Applicable differential geometry / M. Crampin, F.A.E. Pirani.
1986
QA641 .C73 1986eb
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Details
Title
Applicable differential geometry / M. Crampin, F.A.E. Pirani.
Author
ISBN
9780511623905 (electronic bk.)
0511623909 (electronic bk.)
9781107087187 (electronic bk.)
110708718X (electronic bk.)
1316086747
9781316086742
1107099552
9781107099555
1107093392
9781107093393
1107090229
9781107090224
0521231906
9780521231909
0511623909 (electronic bk.)
9781107087187 (electronic bk.)
110708718X (electronic bk.)
1316086747
9781316086742
1107099552
9781107099555
1107093392
9781107093393
1107090229
9781107090224
0521231906
9780521231909
Imprint
Cambridge ; New York : Cambridge University Press, 1986.
Language
English
Language Note
English.
Description
1 online resource (394 pages) : illustrations
Call Number
QA641 .C73 1986eb
System Control No.
(OCoLC)818665368
Summary
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.
Bibliography, etc. Note
Includes bibliographical references (pages 383-385) and index.
Formatted Contents Note
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
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
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
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
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
Source of Description
Print version record.
Added Author
Series
London Mathematical Society lecture note series ; 59. (OCoLC)788783964
Available in Other Form
Print version: Crampin, M. Applicable differential geometry. Cambridge [Cambridgeshire] ; New York : Cambridge University Press, 1986
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