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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 CRC Press is an imprint of the Taylor St Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1996 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been mad e to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 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