MATHEMATICAL METHODS FORhep.fcfm.buap.mx/.../Metodos_Matematicos_Fisicos-Arfken.pdf · 2013. 7. 9. · George B. Arfken Miami University Oxford, OH Hans J. Weber University of Virginia

MATHEMATICALMETHODS FOR

PHYSICISTSSIXTH EDITION

George B. ArfkenMiami University

Oxford, OH

Hans J. WeberUniversity of VirginiaCharlottesville, VA

Amsterdam Boston Heidelberg London New York OxfordParis San Diego San Francisco Singapore Sydney Tokyo

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MATHEMATICALMETHODS FOR

PHYSICISTSSIXTH EDITION

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CONTENTS

Preface xi

1 Vector Analysis 11.1 Definitions, Elementary Approach. . . . . . . . . . . . . . . . . . . . . 11.2 Rotation of the Coordinate Axes. . . . . . . . . . . . . . . . . . . . . . 71.3 Scalar or Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Vector or Cross Product. . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Triple Scalar Product, Triple Vector Product. . . . . . . . . . . . . . . 251.6 Gradient,∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.7 Divergence,∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.8 Curl, ∇× . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.9 Successive Applications of∇ . . . . . . . . . . . . . . . . . . . . . . . 491.10 Vector Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541.11 Gauss’ Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601.12 Stokes’ Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641.13 Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681.14 Gauss’ Law, Poisson’s Equation. . . . . . . . . . . . . . . . . . . . . . 791.15 Dirac Delta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 831.16 Helmholtz’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2 Vector Analysis in Curved Coordinates and Tensors 1032.1 Orthogonal Coordinates inR3 . . . . . . . . . . . . . . . . . . . . . . . 1032.2 Differential Vector Operators . . . . . . . . . . . . . . . . . . . . . . . 1102.3 Special Coordinate Systems: Introduction. . . . . . . . . . . . . . . . 1142.4 Circular Cylinder Coordinates. . . . . . . . . . . . . . . . . . . . . . . 1152.5 Spherical Polar Coordinates. . . . . . . . . . . . . . . . . . . . . . . . 123

v

vi Contents

2.6 Tensor Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1332.7 Contraction, Direct Product. . . . . . . . . . . . . . . . . . . . . . . . 1392.8 Quotient Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1412.9 Pseudotensors, Dual Tensors. . . . . . . . . . . . . . . . . . . . . . . 1422.10 General Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512.11 Tensor Derivative Operators. . . . . . . . . . . . . . . . . . . . . . . . 160


3 Determinants and Matrices 1653.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1763.3 Orthogonal Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1953.4 Hermitian Matrices, Unitary Matrices . . . . . . . . . . . . . . . . . . 2083.5 Diagonalization of Matrices. . . . . . . . . . . . . . . . . . . . . . . . 2153.6 Normal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231


4 Group Theory 2414.1 Introduction to Group Theory. . . . . . . . . . . . . . . . . . . . . . . 2414.2 Generators of Continuous Groups. . . . . . . . . . . . . . . . . . . . . 2464.3 Orbital Angular Momentum. . . . . . . . . . . . . . . . . . . . . . . . 2614.4 Angular Momentum Coupling. . . . . . . . . . . . . . . . . . . . . . . 2664.5 Homogeneous Lorentz Group. . . . . . . . . . . . . . . . . . . . . . . 2784.6 Lorentz Covariance of Maxwell’s Equations. . . . . . . . . . . . . . . 2834.7 Discrete Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2914.8 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304


5 Infinite Series 3215.1 Fundamental Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.2 Convergence Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3255.3 Alternating Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3395.4 Algebra of Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3425.5 Series of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3485.6 Taylor’s Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3525.7 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3635.8 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3705.9 Bernoulli Numbers, Euler–Maclaurin Formula. . . . . . . . . . . . . . 3765.10 Asymptotic Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3895.11 Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396


6 Functions of a Complex Variable I Analytic Properties, Mapping 4036.1 Complex Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4046.2 Cauchy–Riemann Conditions. . . . . . . . . . . . . . . . . . . . . . . 4136.3 Cauchy’s Integral Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 418

Contents vii

6.4 Cauchy’s Integral Formula. . . . . . . . . . . . . . . . . . . . . . . . . 4256.5 Laurent Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4306.6 Singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4386.7 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4436.8 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451


7 Functions of a Complex Variable II 4557.1 Calculus of Residues. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4557.2 Dispersion Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4827.3 Method of Steepest Descents. . . . . . . . . . . . . . . . . . . . . . . . 489


8 The Gamma Function (Factorial Function) 4998.1 Definitions, Simple Properties. . . . . . . . . . . . . . . . . . . . . . . 4998.2 Digamma and Polygamma Functions. . . . . . . . . . . . . . . . . . . 5108.3 Stirling’s Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5168.4 The Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5208.5 Incomplete Gamma Function. . . . . . . . . . . . . . . . . . . . . . . 527


9 Differential Equations 5359.1 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 5359.2 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . 5439.3 Separation of Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . 5549.4 Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5629.5 Series Solutions—Frobenius’ Method. . . . . . . . . . . . . . . . . . . 5659.6 A Second Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5789.7 Nonhomogeneous Equation—Green’s Function. . . . . . . . . . . . . 5929.8 Heat Flow, or Diffusion, PDE. . . . . . . . . . . . . . . . . . . . . . . 611


10 Sturm–Liouville Theory—Orthogonal Functions 62110.1 Self-Adjoint ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62210.2 Hermitian Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63410.3 Gram–Schmidt Orthogonalization. . . . . . . . . . . . . . . . . . . . . 64210.4 Completeness of Eigenfunctions. . . . . . . . . . . . . . . . . . . . . . 64910.5 Green’s Function—Eigenfunction Expansion. . . . . . . . . . . . . . . 662


11 Bessel Functions 67511.1 Bessel Functions of the First Kind,Jν(x) . . . . . . . . . . . . . . . . . 67511.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69411.3 Neumann Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69911.4 Hankel Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70711.5 Modified Bessel Functions,Iν(x) andKν(x) . . . . . . . . . . . . . . . 713

viii Contents

11.6 Asymptotic Expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . 71911.7 Spherical Bessel Functions. . . . . . . . . . . . . . . . . . . . . . . . . 725


12 Legendre Functions 74112.1 Generating Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74112.2 Recurrence Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74912.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75612.4 Alternate Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76712.5 Associated Legendre Functions. . . . . . . . . . . . . . . . . . . . . . 77112.6 Spherical Harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78612.7 Orbital Angular Momentum Operators. . . . . . . . . . . . . . . . . . 79312.8 Addition Theorem for Spherical Harmonics. . . . . . . . . . . . . . . 79712.9 Integrals of Three Y’s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80312.10 Legendre Functions of the Second Kind. . . . . . . . . . . . . . . . . . 80612.11 Vector Spherical Harmonics. . . . . . . . . . . . . . . . . . . . . . . . 813


13 More Special Functions 81713.1 Hermite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81713.2 Laguerre Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83713.3 Chebyshev Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . 84813.4 Hypergeometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . 85913.5 Confluent Hypergeometric Functions. . . . . . . . . . . . . . . . . . . 86313.6 Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869


14 Fourier Series 88114.1 General Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88114.2 Advantages, Uses of Fourier Series. . . . . . . . . . . . . . . . . . . . 88814.3 Applications of Fourier Series. . . . . . . . . . . . . . . . . . . . . . . 89214.4 Properties of Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . 90314.5 Gibbs Phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91014.6 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 91414.7 Fourier Expansions of Mathieu Functions. . . . . . . . . . . . . . . . 919


15 Integral Transforms 93115.1 Integral Transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93115.2 Development of the Fourier Integral. . . . . . . . . . . . . . . . . . . . 93615.3 Fourier Transforms—Inversion Theorem. . . . . . . . . . . . . . . . . 93815.4 Fourier Transform of Derivatives. . . . . . . . . . . . . . . . . . . . . 94615.5 Convolution Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95115.6 Momentum Representation. . . . . . . . . . . . . . . . . . . . . . . . . 95515.7 Transfer Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96115.8 Laplace Transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965

Contents ix

15.9 Laplace Transform of Derivatives. . . . . . . . . . . . . . . . . . . . . 97115.10 Other Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97915.11 Convolution (Faltungs) Theorem. . . . . . . . . . . . . . . . . . . . . 99015.12 Inverse Laplace Transform. . . . . . . . . . . . . . . . . . . . . . . . . 994


16 Integral Equations 100516.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100516.2 Integral Transforms, Generating Functions. . . . . . . . . . . . . . . . 101216.3 Neumann Series, Separable (Degenerate) Kernels. . . . . . . . . . . . 101816.4 Hilbert–Schmidt Theory. . . . . . . . . . . . . . . . . . . . . . . . . . 1029


17 Calculus of Variations 103717.1 A Dependent and an Independent Variable. . . . . . . . . . . . . . . . 103817.2 Applications of the Euler Equation. . . . . . . . . . . . . . . . . . . . 104417.3 Several Dependent Variables. . . . . . . . . . . . . . . . . . . . . . . . 105217.4 Several Independent Variables. . . . . . . . . . . . . . . . . . . . . . . 105617.5 Several Dependent and Independent Variables. . . . . . . . . . . . . . 105817.6 Lagrangian Multipliers. . . . . . . . . . . . . . . . . . . . . . . . . . . 106017.7 Variation with Constraints. . . . . . . . . . . . . . . . . . . . . . . . . 106517.8 Rayleigh–Ritz Variational Technique. . . . . . . . . . . . . . . . . . . 1072


18 Nonlinear Methods and Chaos 107918.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107918.2 The Logistic Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108018.3 Sensitivity to Initial Conditions and Parameters. . . . . . . . . . . . . 108518.4 Nonlinear Differential Equations. . . . . . . . . . . . . . . . . . . . . 1088


19 Probability 110919.1 Definitions, Simple Properties. . . . . . . . . . . . . . . . . . . . . . . 110919.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111619.3 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 112819.4 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113019.5 Gauss’ Normal Distribution. . . . . . . . . . . . . . . . . . . . . . . . 113419.6 Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138

Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150General References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150

Index 1153

PREFACE

Through six editions now,Mathematical Methods for Physicistshas provided all the math-ematical methods that aspirings scientists and engineers are likely to encounter as studentsand beginning researchers. More than enough material is included for a two-semester un-dergraduate or graduate course.

The book is advanced in the sense that mathematical relations are almost always proven,in addition to being illustrated in terms of examples. These proofs are not what a mathe-matician would regard as rigorous, but sketch the ideas and emphasize the relations thatare essential to the study of physics and related fields. This approach incorporates theo-rems that are usually not cited under the most general assumptions, but are tailored to themore restricted applications required by physics. For example, Stokes’ theorem is usuallyapplied by a physicist to a surface with the tacit understanding that it be simply connected.Such assumptions have been made more explicit.

PROBLEM-SOLVING SKILLS

The book also incorporates a deliberate focus on problem-solving skills. This more ad-vanced level of understanding and active learning is routine in physics courses and requirespractice by the reader. Accordingly, extensive problem sets appearing in each chapter forman integral part of the book. They have been carefully reviewed, revised and enlarged forthis Sixth Edition.

PATHWAYS THROUGH THE MATERIAL

Undergraduates may be best served if they start by reviewing Chapter 1 according to thelevel of training of the class. Section 1.2 on the transformation properties of vectors, thecross product, and the invariance of the scalar product under rotations may be postponeduntil tensor analysis is started, for which these sections form the introduction and serve as

xi

xii Preface

examples. They may continue their studies with linear algebra in Chapter 3, then perhapstensors and symmetries (Chapters 2 and 4), and next real and complex analysis (Chap-ters 5–7), differential equations (Chapters 9, 10), and special functions (Chapters 11–13).

In general, the core of a graduate one-semester course comprises Chapters 5–10 and11–13, which deal with real and complex analysis, differential equations, and special func-tions. Depending on the level of the students in a course, some linear algebra in Chapter 3(eigenvalues, for example), along with symmetries (group theory in Chapter 4), and ten-sors (Chapter 2) may be covered as needed or according to taste. Group theory may also beincluded with differential equations (Chapters 9 and 10). Appropriate relations have beenincluded and are discussed in Chapters 4 and 9.

A two-semester course can treat tensors, group theory, and special functions (Chap-ters 11–13) more extensively, and add Fourier series (Chapter 14), integral transforms(Chapter 15), integral equations (Chapter 16), and the calculus of variations (Chapter 17).

CHANGES TO THE SIXTH EDITION

Improvements to the Sixth Edition have been made in nearly all chapters adding examplesand problems and more derivations of results. Numerous left-over typos caused by scan-ning into LaTeX, an error-prone process at the rate of many errors per page, have beencorrected along with mistakes, such as in the Diracγ -matrices in Chapter 3. A few chap-ters have been relocated. The Gamma function is now in Chapter 8 following Chapters 6and 7 on complex functions in one variable, as it is an application of these methods. Dif-ferential equations are now in Chapters 9 and 10. A new chapter on probability has beenadded, as well as new subsections on differential forms and Mathieu functions in responseto persistent demands by readers and students over the years. The new subsections aremore advanced and are written in the concise style of the book, thereby raising its level tothe graduate level. Many examples have been added, for example in Chapters 1 and 2, thatare often used in physics or are standard lore of physics courses. A number of additionshave been made in Chapter 3, such as on linear dependence of vectors, dual vector spacesand spectral decomposition of symmetric or Hermitian matrices. A subsection on the dif-fusion equation emphasizes methods to adapt solutions of partial differential equations toboundary conditions. New formulas have been developed for Hermite polynomials and areincluded in Chapter 13 that are useful for treating molecular vibrations; they are of interestto the chemical physicists.

ACKNOWLEDGMENTS

We have benefited from the advice and help of many people. Some of the revisions are in re-sponse to comments by readers and former students, such as Dr. K. Bodoor and J. Hughes.We are grateful to them and to our Editors Barbara Holland and Tom Singer who organizedaccuracy checks. We would like to thank in particular Dr. Michael Bozoian and Prof. FrankHarris for their invaluable help with the accuracy checking and Simon Crump, ProductionEditor, for his expert management of the Sixth Edition.

CHAPTER 1

VECTOR ANALYSIS

1.1 DEFINITIONS, ELEMENTARY APPROACH

In science and engineering we frequently encounter quantities that have magnitude andmagnitude only: mass, time, and temperature. These we labelscalar quantities, which re-main the same no matter what coordinates we use. In contrast, many interesting physicalquantities have magnitude and, in addition, an associated direction. This second groupincludes displacement, velocity, acceleration, force, momentum, and angular momentum.Quantities with magnitude and direction are labeledvector quantities. Usually, in elemen-tary treatments, a vector is defined as a quantity having magnitude and direction. To dis-tinguish vectors from scalars, we identify vector quantities with boldface type, that is,V.

Our vector may be conveniently represented by an arrow, with length proportional to themagnitude. The direction of the arrow gives the direction of the vector, the positive senseof direction being indicated by the point. In this representation, vector addition

C= A +B (1.1)consists in placing the rear end of vectorB at the point of vectorA. Vector C is thenrepresented by an arrow drawn from the rear ofA to the point ofB. This procedure, thetriangle law of addition, assigns meaning to Eq. (1.1) and is illustrated in Fig. 1.1. Bycompleting the parallelogram, we see that

C= A +B= B+A, (1.2)as shown in Fig. 1.2. In words, vector addition iscommutative.

For the sum of three vectors

D= A +B+C,Fig. 1.3, we may first addA andB:

A +B= E.

1

2 Chapter 1 Vector Analysis

FIGURE 1.1 Triangle law of vectoraddition.

FIGURE 1.2 Parallelogram law ofvector addition.

FIGURE 1.3 Vector addition isassociative.

Then this sum is added toC:

D= E+C.

Similarly, we may first addB andC:

B+C= F.

Then

D= A + F.

In terms of the original expression,

(A +B)+C= A + (B+C).

Vector addition isassociative.A direct physical example of the parallelogram addition law is provided by a weight

suspended by two cords. If the junction point (O in Fig. 1.4) is in equilibrium, the vector

1.1 Definitions, Elementary Approach 3

FIGURE 1.4 Equilibrium of forces:F1+ F2=−F3.

sum of the two forcesF1 andF2 must just cancel the downward force of gravity,F3. Herethe parallelogram addition law is subject to immediate experimental verification.1

Subtraction may be handled by defining the negative of a vector as a vector of the samemagnitude but with reversed direction. Then

A −B= A + (−B).

In Fig. 1.3,

A = E−B.

Note that the vectors are treated as geometrical objects that are independent of any coor-dinate system. This concept of independence of a preferred coordinate system is developedin detail in the next section.

The representation of vectorA by an arrow suggests a second possibility. ArrowA(Fig. 1.5), starting from the origin,2 terminates at the point(Ax,Ay,Az). Thus, if we agreethat the vector is to start at the origin, the positive end may be specified by giving theCartesian coordinates(Ax,Ay,Az) of the arrowhead.

AlthoughA could have represented any vector quantity (momentum, electric field, etc.),one particularly important vector quantity, the displacement from the origin to the point

1Strictly speaking, the parallelogram addition was introduced as a definition. Experiments show that if we assume that theforces are vector quantities and we combine them by parallelogram addition, the equilibrium condition of zero resultant force issatisfied.2We could start from any point in our Cartesian reference frame; we choose the origin for simplicity. This freedom of shiftingthe origin of the coordinate system without affecting the geometry is calledtranslation invariance.


FIGURE 1.5 Cartesian components and direction cosines ofA.

(x, y, z), is denoted by the special symbolr . We then have a choice of referring to the dis-placement as either the vectorr or the collection(x, y, z), the coordinates of its endpoint:

r ↔ (x, y, z). (1.3)Usingr for the magnitude of vectorr , we find that Fig. 1.5 shows that the endpoint coor-dinates and the magnitude are related by

x = r cosα, y = r cosβ, z= r cosγ. (1.4)Here cosα, cosβ, and cosγ are called thedirection cosines, α being the angle between thegiven vector and the positivex-axis, and so on. One further bit of vocabulary: The quan-tities Ax,Ay , andAz are known as the (Cartesian)componentsof A or theprojectionsof A, with cos2α + cos2β + cos2γ = 1.

Thus, any vectorA may be resolved into its components (or projected onto the coordi-nate axes) to yieldAx =Acosα, etc., as in Eq. (1.4). We may choose to refer to the vectoras a single quantityA or to its components(Ax,Ay,Az). Note that the subscriptx in Axdenotes thex component and not a dependence on the variablex. The choice betweenusingA or its components(Ax,Ay,Az) is essentially a choice between a geometric andan algebraic representation. Use either representation at your convenience. The geometric“arrow in space” may aid in visualization. The algebraic set of components is usually moresuitable for precise numerical or algebraic calculations.

Vectors enter physics in two distinct forms. (1) VectorA may represent a single forceacting at a single point. The force of gravity acting at the center of gravity illustrates thisform. (2) VectorA may be defined over some extended region; that is,A and its compo-nents may be functions of position:Ax = Ax(x, y, z), and so on. Examples of this sortinclude the velocity of a fluid varying from point to point over a given volume and electricand magnetic fields. These two cases may be distinguished by referring to the vector de-fined over a region as avector field. The concept of the vector defined over a region and

1.1 Definitions, Elementary Approach 5

being a function of position will become extremely important when we differentiate andintegrate vectors.

At this stage it is convenient to introduce unit vectors along each of the coordinate axes.Let x̂ be a vector of unit magnitude pointing in the positivex-direction,ŷ, a vector of unitmagnitude in the positivey-direction, and̂z a vector of unit magnitude in the positivez-direction. Thenx̂Ax is a vector with magnitude equal to|Ax | and in thex-direction. Byvector addition,

A = x̂Ax + ŷAy + ẑAz. (1.5)

Note that ifA vanishes, all of its components must vanish individually; that is, if

A = 0, thenAx =Ay =Az = 0.

This means that these unit vectors serve as abasis, or complete set of vectors, in the three-dimensional Euclidean space in terms of which any vector can be expanded. Thus, Eq. (1.5)is an assertion that the three unit vectorsx̂, ŷ, andẑ span our real three-dimensional space:Any vector may be written as a linear combination ofx̂, ŷ, and ẑ. Sincex̂, ŷ, and ẑ arelinearly independent (no one is a linear combination of the other two), they form abasisfor the real three-dimensional Euclidean space. Finally, by the Pythagorean theorem, themagnitude of vectorA is

|A| =(A2x +A2y +A2z

)1/2. (1.6)

Note that the coordinate unit vectors are not the only complete set, or basis. This resolutionof a vector into its components can be carried out in a variety of coordinate systems, asshown in Chapter 2. Here we restrict ourselves to Cartesian coordinates, where the unitvectors have the coordinatesx̂= (1,0,0), ŷ= (0,1,0) andẑ= (0,0,1) and are all constantin length and direction, properties characteristic of Cartesian coordinates.

As a replacement of the graphical technique, addition and subtraction of vectors maynow be carried out in terms of their components. ForA = x̂Ax + ŷAy + ẑAz and B =x̂Bx + ŷBy + ẑBz,

A ±B= x̂(Ax ±Bx)+ ŷ(Ay ±By)+ ẑ(Az ±Bz). (1.7)

It should be emphasized here that the unit vectorsx̂, ŷ, andẑ are used for convenience.They are not essential; we can describe vectors and use them entirely in terms of theircomponents:A ↔ (Ax,Ay,Az). This is the approach of the two more powerful, moresophisticated definitions of vector to be discussed in the next section. However,x̂, ŷ, andẑ emphasize thedirection.

So far we have defined the operations of addition and subtraction of vectors. In the nextsections, three varieties of multiplication will be defined on the basis of their applicability:a scalar, or inner, product, a vector product peculiar to three-dimensional space, and adirect, or outer, product yielding a second-rank tensor. Division by a vector is not defined.


Exercises

1.1.1 Show how to findA andB, givenA +B andA −B.1.1.2 The vectorA whose magnitude is 1.732 units makes equal angles with the coordinate

axes. FindAx,Ay , andAz.

1.1.3 Calculate the components of a unit vector that lies in thexy-plane and makes equalangles with the positive directions of thex- andy-axes.

1.1.4 The velocity of sailboatA relative to sailboatB, vrel, is defined by the equationvrel=vA − vB , wherevA is the velocity ofA and vB is the velocity ofB. Determine thevelocity ofA relative toB if

vA = 30 km/hr eastvB = 40 km/hr north.

ANS. vrel= 50 km/hr, 53.1◦ south of east.1.1.5 A sailboat sails for 1 hr at 4 km/hr (relative to the water) on a steady compass heading

of 40◦ east of north. The sailboat is simultaneously carried along by a current. At theend of the hour the boat is 6.12 km from its starting point. The line from its starting pointto its location lies 60◦ east of north. Find thex (easterly) andy (northerly) componentsof the water’s velocity.

ANS. veast= 2.73 km/hr, vnorth≈ 0 km/hr.1.1.6 A vector equation can be reduced to the formA = B. From this show that the one vector

equation is equivalent tothree scalar equations. Assuming the validity of Newton’ssecond law,F=ma, as avector equation, this means thatax depends only onFx andis independent ofFy andFz.

1.1.7 The verticesA,B, andC of a triangle are given by the points(−1,0,2), (0,1,0), and(1,−1,0), respectively. Find pointD so that the figureABCD forms a plane parallel-ogram.

ANS. (0,−2,2) or (2,0,−2).1.1.8 A triangle is defined by the vertices of three vectorsA,B andC that extend from the

origin. In terms ofA,B, andC show that thevector sum of the successive sides of thetriangle(AB +BC +CA) is zero, where the sideAB is fromA to B, etc.

1.1.9 A sphere of radiusa is centered at a pointr1.

(a) Write out the algebraic equation for the sphere.(b) Write out avector equation for the sphere.

ANS. (a)(x − x1)2+ (y − y1)2+ (z− z1)2= a2.(b) r = r1+ a, with r1= center.(a takes on all directions but has a fixed magnitudea.)

1.2 Rotation of the Coordinate Axes 7

1.1.10 A corner reflector is formed by three mutually perpendicular reflecting surfaces. Showthat a ray of light incident upon the corner reflector (striking all three surfaces) is re-flected back along a line parallel to the line of incidence.Hint. Consider the effect of a reflection on the components of a vector describing thedirection of the light ray.

1.1.11 Hubble’s law. Hubble found that distant galaxies are receding with a velocity propor-tional to their distance from where we are on Earth. For theith galaxy,

vi =H0r i,with us at the origin. Show that this recession of the galaxies from us doesnot implythat we are at the center of the universe. Specifically, take the galaxy atr1 as a neworigin and show that Hubble’s law is still obeyed.

1.1.12 Find the diagonal vectors of a unit cube with one corner at the origin and its three sideslying along Cartesian coordinates axes. Show that there are four diagonals with length√

3. Representing these as vectors, what are their components? Show that the diagonalsof the cube’s faces have length

√2 and determine their components.

1.2 ROTATION OF THE COORDINATE AXES3

In the preceding section vectors were defined or represented in two equivalent ways:(1) geometrically by specifying magnitude and direction, as with an arrow, and (2) al-gebraically by specifying the components relative to Cartesian coordinate axes. The sec-ond definition is adequate for the vector analysis of this chapter. In this section two morerefined, sophisticated, and powerful definitions are presented. First, the vector field is de-fined in terms of the behavior of its components under rotation of the coordinate axes. Thistransformation theory approach leads into the tensor analysis of Chapter 2 and groups oftransformations in Chapter 4. Second, the component definition of Section 1.1 is refinedand generalized according to the mathematician’s concepts of vector and vector space. Thisapproach leads to function spaces, including the Hilbert space.

The definition of vector as a quantity with magnitude and direction is incomplete. Onthe one hand, we encounter quantities, such as elastic constants and index of refractionin anisotropic crystals, that have magnitude and directionbut that are not vectors. Onthe other hand, our naïve approach is awkward to generalize to extend to more complexquantities. We seek a new definition of vector field using our coordinate vectorr as aprototype.

There is a physical basis for our development of a new definition. We describe our phys-ical world by mathematics, but it and any physical predictions we may make must beindependentof our mathematical conventions.

In our specific case we assume that space is isotropic; that is, there is no preferred di-rection, or all directions are equivalent. Then the physical system being analyzed or thephysical law being enunciated cannot and must not depend on our choice ororientationof the coordinate axes. Specifically, if a quantityS does not depend on the orientation ofthe coordinate axes, it is called a scalar.

3This section is optional here. It will be essential for Chapter 2.


FIGURE 1.6 Rotation of Cartesian coordinate axes about thez-axis.

Now we return to the concept of vectorr as a geometric object independent of thecoordinate system. Let us look atr in two different systems, one rotated in relation to theother.

For simplicity we consider first the two-dimensional case. If thex-, y-coordinates arerotated counterclockwise through an angleϕ, keeping r, fixed (Fig. 1.6), we get the fol-lowing relations between the components resolved in the original system (unprimed) andthose resolved in the new rotated system (primed):

x′ = x cosϕ + y sinϕ,y′ =−x sinϕ + y cosϕ. (1.8)

We saw in Section 1.1 that a vector could be represented by the coordinates of a point;that is, the coordinates were proportional to the vector components. Hence the componentsof a vector must transform under rotation as coordinates of a point (such asr ). Thereforewhenever any pair of quantitiesAx andAy in thexy-coordinate system is transformed into(A′x,A

′y) by this rotation of the coordinate system with

A′x =Ax cosϕ +Ay sinϕ,A′y =−Ax sinϕ +Ay cosϕ,

(1.9)

wedefine4 Ax andAy as the components of a vectorA. Our vector now is defined in termsof the transformation of its components under rotation of the coordinate system. IfAx andAy transform in the same way asx andy, the components of the general two-dimensionalcoordinate vectorr , they are the components of a vectorA. If Ax andAy do not show this

4A scalar quantity does not depend on the orientation of coordinates;S′ = S expresses the fact that it is invariant under rotationof the coordinates.


form invariance (also calledcovariance) when the coordinates are rotated, they do notform a vector.

The vector field componentsAx andAy satisfying the defining equations, Eqs. (1.9), as-sociate a magnitudeA and a direction with each point in space. The magnitude is a scalarquantity, invariant to the rotation of the coordinate system. The direction (relative to theunprimed system) is likewise invariant to the rotation of the coordinate system (see Exer-cise 1.2.1). The result of all this is that the components of a vector may vary according tothe rotation of the primed coordinate system. This is what Eqs. (1.9) say. But the variationwith the angle is just such that the components in the rotated coordinate systemA′x andA

′y

define a vector with the same magnitude and the same direction as the vector defined bythe componentsAx andAy relative to thex-, y-coordinate axes. (Compare Exercise 1.2.1.)The components ofA in a particular coordinate system constitute therepresentation ofA in that coordinate system. Equations (1.9), the transformation relations, are a guaranteethat the entityA is independent of the rotation of the coordinate system.

To go on to three and, later, four dimensions, we find it convenient to use a more compactnotation. Let

x→ x1y→ x2 (1.10)

a11= cosϕ, a12= sinϕ,a21=−sinϕ, a22= cosϕ.

(1.11)

Then Eqs. (1.8) become

x′1= a11x1+ a12x2,x′2= a21x1+ a22x2.

(1.12)

The coefficientaij may be interpreted as a direction cosine, the cosine of the angle betweenx′i andxj ; that is,

a12= cos(x′1, x2)= sinϕ,a21= cos(x′2, x1)= cos

(ϕ + π2

)=−sinϕ. (1.13)

The advantage of the new notation5 is that it permits us to use the summation symbol∑

and to rewrite Eqs. (1.12) as

x′i =2∑

j=1aijxj , i = 1,2. (1.14)

Note thati remains as a parameter that gives rise to one equation when it is set equal to 1and to a second equation when it is set equal to 2. The indexj , of course, is a summationindex, a dummy index, and, as with a variable of integration,j may be replaced by anyother convenient symbol.

5You may wonder at the replacement of one parameterϕ by four parametersaij . Clearly, theaij do not constitute a minimumset of parameters. For two dimensions the fouraij are subject to the three constraints given in Eq. (1.18). The justification forthis redundant set of direction cosines is the convenience it provides. Hopefully, this convenience will become more apparentin Chapters 2 and 3. For three-dimensional rotations (9aij but only three independent) alternate descriptions are provided by:(1) the Euler angles discussed in Section 3.3, (2) quaternions, and (3) the Cayley–Klein parameters. These alternatives have theirrespective advantages and disadvantages.


The generalization to three, four, orN dimensions is now simple. The set ofN quantitiesVj is said to be the components of anN -dimensional vectorV if and only if their valuesrelative to the rotated coordinate axes are given by

V ′i =N∑

j=1aijVj , i = 1,2, . . . ,N. (1.15)

As before,aij is the cosine of the angle betweenx′i andxj . Often the upper limitN andthe corresponding range ofi will not be indicated. It is taken for granted that you knowhow many dimensions your space has.

From the definition ofaij as the cosine of the angle between the positivex′i directionand the positivexj direction we may write (Cartesian coordinates)6

aij =∂x′i∂xj

. (1.16a)

Using the inverse rotation (ϕ→−ϕ) yields

xj =2∑

i=1aijx

′i or

∂xj

∂x′i= aij . (1.16b)

Note that these arepartial derivatives. By use of Eqs. (1.16a) and (1.16b), Eq. (1.15)becomes

V ′i =N∑

j=1

∂x′i∂xj

Vj =N∑

j=1

∂xj

∂x′iVj . (1.17)

The direction cosinesaij satisfy anorthogonality condition∑

i

aijaik = δjk (1.18)

or, equivalently,∑

i

ajiaki = δjk. (1.19)

Here, the symbolδjk is the Kronecker delta, defined by

δjk = 1 for j = k,δjk = 0 for j = k. (1.20)

It is easily verified that Eqs. (1.18) and (1.19) hold in the two-dimensional case bysubstituting in the specificaij from Eqs. (1.11). The result is the well-known identitysin2ϕ + cos2ϕ = 1 for the nonvanishing case. To verify Eq. (1.18) in general form, wemay use the partial derivative forms of Eqs. (1.16a) and (1.16b) to obtain

∑

i

∂xj

∂x′i

∂xk

∂x′i=∑

i

∂xj

∂x′i

∂x′i∂xk

= ∂xj∂xk

. (1.21)

6Differentiatex′i

with respect toxj . See discussion following Eq. (1.21).


The last step follows by the standard rules for partial differentiation, assuming thatxj isa function ofx′1, x

′2, x

′3, and so on. The final result,∂xj/∂xk , is equal toδjk , sincexj and

xk as coordinate lines (j = k) are assumed to be perpendicular (two or three dimensions)or orthogonal (for any number of dimensions). Equivalently, we may assume thatxj andxk (j = k) are totally independent variables. Ifj = k, the partial derivative is clearly equalto 1.

In redefining a vector in terms of how its components transform under a rotation of thecoordinate system, we should emphasize two points:

1. This definition is developed because it is useful and appropriate in describing ourphysical world. Our vector equations will be independent of any particular coordinatesystem. (The coordinate system need not even be Cartesian.) The vector equation canalways be expressed in some particular coordinate system, and, to obtain numericalresults, we must ultimately express the equation in some specific coordinate system.

2. This definition is subject to a generalization that will open up the branch of mathemat-ics known as tensor analysis (Chapter 2).

A qualification is in order. The behavior of the vector components under rotation of thecoordinates is used in Section 1.3 to prove that a scalar product is a scalar, in Section 1.4to prove that a vector product is a vector, and in Section 1.6 to show that the gradient of ascalarψ, ∇ψ , is a vector. The remainder of this chapter proceeds on the basis of the lessrestrictive definitions of the vector given in Section 1.1.

Summary: Vectors and Vector Space

It is customary in mathematics to label an ordered triple of real numbers (x1, x2, x3) avector x. The numberxn is called thenth component of vectorx. The collection of allsuch vectors (obeying the properties that follow) form a three-dimensional realvectorspace. We ascribe five properties to our vectors: Ifx= (x1, x2, x3) andy= (y1, y2, y3),

1. Vector equality:x= y meansxi = yi , i = 1,2,3.2. Vector addition:x+ y= z meansxi + yi = zi, i = 1,2,3.3. Scalar multiplication:ax↔ (ax1, ax2, ax3) (with a real).4. Negative of a vector:−x= (−1)x↔ (−x1,−x2,−x3).5. Null vector: There exists a null vector0↔ (0,0,0).

Since our vector components are real (or complex) numbers, the following propertiesalso hold:

1. Addition of vectors is commutative:x+ y= y+ x.2. Addition of vectors is associative:(x+ y)+ z= x+ (y+ z).3. Scalar multiplication is distributive:

a(x+ y)= ax+ ay, also (a + b)x= ax+ bx.4. Scalar multiplication is associative:(ab)x= a(bx).


Further, the null vector0 is unique, as is the negative of a given vectorx.So far as the vectors themselves are concerned this approach merely formalizes the com-

ponent discussion of Section 1.1. The importance lies in the extensions, which will be con-sidered in later chapters. In Chapter 4, we show that vectors form both an Abelian groupunder addition and a linear space with the transformations in the linear space described bymatrices. Finally, and perhaps most important, for advanced physics the concept of vectorspresented here may be generalized to (1) complex quantities,7 (2) functions, and (3) an infi-nite number of components. This leads to infinite-dimensional function spaces, the Hilbertspaces, which are important in modern quantum theory. A brief introduction to functionexpansions and Hilbert space appears in Section 10.4.

Exercises

1.2.1 (a) Show that the magnitude of a vectorA, A= (A2x +A2y)1/2, is independent of theorientation of the rotated coordinate system,

(A2x +A2y

)1/2=(A′2x +A′2y

)1/2,

that is, independent of the rotation angleϕ.This independence of angle is expressed by saying thatA is invariant under

rotations.(b) At a given point(x, y), A defines an angleα relative to the positivex-axis and

α′ relative to the positivex′-axis. The angle fromx to x′ is ϕ. Show thatA = A′defines thesamedirection in space when expressed in terms of its primed compo-nents as in terms of its unprimed components; that is,

α′ = α − ϕ.1.2.2 Prove the orthogonality condition

∑i ajiaki = δjk . As a special case of this, the direc-

tion cosines of Section 1.1 satisfy the relation

cos2α + cos2β + cos2γ = 1,a result that also follows from Eq. (1.6).

1.3 SCALAR OR DOT PRODUCT

Having defined vectors, we now proceed to combine them. The laws for combining vectorsmust be mathematically consistent. From the possibilities that are consistent we select twothat are both mathematically and physically interesting. A third possibility is introduced inChapter 2, in which we form tensors.

The projection of a vectorA onto a coordinate axis, which gives its Cartesian compo-nents in Eq. (1.4), defines a special geometrical case of the scalar product ofA and thecoordinate unit vectors:

Ax =Acosα ≡ A · x̂, Ay =Acosβ ≡ A · ŷ, Az =Acosγ ≡ A · ẑ. (1.22)

7Then-dimensional vector space of realn-tuples is often labeledRn and then-dimensional vector space of complexn-tuples islabeledCn.

1.3 Scalar or Dot Product 13

This special case of a scalar product in conjunction with general properties the scalar prod-uct is sufficient to derive the general case of the scalar product.

Just as the projection is linear inA, we want the scalar product of two vectors to belinear inA andB, that is, obey the distributive and associative laws

A · (B+C) = A ·B+A ·C (1.23a)A · (yB) = (yA) ·B= yA ·B, (1.23b)

wherey is a number. Now we can use the decomposition ofB into its Cartesian componentsaccording to Eq. (1.5),B= Bx x̂+By ŷ+Bzẑ, to construct the general scalar or dot productof the vectorsA andB as

A ·B = A · (Bx x̂+By ŷ+Bzẑ)= BxA · x̂+ByA · ŷ+BzA · ẑ upon applying Eqs. (1.23a) and (1.23b)= BxAx +ByAy +BzAz upon substituting Eq. (1.22).

Hence

A ·B≡∑

i

BiAi =∑

i

AiBi = B ·A. (1.24)

If A = B in Eq. (1.24), we recover the magnitudeA = (∑A2i )1/2 of A in Eq. (1.6) fromEq. (1.24).

It is obvious from Eq. (1.24) that the scalar product treatsA and B alike, or is sym-metric in A andB, and is commutative. Thus, alternatively and equivalently, we can firstgeneralize Eqs. (1.22) to the projectionAB of A onto the direction of a vectorB = 0asAB = Acosθ ≡ A · B̂, whereB̂ = B/B is the unit vector in the direction ofB andθis the angle betweenA andB, as shown in Fig. 1.7. Similarly, we projectB onto A asBA = B cosθ ≡ B · Â. Second, we make these projections symmetric inA andB, whichleads to the definition

A ·B≡ABB =ABA =AB cosθ. (1.25)

FIGURE 1.7 Scalar productA ·B=AB cosθ .


FIGURE 1.8 The distributive lawA · (B+C)=ABA +ACA =A(B+C)A, Eq. (1.23a).

The distributive law in Eq. (1.23a) is illustrated in Fig. 1.8, which shows that the sum ofthe projections ofB andC ontoA, BA + CA is equal to the projection ofB+ C ontoA,(B+C)A.

It follows from Eqs. (1.22), (1.24), and (1.25) that the coordinate unit vectors satisfy therelations

x̂ · x̂= ŷ · ŷ= ẑ · ẑ= 1, (1.26a)whereas

x̂ · ŷ= x̂ · ẑ= ŷ · ẑ= 0. (1.26b)If the component definition, Eq. (1.24), is labeled an algebraic definition, then Eq. (1.25)

is a geometric definition. One of the most common applications of the scalar product inphysics is in the calculation ofwork = force·displacement·cosθ , which is interpreted asdisplacement times the projection of the force along the displacement direction, i.e., thescalar product of force and displacement,W = F ·S.

If A · B = 0 and we know thatA = 0 andB = 0, then, from Eq. (1.25), cosθ = 0, orθ = 90◦,270◦, and so on. The vectorsA and B must be perpendicular. Alternately, wemay sayA andB are orthogonal. The unit vectorsx̂, ŷ, andẑ are mutually orthogonal. Todevelop this notion of orthogonality one more step, suppose thatn is a unit vector andr isa nonzero vector in thexy-plane; that is,r = x̂x + ŷy (Fig. 1.9). If

n · r = 0for all choices ofr , thenn must be perpendicular (orthogonal) to thexy-plane.

Often it is convenient to replacêx, ŷ, andẑ by subscripted unit vectorsem,m= 1,2,3,with x̂= e1, and so on. Then Eqs. (1.26a) and (1.26b) become

em · en = δmn. (1.26c)For m = n the unit vectorsem anden are orthogonal. Form = n each vector is normal-ized to unity, that is, has unit magnitude. The setem is said to beorthonormal . A majoradvantage of Eq. (1.26c) over Eqs. (1.26a) and (1.26b) is that Eq. (1.26c) may readily begeneralized toN -dimensional space:m,n = 1,2, . . . ,N . Finally, we are picking sets ofunit vectorsem that are orthonormal for convenience – a very great convenience.


FIGURE 1.9 A normal vector.

Invariance of the Scalar Product Under Rotations

We have not yet shown that the wordscalar is justified or that the scalar product is indeeda scalar quantity. To do this, we investigate the behavior ofA · B under a rotation of thecoordinate system. By use of Eq. (1.15),

A′xB′x +A′yB ′y +A′zB ′z =

∑

i

axiAi∑

j

axjBj +∑

i

ayiAi∑

j

ayjBj

+∑

i

aziAi∑

j

azjBj . (1.27)

Using the indicesk andl to sum overx, y, andz, we obtain∑

k

A′kB′k =

∑

l

∑

i

∑

j

aliAialjBj , (1.28)

and, by rearranging the terms on the right-hand side, we have∑

k

A′kB′k =

∑

l

∑

i

∑

j

(alialj )AiBj =∑

i

∑

j

δijAiBj =∑

i

AiBi . (1.29)

The last two steps follow by using Eq. (1.18), the orthogonality condition of the directioncosines, and Eqs. (1.20), which define the Kronecker delta. The effect of the Kroneckerdelta is to cancel all terms in a summation over either index except the term for which theindices are equal. In Eq. (1.29) its effect is to setj = i and to eliminate the summationover j . Of course, we could equally well seti = j and eliminate the summation overi.


Equation (1.29) gives us

∑

k

A′kB′k =

∑

i

AiBi, (1.30)

which is just our definition of a scalar quantity, one that remains invariant under the rotationof the coordinate system.

In a similar approach that exploits this concept of invariance, we takeC = A + B anddot it into itself:

C ·C = (A +B) · (A +B)= A ·A +B ·B+ 2A ·B. (1.31)

Since

C ·C= C2, (1.32)

the square of the magnitude of vectorC and thus an invariant quantity, we see that

A ·B= 12

(C2−A2−B2

), invariant. (1.33)

Since the right-hand side of Eq. (1.33) is invariant — that is, a scalar quantity — the left-hand side,A · B, must also be invariant under rotation of the coordinate system. HenceA ·B is a scalar.

Equation (1.31) is really another form of the law of cosines, which is

C2=A2+B2+ 2AB cosθ. (1.34)

Comparing Eqs. (1.31) and (1.34), we have another verification of Eq. (1.25), or, if pre-ferred, a vector derivation of the law of cosines (Fig. 1.10).

The dot product, given by Eq. (1.24), may be generalized in two ways. The space neednot be restricted to three dimensions. Inn-dimensional space, Eq. (1.24) applies with thesum running from 1 ton. Moreover,n may be infinity, with the sum then a convergent infi-nite series (Section 5.2). The other generalization extends the concept of vector to embracefunctions. The function analog of a dot, or inner, product appears in Section 10.4.

FIGURE 1.10 The law of cosines.


Exercises

1.3.1 Two unit magnitude vectorsei andej are required to be either parallel or perpendicularto each other. Show thatei · ej provides an interpretation of Eq. (1.18), the directioncosine orthogonality relation.

1.3.2 Given that (1) the dot product of a unit vector with itself is unity and (2) this relation isvalid in all (rotated) coordinate systems, show thatx̂′ · x̂′ = 1 (with the primed systemrotated 45◦ about thez-axis relative to the unprimed) implies thatx̂ · ŷ= 0.

1.3.3 The vectorr , starting at the origin, terminates at and specifies the point in space(x, y, z).Find the surface swept out by the tip ofr if

(a) (r − a) · a= 0. Characterizea geometrically.(b) (r − a) · r = 0. Describe the geometric role ofa.

The vectora is constant (in magnitude and direction).

1.3.4 The interaction energy between two dipoles of momentsµ1 andµ2 may be written inthe vector form

V =−µ1 ·µ2r3

+ 3(µ1 · r)(µ2 · r)r5

and in the scalar form

V = µ1µ2r3

(2 cosθ1 cosθ2− sinθ1 sinθ2 cosϕ).

Hereθ1 andθ2 are the angles ofµ1 andµ2 relative tor , while ϕ is the azimuth ofµ2relative to theµ1–r plane (Fig. 1.11). Show that these two forms are equivalent.Hint: Equation (12.178) will be helpful.

1.3.5 A pipe comes diagonally down the south wall of a building, making an angle of 45◦

with the horizontal. Coming into a corner, the pipe turns and continues diagonally downa west-facing wall, still making an angle of 45◦ with the horizontal. What is the anglebetween the south-wall and west-wall sections of the pipe?

ANS. 120◦.

1.3.6 Find the shortest distance of an observer at the point(2,1,3) from a rocket in freeflight with velocity (1,2,3) m/s. The rocket was launched at timet = 0 from (1,1,1).Lengths are in kilometers.

1.3.7 Prove the law of cosines from the triangle with corners at the point ofC and A inFig. 1.10 and the projection of vectorB onto vectorA.

FIGURE 1.11 Two dipole moments.


1.4 VECTOR OR CROSS PRODUCT

A second form of vector multiplication employs the sine of the included angle insteadof the cosine. For instance, the angular momentum of a body shown at the point of thedistance vector in Fig. 1.12 is defined as

angular momentum= radius arm× linear momentum= distance× linear momentum× sinθ.

For convenience in treating problems relating to quantities such as angular momentum,torque, and angular velocity, we define the vector product, or cross product, as

C= A ×B, with C =AB sinθ. (1.35)

Unlike the preceding case of the scalar product,C is now a vector, and we assign it adirection perpendicular to the plane ofA andB such thatA,B, andC form a right-handedsystem. With this choice of direction we have

A ×B=−B×A, anticommutation. (1.36a)From this definition of cross product we have

x̂× x̂= ŷ× ŷ= ẑ× ẑ= 0, (1.36b)whereas

x̂× ŷ= ẑ, ŷ× ẑ= x̂, ẑ× x̂= ŷ,ŷ× x̂=−ẑ, ẑ× ŷ=−x̂, x̂× ẑ=−ŷ. (1.36c)

Among the examples of the cross product in mathematical physics are the relation betweenlinear momentump and angular momentumL , with L defined as

L = r × p,

FIGURE 1.12 Angular momentum.

1.4 Vector or Cross Product 19

FIGURE 1.13 Parallelogram representation of the vector product.

and the relation between linear velocityv and angular velocityω,

v= ω× r .Vectorsv andp describe properties of the particle or physical system. However, the posi-tion vectorr is determined by the choice of the origin of the coordinates. This means thatω andL depend on the choice of the origin.

The familiar magnetic inductionB is usually defined by the vector product force equa-tion8

FM = qv×B (mks units).Herev is the velocity of the electric chargeq andFM is the resulting force on the movingcharge.

The cross product has an important geometrical interpretation, which we shall use insubsequent sections. In the parallelogram defined byA andB (Fig. 1.13),B sinθ is theheight ifA is taken as the length of the base. Then|A × B| = AB sinθ is thearea of theparallelogram. As a vector,A×B is the area of the parallelogram defined byA andB, withthe area vector normal to the plane of the parallelogram. This suggests that area (with itsorientation in space) may be treated as a vector quantity.

An alternate definition of the vector product can be derived from the special case of thecoordinate unit vectors in Eqs. (1.36c) in conjunction with the linearity of the cross productin both vector arguments, in analogy with Eqs. (1.23) for the dot product,

A × (B+C)= A ×B+A ×C, (1.37a)(A +B)×C= A ×C+B×C, (1.37b)A × (yB)= yA ×B= (yA)×B, (1.37c)

8The electric fieldE is assumed here to be zero.


wherey is a number again. Using the decomposition ofA andB into their Cartesian com-ponents according to Eq. (1.5), we find

A ×B ≡ C= (Cx,Cy,Cz)= (Ax x̂+Ay ŷ+Azẑ)× (Bx x̂+By ŷ+Bzẑ)= (AxBy −AyBx)x̂× ŷ+ (AxBz −AzBx)x̂× ẑ+ (AyBz −AzBy)ŷ× ẑ

upon applying Eqs. (1.37a) and (1.37b) and substituting Eqs. (1.36a), (1.36b), and (1.36c)so that the Cartesian components ofA ×B become

Cx =AyBz −AzBy, Cy =AzBx −AxBz, Cz =AxBy −AyBx, (1.38)or

Ci =AjBk −AkBj , i, j, k all different, (1.39)and with cyclic permutation of the indicesi, j , andk corresponding tox, y, andz, respec-tively. The vector productC may be mnemonically represented by a determinant,9

C=

∣∣∣∣∣∣

x̂ ŷ ẑAx Ay AzBx By Bz

∣∣∣∣∣∣≡ x̂

∣∣∣∣Ay AzBy Bz

∣∣∣∣− ŷ∣∣∣∣Ax AzBx Bz

∣∣∣∣+ ẑ∣∣∣∣Ax AyBx By

∣∣∣∣ , (1.40)

which is meant to be expanded across the top row to reproduce the three components ofClisted in Eqs. (1.38).

Equation (1.35) might be called a geometric definition of the vector product. ThenEqs. (1.38) would be an algebraic definition.

To show the equivalence of Eq. (1.35) and the component definition, Eqs. (1.38), let usform A ·C andB ·C, using Eqs. (1.38). We have

A ·C = A · (A ×B)= Ax(AyBz −AzBy)+Ay(AzBx −AxBz)+Az(AxBy −AyBx)= 0. (1.41)

Similarly,

B ·C= B · (A ×B)= 0. (1.42)Equations (1.41) and (1.42) show thatC is perpendicular to bothA andB (cosθ = 0, θ =±90◦) and therefore perpendicular to the plane they determine. The positive direction isdetermined by considering special cases, such as the unit vectorsx̂× ŷ= ẑ (Cz =+AxBy).

The magnitude is obtained from

(A ×B) · (A ×B) = A2B2− (A ·B)2

= A2B2−A2B2 cos2 θ= A2B2 sin2 θ. (1.43)

9See Section 3.1 for a brief summary of determinants.


Hence

C =AB sinθ. (1.44)

The first step in Eq. (1.43) may be verified by expanding out in component form, usingEqs. (1.38) forA × B and Eq. (1.24) for the dot product. From Eqs. (1.41), (1.42), and(1.44) we see the equivalence of Eqs. (1.35) and (1.38), the two definitions of vector prod-uct.

There still remains the problem of verifying thatC = A × B is indeed a vector, thatis, that it obeys Eq. (1.15), the vector transformation law. Starting in a rotated (primedsystem),

C′i = A′jB ′k −A′kB ′j , i, j, andk in cyclic order,

=∑

l

aj lAl∑

m

akmBm −∑

l

aklAl∑

m

ajmBm

=∑

l,m

(aj lakm − aklajm)AlBm. (1.45)

The combination of direction cosines in parentheses vanishes form= l. We therefore havej and k taking on fixed values, dependent on the choice ofi, and six combinations ofl andm. If i = 3, thenj = 1, k = 2 (cyclic order), and we have the following directioncosine combinations:10

a11a22− a21a12= a33,a13a21− a23a11= a32,a12a23− a22a13= a31

(1.46)

and their negatives. Equations (1.46) are identities satisfied by the direction cosines. Theymay be verified with the use of determinants and matrices (see Exercise 3.3.3). Substitutingback into Eq. (1.45),

C′3 = a33A1B2+ a32A3B1+ a31A2B3− a33A2B1− a32A1B3− a31A3B2= a31C1+ a32C2+ a33C3=∑

n

a3nCn. (1.47)

By permuting indices to pick upC′1 andC′2, we see that Eq. (1.15) is satisfied andC is

indeed a vector. It should be mentioned here that thisvector nature of thecross productis an accident associated with thethree-dimensionalnature of ordinary space.11 It will beseen in Chapter 2 that the cross product may also be treated as a second-rank antisymmetrictensor.

10Equations (1.46) hold for rotations because they preserve volumes. For a more general orthogonal transformation, the r.h.s. ofEqs. (1.46) is multiplied by the determinant of the transformation matrix (see Chapter 3 for matrices and determinants).11Specifically Eqs. (1.46) hold only for three-dimensional space. See D. Hestenes and G. Sobczyk,Clifford Algebra to GeometricCalculus(Dordrecht: Reidel, 1984) for a far-reaching generalization of the cross product.


If we define a vector as an ordered triplet of numbers (or functions), as in the latter partof Section 1.2, then there is no problem identifying the cross product as a vector. The cross-product operation maps the two triplesA andB into a third triple,C, which by definitionis a vector.

We now have two ways of multiplying vectors; a third form appears in Chapter 2. Butwhat about division by a vector? It turns out that the ratioB/A is not uniquely specified(Exercise 3.2.21) unlessA andB are also required to be parallel. Hence division of onevector by another is not defined.

Exercises

1.4.1 Show that the medians of a triangle intersect in the center, which is 2/3 of the median’slength from each corner. Construct a numerical example and plot it.

1.4.2 Prove the law of cosines starting fromA2= (B−C)2.1.4.3 Starting withC= A +B, show thatC×C= 0 leads to

A ×B=−B×A.

1.4.4 Show that

(a) (A −B) · (A +B)=A2−B2,(b) (A −B)× (A +B)= 2A ×B.

The distributive laws needed here,

A · (B+C)= A ·B+A ·C,

and

A × (B+C)= A ×B+A ×C,

may easily be verified (if desired) by expansion in Cartesian components.

1.4.5 Given the three vectors,

P= 3x̂+ 2ŷ− ẑ,Q = −6x̂− 4ŷ+ 2ẑ,R = x̂− 2ŷ− ẑ,

find two that are perpendicular and two that are parallel or antiparallel.

1.4.6 If P= x̂Px + ŷPy andQ= x̂Qx + ŷQy are any two nonparallel (also nonantiparallel)vectors in thexy-plane, show thatP×Q is in thez-direction.

1.4.7 Prove that(A ×B) · (A ×B)= (AB)2− (A ·B)2.


1.4.8 Using the vectors

P= x̂cosθ + ŷ sinθ,Q = x̂cosϕ − ŷsinϕ,R = x̂cosϕ + ŷsinϕ,

prove the familiar trigonometric identities

sin(θ + ϕ) = sinθ cosϕ + cosθ sinϕ,cos(θ + ϕ) = cosθ cosϕ − sinθ sinϕ.

1.4.9 (a) Find a vectorA that is perpendicular to

U = 2x̂+ ŷ− ẑ,V = x̂− ŷ+ ẑ.

(b) What isA if, in addition to this requirement, we demand that it have unit magni-tude?

1.4.10 If four vectorsa,b,c, andd all lie in the same plane, show that

(a× b)× (c× d)= 0.

Hint. Consider the directions of the cross-product vectors.

1.4.11 The coordinates of the three vertices of a triangle are(2,1,5), (5,2,8), and(4,8,2).Compute its area by vector methods, its center and medians. Lengths are in centimeters.Hint. See Exercise 1.4.1.

1.4.12 The vertices of parallelogramABCDare(1,0,0), (2,−1,0), (0,−1,1), and(−1,0,1)in order. Calculate the vector areas of triangleABD and of triangleBCD. Are the twovector areas equal?

ANS. AreaABD=−12(x̂+ ŷ+ 2ẑ).1.4.13 The origin and the three vectorsA, B, andC (all of which start at the origin) define a

tetrahedron. Taking the outward direction as positive, calculate the total vector area ofthe four tetrahedral surfaces.Note. In Section 1.11 this result is generalized to any closed surface.

1.4.14 Find the sides and angles of the spherical triangleABCdefined by the three vectors

A = (1,0,0),

B =(

1√2,0,

1√2

),

C =(

0,1√2,

1√2

).

Each vector starts from the origin (Fig. 1.14).


FIGURE 1.14 Spherical triangle.

1.4.15 Derive the law of sines (Fig. 1.15):

sinα

|A| =sinβ

|B| =sinγ

|C| .

1.4.16 The magnetic inductionB is definedby the Lorentz force equation,

F= q(v×B).Carrying out three experiments, we find that if

v = x̂, Fq= 2ẑ− 4ŷ,

v = ŷ, Fq= 4x̂− ẑ,

v = ẑ, Fq= ŷ− 2x̂.

From the results of these three separate experiments calculate the magnetic inductionB.

1.4.17 Define a cross product of two vectors in two-dimensional space and give a geometricalinterpretation of your construction.

1.4.18 Find the shortest distance between the paths of two rockets in free flight. Take the firstrocket path to ber = r1+ t1v1 with launch atr1= (1,1,1) and velocityv1= (1,2,3)

1.5 Triple Scalar Product, Triple Vector Product 25

FIGURE 1.15 Law of sines.

and the second rocket path asr = r2+ t2v2 with r2 = (5,2,1) andv2 = (−1,−1,1).Lengths are in kilometers, velocities in kilometers per hour.

1.5 TRIPLE SCALAR PRODUCT, TRIPLE VECTOR PRODUCT

Triple Scalar Product

Sections 1.3 and 1.4 cover the two types of multiplication of interest here. However, thereare combinations of three vectors,A · (B×C) andA × (B×C), that occur with sufficientfrequency to deserve further attention. The combination

A · (B×C)

is known as thetriple scalar product . B× C yields a vector that, dotted intoA, gives ascalar. We note that(A ·B)×C represents a scalar crossed into a vector, an operation thatis not defined. Hence, if we agree to exclude this undefined interpretation, the parenthesesmay be omitted and the triple scalar product writtenA ·B×C.

Using Eqs. (1.38) for the cross product and Eq. (1.24) for the dot product, we obtain

A ·B×C = Ax(ByCz −BzCy)+Ay(BzCx −BxCz)+Az(BxCy −ByCx)= B ·C×A =C ·A ×B= −A ·C×B=−C ·B×A =−B ·A ×C, and so on. (1.48)

There is a high degree of symmetry in the component expansion. Every term contains thefactorsAi , Bj , andCk . If i, j , andk are in cyclic order(x, y, z), the sign is positive. If theorder is anticyclic, the sign is negative. Further, the dot and the cross may be interchanged,

A ·B×C= A ×B ·C. (1.49)


FIGURE 1.16 Parallelepiped representation of triple scalar product.

A convenient representation of the component expansion of Eq. (1.48) is provided by thedeterminant

A ·B×C=

∣∣∣∣∣∣

Ax Ay AzBx By BzCx Cy Cz

∣∣∣∣∣∣. (1.50)

The rules for interchanging rows and columns of a determinant12 provide an immediateverification of the permutations listed in Eq. (1.48), whereas the symmetry ofA, B, andC in the determinant form suggests the relation given in Eq. (1.49). The triple productsencountered in Section 1.4, which showed thatA ×B was perpendicular to bothA andB,were special cases of the general result (Eq. (1.48)).

The triple scalar product has a direct geometrical interpretation. The three vectorsA, B,andC may be interpreted as defining a parallelepiped (Fig. 1.16):

|B×C| = BC sinθ= area of parallelogram base. (1.51)

The direction, of course, is normal to the base. DottingA into this means multiplying thebase area by the projection ofA onto the normal, or base times height. Therefore

A ·B×C= volume of parallelepiped defined byA,B, andC.The triple scalar product finds an interesting and important application in the construc-

tion of a reciprocal crystal lattice. Leta, b, andc (not necessarily mutually perpendicular)

12See Section 3.1 for a summary of the properties of determinants.


represent the vectors that define a crystal lattice. The displacement from one lattice pointto another may then be written

r = naa+ nbb+ ncc, (1.52)with na, nb, andnc taking on integral values. With these vectors we may form

a′ = b× ca · b× c, b

′ = c× aa · b× c, c

′ = a× ba · b× c. (1.53a)

We see thata′ is perpendicular to the plane containingb andc, and we can readily showthat

a′ · a= b′ · b= c′ · c= 1, (1.53b)whereas

a′ · b= a′ · c= b′ · a= b′ · c= c′ · a= c′ · b= 0. (1.53c)It is from Eqs. (1.53b) and (1.53c) that the namereciprocal lattice is associated with thepointsr ′ = n′aa′+ n′bb′+ n′cc′. The mathematical space in which this reciprocal lattice ex-ists is sometimes called aFourier space, on the basis of relations to the Fourier analysis ofChapters 14 and 15. This reciprocal lattice is useful in problems involving the scattering ofwaves from the various planes in a crystal. Further details may be found in R. B. Leighton’sPrinciples of Modern Physics, pp. 440–448 [New York: McGraw-Hill (1959)].

Triple Vector Product

The second triple product of interest isA×(B×C), which is a vector. Here the parenthesesmust be retained, as may be seen from a special case(x̂× x̂)× ŷ= 0, while x̂× (x̂× ŷ)=x̂× ẑ=−ŷ.

Example 1.5.1 A TRIPLE VECTOR PRODUCT

For the vectors

A = x̂+ 2ŷ− ẑ= (1,2,−1), B= ŷ+ ẑ= (0,1,1), C= x̂− ŷ= (0,1,1),

B×C=

∣∣∣∣∣∣

x̂ ŷ ẑ0 1 11 −1 0

∣∣∣∣∣∣= x̂+ ŷ− ẑ,

and

A × (B×C)=

∣∣∣∣∣∣

x̂ ŷ ẑ1 2 −11 1 −1

∣∣∣∣∣∣= −x̂− ẑ=−(ŷ+ ẑ)− (x̂− ŷ)

= −B−C. �By rewriting the result in the last line of Example 1.5.1 as a linear combination ofB and

C, we notice that, taking a geometric approach, the triple vector product is perpendicular


FIGURE 1.17 B andC are in thexy-plane.B×C is perpendicular to thexy-plane and

is shown here along thez-axis. ThenA × (B×C) is perpendicular to thez-axis

and therefore is back in thexy-plane.

to A and toB× C. The plane defined byB andC is perpendicular toB× C, and so thetriple product lies in this plane (see Fig. 1.17):

A × (B×C)= uB+ vC. (1.54)Taking the scalar product of Eq. (1.54) withA gives zero for the left-hand side, souA · B + vA · C = 0. Henceu = wA · C andv = −wA · B for a suitablew. Substitut-ing these values into Eq. (1.54) gives

A × (B×C)=w[B(A ·C)−C(A ·B)

]; (1.55)

we want to show that

w = 1in Eq. (1.55), an important relation sometimes known as theBAC–CAB rule. SinceEq. (1.55) is linear inA, B, andC, w is independent of these magnitudes. That is, weonly need to show thatw = 1 for unit vectorsÂ, B̂, Ĉ. Let us denoteB̂ · Ĉ = cosα,Ĉ · Â = cosβ, Â · B̂= cosγ , and square Eq. (1.55) to obtain

[Â × (B̂× Ĉ)

]2 = Â2(B̂× Ĉ)2−[Â · (B̂× Ĉ)

]2

= 1− cos2α −[Â · (B̂× Ĉ)

]2

= w2[(Â · Ĉ)2+ (Â · B̂)2− 2(Â · B̂)(Â · Ĉ)(B̂ · Ĉ)

]

= w2(cos2β + cos2γ − 2 cosα cosβ cosγ

), (1.56)


using(Â × B̂)2= Â2B̂2− (Â · B̂)2 repeatedly (see Eq. (1.43) for a proof). Consequently,the (squared) volume spanned byÂ, B̂, Ĉ that occurs in Eq. (1.56) can be written as

[Â · (B̂× Ĉ)

]2= 1− cos2α −w2(cos2β + cos2γ − 2 cosα cosβ cosγ

).

Herew2 = 1, since this volume is symmetric inα,β, γ . That is,w = ±1 and is inde-pendent ofÂ, B̂, Ĉ. Using again the special casex̂× (x̂× ŷ) = −ŷ in Eq. (1.55) finallygivesw = 1. (An alternate derivation using the Levi-Civita symbolεijk of Chapter 2 is thetopic of Exercise 2.9.8.)

It might be noted here that just as vectors are independent of the coordinates, so a vectorequation is independent of the particular coordinate system. The coordinate system onlydetermines the components. If the vector equation can be established in Cartesian coor-dinates, it is established and valid in any of the coordinate systems to be introduced inChapter 2. Thus, Eq. (1.55) may be verified by a direct though not very elegant method ofexpanding into Cartesian components (see Exercise 1.5.2).

Exercises

1.5.1 One vertex of a glass parallelepiped is at the origin (Fig. 1.18). The three adjacentvertices are at(3,0,0), (0,0,2), and(0,3,1). All lengths are in centimeters. Calculatethe number of cubic centimeters of glass in the parallelepiped using the triple scalarproduct.

1.5.2 Verify the expansion of the triple vector product

A × (B×C)= B(A ·C)−C(A ·B)

FIGURE 1.18 Parallelepiped: triple scalar product.


by direct expansion in Cartesian coordinates.

1.5.3 Show that the first step in Eq. (1.43), which is

(A ×B) · (A ×B)=A2B2− (A ·B)2,is consistent with theBAC–CABrule for a triple vector product.

1.5.4 You are given the three vectorsA, B, andC,

A = x̂+ ŷ,B = ŷ+ ẑ,C = x̂− ẑ.

(a) Compute the triple scalar product,A · B×C. Noting thatA = B+C, give a geo-metric interpretation of your result for the triple scalar product.

(b) ComputeA × (B×C).

1.5.5 The orbital angular momentumL of a particle is given byL = r × p=mr × v, wherep is the linear momentum. With linear and angular velocity related byv= ω× r , showthat

L =mr2[ω− r̂(r̂ ·ω)

].

Herer̂ is a unit vector in ther -direction. Forr · ω= 0 this reduces toL = Iω, with themoment of inertiaI given bymr2. In Section 3.5 this result is generalized to form aninertia tensor.

1.5.6 The kinetic energy of a single particle is given byT = 12mv2. For rotational motion thisbecomes12m(ω× r)2. Show that

T = 12m[r2ω2− (r ·ω)2

].

For r ·ω= 0 this reduces toT = 12Iω2, with the moment of inertiaI given bymr2.1.5.7 Show that13

a× (b× c)+ b× (c× a)+ c× (a× b)= 0.1.5.8 A vector A is decomposed into a radial vectorAr and a tangential vectorAt . If r̂ is a

unit vector in the radial direction, show that

(a) Ar = r̂(A · r̂) and(b) At =−r̂ × (r̂ ×A).

1.5.9 Prove that a necessary and sufficient condition for the three (nonvanishing) vectorsA,B, andC to be coplanar is the vanishing of the triple scalar product

A ·B×C= 0.13This is Jacobi’s identity for vector products; for commutators it is important in the context of Lie algebras (see Eq. (4.16) inSection 4.2).


1.5.10 Three vectorsA, B, andC are given by

A = 3x̂− 2ŷ+ 2ẑ,B = 6x̂+ 4ŷ− 2ẑ,C = −3x̂− 2ŷ− 4ẑ.

Compute the values ofA ·B×C andA × (B×C),C× (A ×B) andB× (C×A).1.5.11 VectorD is a linear combination of three noncoplanar (and nonorthogonal) vectors:

D= aA + bB+ cC.Show that the coefficients are given by a ratio of triple scalar products,

a = D ·B×CA ·B×C , and so on.

1.5.12 Show that

(A ×B) · (C×D)= (A ·C)(B ·D)− (A ·D)(B ·C).1.5.13 Show that

(A ×B)× (C×D)= (A ·B×D)C− (A ·B×C)D.1.5.14 For aspherical triangle such as pictured in Fig. 1.14 show that

sinA

sinBC= sinB

sinCA= sinC

sinAB.

Here sinA is the sine of the included angle atA, while BC is the side opposite (inradians).

1.5.15 Given

a′ = b× ca · b× c, b

′ = c× aa · b× c, c

′ = a× ba · b× c,

anda · b× c = 0, show that

(a) x · y′ = δxy, (x,y= a,b,c),(b) a′ · b′ × c′ = (a · b× c)−1,(c) a= b

′ × c′a′ · b′ × c′ .

1.5.16 If x · y′ = δxy, (x,y= a,b,c), prove that

a′ = b× ca · b× c.

(This is the converse of Problem 1.5.15.)

1.5.17 Show that any vectorV may be expressed in terms of the reciprocal vectorsa′, b′, c′ (ofProblem 1.5.15) by

V = (V · a)a′ + (V · b)b′ + (V · c)c′.


1.5.18 An electric chargeq1 moving with velocityv1 produces a magnetic inductionB givenby

B= µ04π

q1v1× r̂r2

(mks units),

wherer̂ points fromq1 to the point at whichB is measured (Biot and Savart law).

(a) Show that the magnetic force on a second chargeq2, velocity v2, is given by thetriple vector product

F2=µ0

4π

q1q2

r2v2× (v1× r̂).

(b) Write out the corresponding magnetic forceF1 thatq2 exerts onq1. Define yourunit radial vector. How doF1 andF2 compare?

(c) CalculateF1 andF2 for the case ofq1 andq2 moving along parallel trajectoriesside by side.

ANS.

(b) F1=−µ0

4π

q1q2

r2v1× (v2× r̂).

In general, there is no simple relation betweenF1 andF2. Specifically, Newton’s third law,F1=−F2,does not hold.

(c) F1=µ0

4π

q1q2

r2v2r̂ =−F2.

Mutual attraction.

1.6 GRADIENT, ∇

To provide a motivation for the vector nature of partial derivatives, we now introduce thetotal variation of a function F(x, y),

dF = ∂F∂x

dx + ∂F∂y

dy.

It consists of independent variations in thex- andy-directions. We writedF as a sum oftwo increments, one purely in thex- and the other in they-direction,

dF(x, y) ≡ F(x + dx, y + dy)− F(x, y)=[F(x + dx, y + dy)− F(x, y + dy)

]+[F(x, y + dy)− F(x, y)

]

= ∂F∂x

dx + ∂F∂y

dy,

by adding and subtractingF(x, y+ dy). The mean value theorem (that is, continuity ofF )tells us that here∂F/∂x, ∂F/∂y are evaluated at some pointξ, η betweenx andx + dx, y

1.6 Gradient, ∇ 33

andy + dy, respectively. Asdx→ 0 anddy→ 0, ξ → x andη→ y. This result general-izes to three and higher dimensions. For example, for a functionϕ of three variables,

dϕ(x, y, z) ≡[ϕ(x + dx, y + dy, z+ dz)− ϕ(x, y + dy, z+ dz)

]

+[ϕ(x, y + dy, z+ dz)− ϕ(x, y, z+ dz)

]

+[ϕ(x, y, z+ dz)− ϕ(x, y, z)

](1.57)

= ∂ϕ∂x

dx + ∂ϕ∂y

dy + ∂ϕ∂z

dz.

Algebraically,dϕ in the total variation is a scalar product of the change in positiondr andthe directional change ofϕ. And now we are ready to recognize the three-dimensionalpartial derivative as a vector, which leads us to the concept of gradient.

Suppose thatϕ(x, y, z) is a scalar point function, that is, a function whose value dependson the values of the coordinates(x, y, z). As a scalar, it must have the same value at a givenfixed point in space, independent of the rotation of our coordinate system, or

ϕ′(x′1, x′2, x

′3)= ϕ(x1, x2, x3). (1.58)

By differentiating with respect tox′i we obtain

∂ϕ′(x′1, x′2, x

′3)

∂x′i= ∂ϕ(x1, x2, x3)

∂x′i=∑

j

∂ϕ

∂xj

∂xj

∂x′i=∑

j

aij∂ϕ

∂xj(1.59)

by the rules of partial differentiation and Eqs. (1.16a) and (1.16b). But comparison withEq. (1.17), the vector transformation law, now shows that we haveconstructed a vectorwith components∂ϕ/∂xj . This vector we label the gradient ofϕ.

A convenient symbolism is

∇ϕ = x̂∂ϕ∂x+ ŷ∂ϕ

∂y+ ẑ∂ϕ

∂z(1.60)

or

∇ = x̂ ∂∂x+ ŷ ∂

∂y+ ẑ ∂

∂z. (1.61)

∇ϕ (or delϕ) is our gradient of the scalarϕ, whereas∇ (del) itself is a vector differentialoperator (available to operate on or to differentiate a scalarϕ). All the relationships for∇(del) can be derived from the hybrid nature of del in terms of both the partial derivativesand its vector nature.

The gradient of a scalar is extremely important in physics and engineering in expressingthe relation between a force field and a potential field,

forceF=−∇(potentialV ), (1.62)which holds for both gravitational and electrostatic fields, among others. Note that theminus sign in Eq. (1.62) results in water flowing downhill rather than uphill! If a force canbe described, as in Eq. (1.62), by a single functionV (r) everywhere, we call the scalarfunctionV its potential. Because the force is the directional derivative of the potential, wecan find the potential, if it exists, by integrating the force along a suitable path. Because the


total variationdV =∇V · dr =−F · dr is the work done against the force along the pathdr , we recognize the physical meaning of the potential (difference) as work and energy.Moreover, in a sum of path increments the intermediate points cancel,[V (r + dr1+ dr2)− V (r + dr1)

]+[V (r + dr1)− V (r)

]= V (r + dr2+ dr1)− V (r),

so the integrated work along some path from an initial pointr i to a final pointr is given bythe potential differenceV (r)− V (r i) at the endpoints of the path. Therefore, such forcesare especially simple and well behaved: They are calledconservative. When there is loss ofenergy due to friction along the path or some other dissipation, the work will depend on thepath, and such forces cannot be conservative: No potential exists. We discuss conservativeforces in more detail in Section 1.13.

Example 1.6.1 THE GRADIENT OF A POTENTIAL V (r)

Let us calculate the gradient ofV (r)= V (√x2+ y2+ z2 ), so

∇V (r)= x̂∂V (r)∂x

+ ŷ∂V (r)∂y

+ ẑ∂V (r)∂z

.

Now,V (r) depends onx through the dependence ofr onx. Therefore14

∂V (r)

∂x= dV (r)

dr· ∂r∂x

.

Fromr as a function ofx, y, z,

∂r

∂x= ∂(x

2+ y2+ z2)1/2∂x

= x(x2+ y2+ z2)1/2 =

x

r.

Therefore∂V (r)

∂x= dV (r)

dr· xr.

Permuting coordinates(x→ y, y→ z, z→ x) to obtain they andz derivatives, we get

∇V (r) = (x̂x + ŷy + ẑz)1r

dV

dr

= rr

dV

dr= r̂ dV

dr.

Here r̂ is a unit vector(r/r) in thepositive radial direction. The gradient of a function ofr is a vector in the (positive or negative) radial direction. In Section 2.5,r̂ is seen as oneof the three orthonormal unit vectors of spherical polar coordinates andr̂∂/∂r as the radialcomponent of∇. �

14This is a special case of thechain rule of partial differentiation:

∂V (r, θ,ϕ)

∂x= ∂V

∂r

∂r

∂x+ ∂V

∂θ

∂θ

∂x+ ∂V

∂ϕ

∂ϕ

∂x,

where∂V/∂θ = ∂V/∂ϕ = 0, ∂V/∂r→ dV/dr.


A Geometrical Interpretation

One immediate application of∇ϕ is to dot it into an increment of length

dr = x̂dx + ŷdy + ẑdz.Thus we obtain

∇ϕ · dr = ∂ϕ∂x

dx + ∂ϕ∂y

dy + ∂ϕ∂z

dz= dϕ,

the change in the scalar functionϕ corresponding to a change in positiondr . Now considerP andQ to be two points on a surfaceϕ(x, y, z)= C, a constant. These points are chosenso thatQ is a distancedr fromP . Then, moving fromP toQ, the change inϕ(x, y, z)= Cis given by

dϕ = (∇ϕ) · dr = 0 (1.63)since we stay on the surfaceϕ(x, y, z) = C. This shows that∇ϕ is perpendicular todr .Sincedr may have any direction fromP as long as it stays in the surfaceof constantϕ,pointQ being restricted to the surface but having arbitrary direction,∇ϕ is seen as normalto the surfaceϕ = constant (Fig. 1.19).

If we now permitdr to take us from one surfaceϕ = C1 to an adjacent surfaceϕ = C2(Fig. 1.20),

dϕ = C1−C2=�C = (∇ϕ) · dr . (1.64)For a givendϕ, |dr | is a minimum when it is chosen parallel to∇ϕ (cosθ = 1); or, fora given|dr |, the change in the scalar functionϕ is maximized by choosingdr parallel to

FIGURE 1.19 The length incrementdr has to stay on the surfaceϕ = C.


FIGURE 1.20 Gradient.

∇ϕ. This identifies ∇ϕ as a vector having the direction of the maximum space rateof change ofϕ, an identification that will be useful in Chapter 2 when we consider non-Cartesian coordinate systems. This identification of∇ϕ may also be developed by usingthe calculus of variations subject to a constraint, Exercise 17.6.9.

Example 1.6.2 FORCE AS GRADIENT OF A POTENTIAL

As a specific example of the foregoing, and as an extension of Example 1.6.1, we considerthe surfaces consisting of concentric spherical shells, Fig. 1.21. We have

ϕ(x, y, z)=(x2+ y2+ z2

)1/2= r = C,wherer is the radius, equal toC, our constant.�C =�ϕ =�r , the distance between twoshells. From Example 1.6.1

∇ϕ(r)= r̂ dϕ(r)dr

= r̂ .

The gradient is in the radial direction and is normal to the spherical surfaceϕ = C. �

Example 1.6.3 INTEGRATION BY PARTS OF GRADIENT

Let us prove the formula∫

A(r) ·∇f (r) d3r =−∫f (r)∇ ·A(r) d3r , whereA or f or both

vanish at infinity so that the integrated parts vanish. This condition is satisfied if, for exam-ple,A is the electromagnetic vector potential andf is a bound-state wave functionψ(r).


FIGURE 1.21 Gradient forϕ(x, y, z)= (x2+ y2+ z2)1/2, spherical

shells:(x22 + y22 + z22)1/2= r2= C2,(x21 + y21 + z21)1/2= r1= C1.

Writing the inner product in Cartesian coordinates, integrating each one-dimensionalintegral by parts, and dropping the integrated terms, we obtain

∫A(r) ·∇f (r) d3r =

∫∫ [Axf |∞x=−∞ −

∫f∂Ax

∂xdx

]dy dz+ · · ·

= −∫∫∫

f∂Ax

∂xdx dy dz−

∫∫∫f∂Ay

∂ydy dx dz−

∫∫∫f∂Az

∂zdz dx dy

=−∫

f (r)∇ ·A(r) d3r.

If A = eikzê describes an outgoing photon in the

MATHEMATICAL METHODS FORhep.fcfm.buap.mx/.../Metodos_Matematicos_Fisicos-Arfken.pdf · 2013. 7. 9. · George B. Arfken Miami University Oxford, OH Hans J. Weber University of Virginia

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