Vectors in physics and engineering

811 Pages·39.086 MB·English

by  Alan Durrant

Vectors in Physics and
Engineering

Vectors in Physics and
Engineering
A.V. Durrant
Senior Lecturer in Physics
The Open University
UK
CRC Press
(M£\
\Cf^ J Taylor & Francis Croup
Boca Raton London New York
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Contents
Preface IX
1 Vector algebra I: Scaling and adding vectors 1
1.1 INTRODUCTION TO SCALARS,NUMBERS AND VECTORS 1
1.1.1 Scalars and numbers 2
1.1.2 Introducing vectors 3
1.1.3 Displacements and arrows 4
1.1.4 Vector notation 5
1.2 SCALING VECTORS AND UNIT VECTORS 7
1.2.1 Scaling a vector or multiplication of a vector by a number 7
1.2.2 Unit vectors 9
1.3 VECTOR ADDITION-THE TRIANGLE ADDITION RULE 11
1.4 LINEAR COMBINATIONS OF VECTORS 17
1.5 CARTESIAN VECTORS 22
1.5.1 Cartesian coordinates of a point - a review 22
1.5.2 Cartesian unit vectors and cartesian components of a vector 23
1.6 MAGNITUDES AND DIRECTIONS OF CARTESIAN VECTORS 28
1.7 SCALING AND ADDING CARTESIAN VECTORS 33
1.8 VECTORS IN SCIENCE AND ENGINEERING 36
1.8.1 Definition of a vector and evidence for vector behaviour 36
1.8.2 Vector problems in science and engineering 39
2 Vector algebra II: Scalar products and vector
products 47
2.1 THE SCALAR PRODUCT 48
2.1.1 Definition of the scalar product and projections 48
2.1.2 The scalar product in vector algebra 50
2.2 CARTESIAN FORM OF THE SCALAR PRODUCT 52
2.3 THE ANGLE BETWEEN TWO VECTORS 55
vi Contents
2.4 THE VECTOR PRODUCT 60
2.4.1 Definition of the vector product 61
2.4.2 The vector product in vector algebra 62
2.5 CARTESIAN FORM OF THE VECTOR PRODUCT 66
2.6 TRIPLE PRODUCTS OF VECTORS 70
2.6.1 The scalar triple product 70
2.6.2 The vector triple product 72
2.7 SCALAR AND VECTOR PRODUCTS IN SCIENCE
AND ENGINEERING 74
2.7.1 Background summary: Forces, torque and equilibrium 74
2.7.2 Background summary: Work and energy 79
2.7.3 Background summary: Energy and torque on dipoles in
electric and magnetic fields 84
3 Time-dependent vectors 89
3.1 INTRODUCING VECTOR FUNCTIONS 90
3.1.1 Scalar functions - a review 90
3.1.2 Vector functions of time 91
3.2 DIFFERENTIATING VECTOR FUNCTIONS - DEFINITIONS
OF VELOCITY AND ACCELERATION 95
3.2.1 Differentiation of a scalar function - a review 95
3.2.2 Differentiation of a vector function 96
3.2.3 Definitions of velocity and acceleration 97
3.3 RULES OF DIFFERENTIATION OF VECTOR FUNCTIONS 102
3.4 ROTATIONAL MOTION-THE ANGULAR VELOCITY VECTOR 109
3.5 ROTATING VECTORS OF CONSTANT MAGNITUDE 114
3.6 APPLICATION TO RELATIVE MOTION AND INERTIAL
FORCES 117
3.6.1 Relative translational motion and inertial forces 118
3.6.2 Relative rotational motion and inertial forces 119
Contents vii
4 Scalar and vector fields 127
4.1 PICTORIAL REPRESENTATIONS OF FIELDS 128
4.1.1 Scalar field contours 128
4.1.2 Vector field lines 129
4.2 SCALAR FIELD FUNCTIONS 132
4.2.1 Specifying scalar field functions 132
4.2.2 Cartesian scalar fields 133
4.2.3 Graphs and contours 134
4.3 VECTOR FIELD FUNCTIONS 140
4.3.1 Specifying vector field functions 141
4.3.2 Cartesian vector fields 141
4.3.3 Equation of a field line 142
4.4 POLAR COORDINATE SYSTEMS 148
4.4.1 Symmetries and coordinate systems 148
4.4.2 Cylindrical polar coordinate systems 149
4.4.3 Spherical polar coordinate systems 151
4.5 INTRODUCING FLUX AND CIRCULATION 160
4.5.1 Flux of a vector field 160
4.5.2 Circulation of a vector field 163
5 Differentiating fields 171
5.1 DIRECTIONAL DERIVATIVES AND PARTIAL DERIVATIVES 172
5.2 GRADIENT OF A SCALAR FIELD 177
5.2.1 Introducing gradient 178
5.2.2 Calculating gradients 179
5.2.3 Gradient and physical law 180
5.3 DIVERGENCE OF A VECTOR HELD 187
5.3.1 Introducing divergence 188
5.3.2 Calculating divergence 190
5.3.3 Divergence and physical law 191
5.4 CURL OF A VECTOR FIELD 197
5.4.1 Introducing curl 198
5.4.2 Calculating curl 199
5.4.3 Curl and physical law 201
viii Contents
5.5 THE VECTOR DIFFERENTIAL OPERATOR "DEL" 208
5.5.1 Introducing differential operators 208
5.5.2 The "del" operator 208
5.5.3 The Laplacian operator 210
5.5.4 Vector-field identities 211
6 Integrating fields 219
6.1 DEFINITE INTEGRALS-A REVIEW 220
6.2 LINE INTEGRALS 223
6.2.1 Defining the scalar line integral 223
6.2.2 Evaluating simple line integrals 226
6.3 LINE INTEGRALS ALONG PARAMETERISED CURVES 232
6.3.1 Parameterisation of a curve 232
6.3.2 A systematic technique for evaluating line integrals 233
6.4 CONSERVATIVE FIELDS 238
6.5 SURFACE INTEGRALS 243
6.5.1 Introducing surface integrals 243
6.5.2 Expressing surface integrals as double integrals and
evaluating them 245
6.6 STOKES'S THEOREM 254
6.6.1 An integral form of curl 254
6.6.2 Deriving Stokes's theorem 255
6.6.3 Using Stokes's theorem 256
6.7 VOLUME INTEGRALS 260
6.8 GAUSS'S THEOREM (THE DIVERGENCE THEOREM) 265
Appendix A SI units and physical constants 270
Appendix B Mathematical conventions and useful results 273
Answers to selected Problems 276
Index 283
Preface
This book is intended as a self-study text for students following courses in
science and engineering where vectors are used. The material covered and the
level of treatment should be sufficient to provide the vector algebra and vector
calculus skills required for most honours courses in mechanics,
electromagnetism, fluid mechanics, aerodynamics, applied mathematics and
mathematical modelling. It is assumed that the student begins with minimal
(school-level) skills in algebra, geometry and calculus and has no previous
knowledge of vectors. There are brief reviews at appropriate points in the text
on elementary mathematical topics: the definition of a function, the derivative of
a function, the definite integral and partial differentiation.
The text is characterised by short two or three page sections where new
concepts, terminologies and skills are introduced, followed by detailed
summaries and consolidation in the form of Examples and Problems that test the
objectives listed at the beginning of each chapter. Each Example is followed by
a fully worked out solution, but the student is well advised to have a go at each
Example before looking at the solution. Many of the Examples and Problems
are set in the context of mechanics and electromagnetism but no significant
previous knowledge of these subjects is assumed. Bare answers to selected
Problems are given at the back of the book. Full solutions to all Problems can be
found on www at http://physics.open.uk/~avdurran/vectors.html.
Although the material covered makes relatively little demand on previously
acquired mathematical skills, the newcomer will find that there are many new
concepts to grapple with, new notations and skills to master and a large number
of technical terms to assimilate. Each new technical term is highlighted by
heavy print at the point in the text where it is most fully described or defined.
Vector algebra is developed in the first two chapters as a way of describing
elementary two and three dimensional spatial relationships and geometrical
figures in terms of displacement vectors, and is then applied to problems
involving velocities, forces and other physical vectors. Chapter 3 introduces
vector functions of time and the derivatives of vector functions, with applications
to circular motion, projectile motion and inertial forces in accelerating and
rotating coordinate systems. Chapter 4 introduces scalar and vector fields
initially in terms of contour surfaces and vector field lines and then as scalar and
vector functions of position. Spatial symmetries of fields are briefly discussed,
and cylindrical polar and spherical polar coordinate systems are introduced and
then used where appropriate throughout the book. Chapter 5 introduces the
differential calculus of scalar and vector fields. The concepts of gradient,
divergence and curl are dealt with informally, and their role in the expression of
physical laws is described. A section on the use of the "del" operator is included.
The integral calculus of fields culminating in Stokes's and Gauss's theorems is
dealt with in Chapter 6.
The approach throughout is physical and intuitive and there are no formal
proofs. The emphasis is on developing calculation skills, understanding the
concepts and seeing the relevance to physical processes. However, an attempt