UID:
edocfu_9959238162302883
Format:
1 online resource (394 pages) :
,
digital, PDF file(s).
ISBN:
1-316-08674-7
,
0-511-62390-9
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1-107-08718-X
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1-107-09955-2
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1-107-09339-2
,
1-107-09022-9
Series Statement:
London Mathematical Society lecture note series ;
Content:
This is an introduction to geometrical topics that are useful in applied mathematics and theoretical physics, including manifolds, metrics, connections, Lie groups, spinors and bundles, preparing readers for the study of modern treatments of mechanics, gauge fields theories, relativity and gravitation. The order of presentation corresponds to that used for the relevant material in theoretical physics: the geometry of affine spaces, which is appropriate to special relativity theory, as well as to Newtonian mechanics, is developed in the first half of the book, and the geometry of manifolds, which is needed for general relativity and gauge field theory, in the second half. Analysis is included not for its own sake, but only where it illuminates geometrical ideas. The style is informal and clear yet rigorous; each chapter ends with a summary of important concepts and results. In addition there are over 650 exercises, making this a book which is valuable as a text for advanced undergraduate and postgraduate students.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover; Title; Copyright; CONTENTS; Preface; 0. THE BACKGROUND: VECTOR CALCULUS; 1. Vectors; 2. Derivatives; 3. Coordinates; 4. The Range and Summation Conventions; Note to Chapter 0; 1. AFFINE SPACES; 1. Affine Spaces; 2. Lines and Planes; 3. Affine Spaces Modelled on Quotients and Direct Sums; 4. Affine Maps; 5. Affine Maps of Lines and Hyperplanes; Summary of Chapter 1; Notes to Chapter 1; 2. CURVES, FUNCTIONS AND DERIVATIVES; 1. Curves and Functions; 2. Tangent Vectors; 3. Directional Derivatives; 4. Cotangent Vectors; 5. Induced Maps; 6. Curvilinear Coordinates; 7. Smooth Maps
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8. Parallelism9. Covariant Derivatives; Summary of Chapter 2; Notes to Chapter 2; 3. VECTOR FIELDS AND FLOWS; 1. One-parameter Affine Groups; 2. One-parameter Groups: the General Case; 3. Flows; 4. Flows Associated with Vector Fields; 5. Lie Transport; 6. Lie Difference and Lie Derivative; 7. The Lie Derivative of a Vector Field as a Directional Derivative; 8. Vector Fields as Differential Operators; 9. Brackets and Commutators; 10. Covector Fields and the Lie Derivative; 11. Lie Derivative and Covariant Derivative Compared; 12. The Geometrical Significance of the Bracket
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Summary of Chapter 3Note to Chapter 3; 4. VOLUMES AND SUBSPACES: EXTERIOR ALGEBRA; 1. Volume of a Parallelepiped; 2. Volume as an Alternating Multilinear Function; 3. Transformation of Volumes; 4. Subspaces; 5. The Correspondence Between Multivectors and Forms; 6. Sums and Intersections of Subspaces; 7. Volume in a Subspace; 8. Exterior Algebra; 9. Bases and Dimensions; 10. The Interior Product; 11. Induced Maps of Forms; 12. Decomposable Forms; 13. An Extension Principle for Constructing Linear Maps of Forms; Summary of Chapter 4; Notes to Chapter 4; 5. CALCULUS OF FORMS; 1. Fields of Forms
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2. The Exterior Derivative3. Properties of the Exterior Derivative; 4. Lie Derivatives of Forms; 5. Volume Forms and the Divergence of a Vector Field; 6. A Formula Relating Lie and Exterior Derivatives; 8. Closed and Exact Forms; Summary of Chapter 5; 6. FROBENIUS'S THEOREM; 1. Distributions and Integral Submanifolds; Section 1; Section 2; 2. Necessary Conditions for Integrability; 3. Sufficient Conditions for Integrability; 4. Special Coordinate Systems; 5. Applications: Partial Differential Equations; 6. Application: Darboux's Theorem; 7. Application: Hamilton-Jacobi Theory
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Summary of Chapter 67. METRICS ON AFFINE SPACES; 1. Scalar Products on Vector Spaces; 2. Euclidean and Pseudo-Euclidean Spaces; 3. Scalar Products and Dual Spaces; 4. The Star Operator; 5. Metrics on Affine Spaces; 6. Parallelism in Affine Metric Space; 7. Vector Calculus; Summary of Chapter 7; 8. ISOMETRIES; 1. Isometries Defined; 2. Infinitesimal Isometries; 3. Killing's Equation and Killing Fields; 4. Conformal Transformations; 5. The Rotation Group; 6. Parametrising Rotations; 7. The Lorentz Group; 8. The Celestial Sphere; Summary of Chapter 8; 9. GEOMETRY OF SURFACES; 1. Surfaces
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2. Differential Geometry on a Surface
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English
Additional Edition:
ISBN 0-521-23190-6
Additional Edition:
ISBN 1-299-70608-8
Language:
English
