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Applicable differential geometry / M. Crampin, F.A.E. Pirani.

By: Crampin, MContributor(s): Pirani, F. A. E. (Felix Arnold Edward), 1928-2015Material type: TextTextSeries: London Mathematical Society lecture note series ; 59.Publication details: Cambridge ; New York : Cambridge University Press, 1986. Description: 1 online resource (394 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9780511623905; 0511623909; 9781107087187; 110708718X; 1316086747; 9781316086742; 1107099552; 9781107099555; 1107093392; 9781107093393; 1107090229; 9781107090224Subject(s): Geometry, Differential | Mechanics | Géométrie différentielle | Mécanique | MATHEMATICS -- Geometry -- Differential | Geometry, Differential | Mechanics | Differentialgeometrie | Einführung | Differentiaalmeetkunde | Toepassingen | Géométrie différentielle | DifferensialgeometriGenre/Form: Electronic books. | Einführung. | Electronic books. Additional physical formats: Print version:: Applicable differential geometry.DDC classification: 516.3/6 LOC classification: QA641 | .C73 1986ebOther classification: 31.52 | *53-01 | 22-01 | 53A45 | 53C05 | 53C80 | SI 320 | SK 370 | MAT 530f Online resources: Click here to access online
Contents:
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
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.
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Includes bibliographical references (pages 383-385) and index.

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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.

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

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