# An Introduction to Vectors, Vector Operators and Vector Analysis

## Transcript Of An Introduction to Vectors, Vector Operators and Vector Analysis

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

An Introduction to Vectors, Vector Operators and Vector Analysis

Conceived as s a supplementary text and reference book for undergraduate and graduate students of science and engineering, this book intends communicating the fundamental concepts of vectors and their applications. It is divided into three units. The ﬁrst unit deals with basic formulation: both conceptual and theoretical. It discusses applications of algebraic operations, Levi-Civita notation and curvilinear coordinate systems like spherical polar and parabolic systems. Structures and analytical geometry of curves and surfaces is covered in detail. The second unit discusses algebra of operators and their types. It explains the equivalence between the algebra of vector operators and the algebra of matrices. Formulation of eigenvectors and eigenvalues of a linear vector operator are discussed using vector algebra. Topics including Mohr’s algorithm, Hamilton’s theorem and Euler’s theorem are discussed in detail. The unit ends with a discussion on transformation groups, rotation group, group of isometries and the Euclidean group, with applications to rigid displacements. The third unit deals with vector analysis. It discusses important topics including vector valued functions of a scalar variable, functions of vector argument (both scalar valued and vector valued): thus covering both the scalar and vector ﬁelds and vector integration Pramod S. Joag is presently working as CSIR Emeritus Scientist at the Savitribai Phule University of Pune, India. For over 30 years he has been teaching classical mechanics, quantum mechanics, electrodynamics, solid state physics, thermodynamics and statistical mechanics at undergraduate and graduate levels. His research interests include quantum information, and more speciﬁcally measures of quantum entanglement and quantum discord, production of multipartite entangled states, entangled Fermion systems, models of quantum nonlocality etc.

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

An Introduction to Vectors, Vector Operators and Vector Analysis

Pramod S. Joag

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi - 110002, India Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107154438 © Pramod S. Joag 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in India A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Joag, Pramod S., 1951- author. Title: An introduction to vectors, vector operators and vector analysis /

Pramod S. Joag. Description: Daryaganj, Delhi, India : Cambridge University Press, 2016. |

Includes bibliographical references and index. Identiﬁers: LCCN 2016019490| ISBN 9781107154438 (hardback) | ISBN 110715443X

(hardback) Subjects: LCSH: Vector analysis. | Mathematical physics. Classiﬁcation:LCC QC20.7.V4 J63 2016 | DDC 512/.5–dc23 LC record available at https://lccn.loc.gov/2016019490 ISBN 978-1-107-15443-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

To Ela and Ninad who made me write this document

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

Contents

Figures

xiii

Tables

xx

Preface

xxi

Nomenclature

xxv

I Basic Formulation

1 Getting Concepts and Gathering Tools

3

1.1 Vectors and Scalars

3

1.2 Space and Direction

4

1.3 Representing Vectors in Space

6

1.4 Addition and its Properties

8

1.4.1 Decomposition and resolution of vectors

13

1.4.2 Examples of vector addition

16

1.5 Coordinate Systems

18

1.5.1 Right-handed (dextral) and left-handed coordinate systems

18

1.6 Linear Independence, Basis

19

1.7 Scalar and Vector Products

22

1.7.1 Scalar product

22

1.7.2 Physical applications of the scalar product

30

1.7.3 Vector product

32

1.7.4 Generalizing the geometric interpretation of the vector product

36

1.7.5 Physical applications of the vector product

38

1.8 Products of Three or More Vectors

39

1.8.1 The scalar triple product

39

1.8.2 Physical applications of the scalar triple product

43

1.8.3 The vector triple product

45

1.9 Homomorphism and Isomorphism

45

© in this web service Cambridge University Press

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Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

viii Contents

1.10 Isomorphism with R3

45

1.11 A New Notation: Levi-Civita Symbols

48

1.12 Vector Identities

52

1.13 Vector Equations

54

1.14 Coordinate Systems Revisited: Curvilinear Coordinates

57

1.14.1 Spherical polar coordinates

57

1.14.2 Parabolic coordinates

60

1.15 Vector Fields

67

1.16 Orientation of a Triplet of Non-coplanar Vectors

68

1.16.1 Orientation of a plane

72

2 Vectors and Analytic Geometry

74

2.1 Straight Lines

74

2.2 Planes

83

2.3 Spheres

89

2.4 Conic Sections

90

3 Planar Vectors and Complex Numbers

94

3.1 Planar Curves on the Complex Plane

94

3.2 Comparison of Angles Between Vectors

99

3.3 Anharmonic Ratio: Parametric Equation to a Circle

100

3.4 Conformal Transforms, Inversion

101

3.5 Circle: Constant Angle and Constant Power Theorems

103

3.6 General Circle Formula

105

3.7 Circuit Impedance and Admittance

106

3.8 The Circle Transformation

107

II Vector Operators

4 Linear Operators

115

4.1 Linear Operators on E3

115

4.1.1 Adjoint operators

117

4.1.2 Inverse of an operator

117

4.1.3 Determinant of an invertible linear operator

119

4.1.4 Non-singular operators

121

4.1.5 Examples

121

4.2 Frames and Reciprocal Frames

124

4.3 Symmetric and Skewsymmetric Operators

126

4.3.1 Vector product as a skewsymmetric operator

128

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

Contents ix

4.4 Linear Operators and Matrices

129

4.5 An Equivalence Between Algebras

130

4.6 Change of Basis

132

5 Eigenvalues and Eigenvectors

134

5.1 Eigenvalues and Eigenvectors of a Linear Operator

134

5.1.1 Examples

138

5.2 Spectrum of a Symmetric Operator

141

5.3 Mohr’s Algorithm

147

5.3.1 Examples

151

5.4 Spectrum of a 2 × 2 Symmetric Matrix

155

5.5 Spectrum of Sn

156

6 Rotations and Reﬂections

158

6.1 Orthogonal Transformations: Rotations and Reﬂections

158

6.1.1 The canonical form of the orthogonal operator for reﬂection

161

6.1.2 Hamilton’s theorem

164

6.2 Canonical Form for Linear Operators

165

6.2.1 Examples

168

6.3 Rotations

170

6.3.1 Matrices representing rotations

176

6.4 Active and Passive Transformations: Symmetries

180

6.5 Euler Angles

184

6.6 Euler’s Theorem

188

7 Transformation Groups

191

7.1 Deﬁnition and Examples

191

7.2 The Rotation Group O +(3)

196

7.3 The Group of Isometries and the Euclidean Group

199

7.3.1 Chasles theorem

204

7.4 Similarities and Collineations

205

III Vector Analysis

8 Preliminaries

215

8.1 Fundamental Notions

215

8.2 Sets and Mappings

216

8.3 Convergence of a Sequence

217

8.4 Continuous Functions

220

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

x Contents

9 Vector Valued Functions of a Scalar Variable

221

9.1 Continuity and Differentiation

221

9.2 Geometry and Kinematics: Space Curves and Frenet–Seret Formulae

225

9.2.1 Normal, rectifying and osculating planes

236

9.2.2 Order of contact

238

9.2.3 The osculating circle

239

9.2.4 Natural equations of a space curve

240

9.2.5 Evolutes and involutes

243

9.3 Plane Curves

248

9.3.1 Three different parameterizations of an ellipse

248

9.3.2 Cycloids, epicycloids and trochoids

253

9.3.3 Orientation of curves

258

9.4 Chain Rule

263

9.5 Scalar Integration

263

9.6 Taylor Series

264

10 Functions with Vector Arguments

266

10.1 Need for the Directional Derivative

266

10.2 Partial Derivatives

266

10.3 Chain Rule

269

10.4 Directional Derivative and the Grad Operator

271

10.5 Taylor series

278

10.6 The Differential

279

10.7 Variation on a Curve

281

10.8 Gradient of a Potential

282

10.9 Inverse Maps and Implicit Functions

283

10.9.1 Inverse mapping theorem

284

10.9.2 Implicit function theorem

285

10.9.3 Algorithm to construct the inverse of a map

287

10.10 Differentiating Inverse Functions

291

10.11 Jacobian for the Composition of Maps

294

10.12 Surfaces

297

10.13 The Divergence and the Curl of a Vector Field

304

10.14 Differential Operators in Curvilinear Coordinates

313

11 Vector Integration

323

11.1 Line Integrals and Potential Functions

323

11.1.1 Curl of a vector ﬁeld and the line integral

341

© in this web service Cambridge University Press

www.cambridge.org

An Introduction to Vectors, Vector Operators and Vector Analysis

Conceived as s a supplementary text and reference book for undergraduate and graduate students of science and engineering, this book intends communicating the fundamental concepts of vectors and their applications. It is divided into three units. The ﬁrst unit deals with basic formulation: both conceptual and theoretical. It discusses applications of algebraic operations, Levi-Civita notation and curvilinear coordinate systems like spherical polar and parabolic systems. Structures and analytical geometry of curves and surfaces is covered in detail. The second unit discusses algebra of operators and their types. It explains the equivalence between the algebra of vector operators and the algebra of matrices. Formulation of eigenvectors and eigenvalues of a linear vector operator are discussed using vector algebra. Topics including Mohr’s algorithm, Hamilton’s theorem and Euler’s theorem are discussed in detail. The unit ends with a discussion on transformation groups, rotation group, group of isometries and the Euclidean group, with applications to rigid displacements. The third unit deals with vector analysis. It discusses important topics including vector valued functions of a scalar variable, functions of vector argument (both scalar valued and vector valued): thus covering both the scalar and vector ﬁelds and vector integration Pramod S. Joag is presently working as CSIR Emeritus Scientist at the Savitribai Phule University of Pune, India. For over 30 years he has been teaching classical mechanics, quantum mechanics, electrodynamics, solid state physics, thermodynamics and statistical mechanics at undergraduate and graduate levels. His research interests include quantum information, and more speciﬁcally measures of quantum entanglement and quantum discord, production of multipartite entangled states, entangled Fermion systems, models of quantum nonlocality etc.

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

An Introduction to Vectors, Vector Operators and Vector Analysis

Pramod S. Joag

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi - 110002, India Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107154438 © Pramod S. Joag 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in India A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Joag, Pramod S., 1951- author. Title: An introduction to vectors, vector operators and vector analysis /

Pramod S. Joag. Description: Daryaganj, Delhi, India : Cambridge University Press, 2016. |

Includes bibliographical references and index. Identiﬁers: LCCN 2016019490| ISBN 9781107154438 (hardback) | ISBN 110715443X

(hardback) Subjects: LCSH: Vector analysis. | Mathematical physics. Classiﬁcation:LCC QC20.7.V4 J63 2016 | DDC 512/.5–dc23 LC record available at https://lccn.loc.gov/2016019490 ISBN 978-1-107-15443-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

To Ela and Ninad who made me write this document

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

Contents

Figures

xiii

Tables

xx

Preface

xxi

Nomenclature

xxv

I Basic Formulation

1 Getting Concepts and Gathering Tools

3

1.1 Vectors and Scalars

3

1.2 Space and Direction

4

1.3 Representing Vectors in Space

6

1.4 Addition and its Properties

8

1.4.1 Decomposition and resolution of vectors

13

1.4.2 Examples of vector addition

16

1.5 Coordinate Systems

18

1.5.1 Right-handed (dextral) and left-handed coordinate systems

18

1.6 Linear Independence, Basis

19

1.7 Scalar and Vector Products

22

1.7.1 Scalar product

22

1.7.2 Physical applications of the scalar product

30

1.7.3 Vector product

32

1.7.4 Generalizing the geometric interpretation of the vector product

36

1.7.5 Physical applications of the vector product

38

1.8 Products of Three or More Vectors

39

1.8.1 The scalar triple product

39

1.8.2 Physical applications of the scalar triple product

43

1.8.3 The vector triple product

45

1.9 Homomorphism and Isomorphism

45

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

viii Contents

1.10 Isomorphism with R3

45

1.11 A New Notation: Levi-Civita Symbols

48

1.12 Vector Identities

52

1.13 Vector Equations

54

1.14 Coordinate Systems Revisited: Curvilinear Coordinates

57

1.14.1 Spherical polar coordinates

57

1.14.2 Parabolic coordinates

60

1.15 Vector Fields

67

1.16 Orientation of a Triplet of Non-coplanar Vectors

68

1.16.1 Orientation of a plane

72

2 Vectors and Analytic Geometry

74

2.1 Straight Lines

74

2.2 Planes

83

2.3 Spheres

89

2.4 Conic Sections

90

3 Planar Vectors and Complex Numbers

94

3.1 Planar Curves on the Complex Plane

94

3.2 Comparison of Angles Between Vectors

99

3.3 Anharmonic Ratio: Parametric Equation to a Circle

100

3.4 Conformal Transforms, Inversion

101

3.5 Circle: Constant Angle and Constant Power Theorems

103

3.6 General Circle Formula

105

3.7 Circuit Impedance and Admittance

106

3.8 The Circle Transformation

107

II Vector Operators

4 Linear Operators

115

4.1 Linear Operators on E3

115

4.1.1 Adjoint operators

117

4.1.2 Inverse of an operator

117

4.1.3 Determinant of an invertible linear operator

119

4.1.4 Non-singular operators

121

4.1.5 Examples

121

4.2 Frames and Reciprocal Frames

124

4.3 Symmetric and Skewsymmetric Operators

126

4.3.1 Vector product as a skewsymmetric operator

128

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

Contents ix

4.4 Linear Operators and Matrices

129

4.5 An Equivalence Between Algebras

130

4.6 Change of Basis

132

5 Eigenvalues and Eigenvectors

134

5.1 Eigenvalues and Eigenvectors of a Linear Operator

134

5.1.1 Examples

138

5.2 Spectrum of a Symmetric Operator

141

5.3 Mohr’s Algorithm

147

5.3.1 Examples

151

5.4 Spectrum of a 2 × 2 Symmetric Matrix

155

5.5 Spectrum of Sn

156

6 Rotations and Reﬂections

158

6.1 Orthogonal Transformations: Rotations and Reﬂections

158

6.1.1 The canonical form of the orthogonal operator for reﬂection

161

6.1.2 Hamilton’s theorem

164

6.2 Canonical Form for Linear Operators

165

6.2.1 Examples

168

6.3 Rotations

170

6.3.1 Matrices representing rotations

176

6.4 Active and Passive Transformations: Symmetries

180

6.5 Euler Angles

184

6.6 Euler’s Theorem

188

7 Transformation Groups

191

7.1 Deﬁnition and Examples

191

7.2 The Rotation Group O +(3)

196

7.3 The Group of Isometries and the Euclidean Group

199

7.3.1 Chasles theorem

204

7.4 Similarities and Collineations

205

III Vector Analysis

8 Preliminaries

215

8.1 Fundamental Notions

215

8.2 Sets and Mappings

216

8.3 Convergence of a Sequence

217

8.4 Continuous Functions

220

© in this web service Cambridge University Press

www.cambridge.org

Cambridge University Press 978-1-107-15443-8 - An Introduction to Vectors, Vector Operators and Vector Analysis Pramod S. Joag Frontmatter More information

x Contents

9 Vector Valued Functions of a Scalar Variable

221

9.1 Continuity and Differentiation

221

9.2 Geometry and Kinematics: Space Curves and Frenet–Seret Formulae

225

9.2.1 Normal, rectifying and osculating planes

236

9.2.2 Order of contact

238

9.2.3 The osculating circle

239

9.2.4 Natural equations of a space curve

240

9.2.5 Evolutes and involutes

243

9.3 Plane Curves

248

9.3.1 Three different parameterizations of an ellipse

248

9.3.2 Cycloids, epicycloids and trochoids

253

9.3.3 Orientation of curves

258

9.4 Chain Rule

263

9.5 Scalar Integration

263

9.6 Taylor Series

264

10 Functions with Vector Arguments

266

10.1 Need for the Directional Derivative

266

10.2 Partial Derivatives

266

10.3 Chain Rule

269

10.4 Directional Derivative and the Grad Operator

271

10.5 Taylor series

278

10.6 The Differential

279

10.7 Variation on a Curve

281

10.8 Gradient of a Potential

282

10.9 Inverse Maps and Implicit Functions

283

10.9.1 Inverse mapping theorem

284

10.9.2 Implicit function theorem

285

10.9.3 Algorithm to construct the inverse of a map

287

10.10 Differentiating Inverse Functions

291

10.11 Jacobian for the Composition of Maps

294

10.12 Surfaces

297

10.13 The Divergence and the Curl of a Vector Field

304

10.14 Differential Operators in Curvilinear Coordinates

313

11 Vector Integration

323

11.1 Line Integrals and Potential Functions

323

11.1.1 Curl of a vector ﬁeld and the line integral

341

© in this web service Cambridge University Press

www.cambridge.org