will learn from this presentation.

Product Details

ISBN-13:	9780486139234
Publisher:	Dover Publications
Publication date:	03/29/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	288
File size:	8 MB

Table of Contents

Chapter 0/Set Theory and Topology
0.1. SET THEORY
0.1.1. Sets
0.1.2. Set Operations
0.1.3. Cartesian Products
0.1.4. Functions
0.1.5. Functions and Set Operations
0.1.6. Equivalence Relations
0.2. TOPOLOGY
0.2.1. Topologies
0.2.2. Metric Spaces
0.2.3. Subspaces
0.2.4. Product Topologies
0.2.5. Hausdorff Spaces
0.2.6. Continuity
0.2.7. Connectedness
0.2.8. Compactness
0.2.9. Local Compactness
0.2.10. Separability
0.2.11 Paracompactness
Chapter 1/Manifolds
1.1. Definition of a Mainifold
1.2. Examples of Manifolds
1.3. Differentiable Maps
1.4. Submanifolds
1.5. Differentiable Maps
1.6. Tangents
1.7. Coordinate Vector Fields
1.8. Differential of a Map
Chapter 2/Tensor Algebra
2.1. Vector Spaces
2.2. Linear Independence
2.3. Summation Convention
2.4. Subspaces
2.5. Linear Functions
2.6. Spaces of Linear Functions
2.7. Dual Space
2.8. Multilinear Functions
2.9. Natural Pairing
2.10. Tensor Spaces
2.11. Algebra of Tensors
2.12. Reinterpretations
2.13. Transformation Laws
2.14. Invariants
2.15. Symmetric Tensors
2.16. Symmetric Algebra
2.17. Skew-Symmetric Tensors
2.18. Exterior Algebra
2.19. Determinants
2.20. Bilinear Forms
2.21. Quadratic Forms
2.22. Hodge Duality
2.23. Symplectic Forms
Chapter 3/Vector Analysis on Manifolds
3.1. Vector Fields
3.2. Tensor Fields
3.3. Riemannian Metrics
3.4. Integral Curves
3.5. Flows
3.6. Lie Derivatives
3.7. Bracket
3.8. Geometric Interpretation of Brackets
3.9. Action of Maps
3.10. Critical Point Theory
3.11. First Order Partial Differential Equations
3.12. Frobenius' Theorem
Appendix to Chapter 3
3A. Tensor Bundles
3B. Parallelizable Manifolds
3C. Orientability
Chapter 4/Integration Theory
4.1. Introduction
4.2. Differential Forms
4.3. Exterior Derivatives
4.4. Interior Products
4.5. Converse of the Poincaré Lemma
4.6. Cubical Chains
4.7. Integration on Euclidean Spaces
4.8. Integration of Forms
4.9. Strokes' Theorem
4.10. Differential Systems
Chapter 5/Riemannian and Semi-riemannian Manifolds
5.1. Introduction
5.2. Riemannian and Semi-riemannian Metrics
5.3. "Lengeth, Angle, Distance, and Energy"
5.4. Euclidean Space
5.5. Variations and Rectangles
5.6. Flat Spaces
5.7. Affine connexions
5.8 Parallel Translation
5.9. Covariant Differentiation of Tensor Fields
5.10. Curvature and Torsion Tensors
5.11. Connexion of a Semi-riemannian Structure
5.12. Geodesics
5.13. Minimizing Properties of Geodesics
5.14. Sectional Curvature
Chapter 6/Physical Application
6.1 Introduction
6.2. Hamiltonian Manifolds
6.3. Canonical Hamiltonian Structure on the Cotangent Bundle
6.4. Geodesic Spray of a Semi-riemannian Manifold
6.5. Phase Space
6.6. State Space
6.7. Contact Coordinates
6.8. Contact Manifolds
Bibliography
Index