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
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
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