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MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES Third Edition MARY L BOAS DePaul University MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES Third Edition MARY L BOAS DePaul University PUBLISHER SENIOR ACQUISITIONS Editor PRODUCTION MANAGER PRODUCTION EDITOR MARKETING MANAGER SENIOR DESIGNER EDITORIAL ASSISTANT PRODUCTION MANAGER Kaye Pace Stuart Johnson Pam Kennedy Sarah Wolfman-Robichaud Amanda Wygal Dawn Stanley Krista Jarmas/Alyson Rentrop Jan Fisher/Publication Services This book was set in 10/12 Computer Modern by Publication Services and printed and bound by R.R Donnelley-Willard The cover was printed by Lehigh Press This book is printed on acid free paper Copyright  2006 John Wiley & Sons, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978)750-8400, fax (978)750-4470 or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748-6008, or online at http://www.wiley.com/go/permissions To order books or for customer service please, call 1-800-CALL WILEY (225-5945) ISBN 0-471-19826-9 ISBN-13 978-0-471-19826-0 ISBN-WIE 0-471-36580-7 ISBN-WIE-13 978-0-471-36580-8 Printed in the United States of America 10 To the memory of RPB PREFACE This book is particularly intended for the student with a year (or a year and a half) of calculus who wants to develop, in a short time, a basic competence in each of the many areas of mathematics needed in junior to senior-graduate courses in physics, chemistry, and engineering Thus it is intended to be accessible to sophomores (or freshmen with AP calculus from high school) It may also be used effectively by a more advanced student to review half-forgotten topics or learn new ones, either by independent study or in a class Although the book was written especially for students of the physical sciences, students in any field (say mathematics or mathematics for teaching) may find it useful to survey many topics or to obtain some knowledge of areas they not have time to study in depth Since theorems are stated carefully, such students should not need to unlearn anything in their later work The question of proper mathematical training for students in the physical sciences is of concern to both mathematicians and those who use mathematics in applications Some instructors may feel that if students are going to study mathematics at all, they should study it in careful and thorough detail For the undergraduate physics, chemistry, or engineering student, this means either (1) learning more mathematics than a mathematics major or (2) learning a few areas of mathematics thoroughly and the others only from snatches in science courses The second alternative is often advocated; let me say why I think it is unsatisfactory It is certainly true that motivation is increased by the immediate application of a mathematical technique, but there are a number of disadvantages: The discussion of the mathematics is apt to be sketchy since that is not the primary concern Students are faced simultaneously with learning a new mathematical method and applying it to an area of science that is also new to them Frequently the vii viii Preface difficulty in comprehending the new scientific area lies more in the distraction caused by poorly understood mathematics than it does in the new scientific ideas Students may meet what is actually the same mathematical principle in two different science courses without recognizing the connection, or even learn apparently contradictory theorems in the two courses! For example, in thermodynamics students learn that the integral of an exact differential around a closed 2π path is always zero In electricity or hydrodynamics, they run into dθ, which is certainly the integral of an exact differential around a closed path but is not equal to zero! Now it would be fine if every science student could take the separate mathematics courses in differential equations (ordinary and partial), advanced calculus, linear algebra, vector and tensor analysis, complex variables, Fourier series, probability, calculus of variations, special functions, and so on However, most science students have neither the time nor the inclination to study that much mathematics, yet they are constantly hampered in their science courses for lack of the basic techniques of these subjects It is the intent of this book to give these students enough background in each of the needed areas so that they can cope successfully with junior, senior, and beginning graduate courses in the physical sciences I hope, also, that some students will be sufficiently intrigued by one or more of the fields of mathematics to pursue it futher It is clear that something must be omitted if so many topics are to be compressed into one course I believe that two things can be left out without serious harm at this stage of a student’s work: generality, and detailed proofs Stating and proving a theorem in its most general form is important to the mathematician and to the advanced student, but it is often unnecessary and may be confusing to the more elementary student This is not in the least to say that science students have no use for careful mathematics Scientists, even more than pure mathematicians, need careful statements of the limits of applicability of mathematical processes so that they can use them with confidence without having to supply proof of their validity Consequently I have endeavored to give accurate statements of the needed theorems, although often for special cases or without proof Interested students can easily find more detail in textbooks in the special fields Mathematical physics texts at the senior-graduate level are able to assume a degree of mathematical sophistication and knowledge of advanced physics not yet attained by students at the sophomore level Yet such students, if given simple and clear explanations, can readily master the techniques we cover in this text (They not only can, but will have to in one way or another, if they are going to pass their junior and senior physics courses!) These students are not ready for detailed applications—these they will get in their science courses—but they need and want to be given some idea of the use of the methods they are studying, and some simple applications This I have tried to for each new topic For those of you familiar with the second edition, let me outline the changes for the third: Prompted by several requests for matrix diagonalization in Chapter 3, I have moved the first part of Chapter 10 to Chapter and then have amplified the treatment of tensors in Chapter 10 I have also changed Chapter to include more detail about linear vector spaces and then have continued the discussion of basis functions in Chapter (Fourier series), Chapter (Differential equations), INDEX raising and lowering operators, 607, 618 recursion relations, 570 Rodrigues’ formula, 568 Legendre series, 580 ff, 649 Legendre transformation, 231, 233 (Problems 10 to 13) Leibniz’ rule: for differentiating integrals, 233 ff for differentiating products, 567 lengthening pendulum, 598 length of arc See arc length length of a vector, 96 lenses, 198 level lines, 290 level surfaces, 290 lever problem, 277 Levi-Civita symbol ijk , 508 ff, 515 in determinant formula, 509 isotropic tensor, 510 products of, 510 L’Hˆopital’s rule, 38, 685 light See also optics applications absorption, 399 color, 346, 378 continuous spectrum, 378 frequency, 346, 378 parabolic mirror, 407 pulse, 382 scattering, 389 waves, 346, 378 limits on an integral, 243 ff line(s) distance between skew, 111 distance from point to, 110–111 equations of, 106 ff parametric, 108 symmetric, 107–108 normal, 109, 293 perpendicular to plane, 109 tangent to curve, 203 linear absorption coefficient, 399 linear algebra, Chapter linear algebraic equations, 82 ff See also determinants; matrices; matrix Cramer’s rule, 93–94 homogeneous, 134 matrix solution of, 119–120 825 solution in vector form, 134 standard form, 84 linear combination, 124 linear dependence, 132 linear differential equations, 401, 408, 417 See also differential equations linear functions, 124 linearly independent functions, 133 linearly independent vectors, 144 linear operators, 124, 425, 438 linear space See linear vector space linear transformations, 125 See also transformations; orthogonal transformations deformation of membrane, 148 ff eigenvectors of, 148 ff matrix of, 125, 148 orthogonal, 126 linear triatomic molecule, 167 ff linear vector space, 143 ff, 179 ff basis, 143 ff (also see) basis complex Euclidean, 146 ff definition, 142, 179 dimension, 144 Euclidean, 142 ff examples, 143 ff, 180–181 function space, 180, 414–415 (Problems 13 to 18) general, 179 Gram-Schmidt process, 145, 182–183 infinite dimensional, 183 inner product, 144, 146, 181 inner product space, 181 norm, 144, 146, 181 orthogonality, 144, 146, 181 orthonormal basis, 145, 182 Schwarz inequality, 145, 146, 182 span, 143 subspace, 143 line element, 521 See also arc length line integrals, 299 ff lines of force, 396 Liouville’s theorem, 718 ln, 72 See also Logarithm Logarithm(s): to base e, 72 branch of, 693 of complex numbers, 72 826 INDEX and inverse hyperbolic functions, 74 of negative numbers, 72 series for, 26 Lorenz equations, 96 loudness, 373 lowering and raising operators (ladder operators), 607, 614, 618 lowering and raising tensor indices, 532 loxodrome, 269 Maclaurin series, 24 ff See also series; power series magnetic field, 329, 714 magnitude of a vector, 96 main diagonal, 118 mapping functions of z, 705 ff, 710 of the plane, 125 ff mass, center of, 251 ff matrices, Chapter See also matrix addition of, 115 commutator of, 117 conformable, 116 and determinants compared, 115 elementary row operations, 86 equal, 86, 114 inverse of product of, 140 multiplication of, 115, 116 operations on, 114 ff row reduction, 83 ff similar, 150 table of special, 137–138 transpose of product of, 139 and vectors or tensors, 114, 503 matrix, Chapter 3, 83ff See also matrices adjoint, 137 anti-Hermitian, 138 anti-symmetric, 138 augmented, 85 block diagonalized, 176 characteristic equation of, 148 of coefficients, 84 column, 114 dagger, 137 determinant of, 89 diagonalizing, 148 ff eigenvalues of, 148 ff of eigenvectors, 150 equations, 114 functions of, 121 Hermitian, 137, 138 Hermitian conjugate, 137 identity, 118 index notation, 84, 116, 138 ff inverse, 119, 137 invertible, 119 multiplication by a number, 114 normal, 138 null, 117 operators, 125 orthogonal, 126, 127, 138 power of, 121 pure imaginary, 138 rank of, 86, 94 real, 138 reflection, 128 ff, 155 ff rotation, 120, 127, 129, 155 ff row, 114 singular, 119 skew-symmetric, 138 special, 137 symmetric, 138 trace of, 140 of a transformation, 125, 148 transpose conjugate of, 137 transpose of, 84, 137 unit, 118 unitary, 138, 154 zero, 117 maxima and minima, 211 ff boundary point or endpoint problems, 223 ff in calculus of variations, 472 ff with constraints, 214 of harmonic functions, 720 interior, 224 ff using Lagrange multipliers, 214 ff second derivative tests, 213 (problems 1, 2) Maxwell-Boltzmann statistics, 742 Maxwell equations, 330, 621 (Problem 3) mean free path, 754 mean value of See also average value a product of random variables, 756 (Problem 14) a random variable, 746,752,763,768 a set of measurements, 771 ff INDEX 827 of a charge or mass distribution, 573 a sum of random variables, 749 (Probelectric dipole, 573–575 lem 9) first, 277 mean value theorem, 195 of a force, 277 measurements, 770 ff of inertia, 252, 265, 277, 505 (see mechanics applications See acceleraalso inertia tensor) tion; energy; motion of a parproblems, 255-257, 267–270, ticle; velocity 273–275, 508, 520 median, 771 quadrupole, 573–575 medians of triangle, 98 second, 277 membrane, deformation of, 148 monogenic function, 667 membrane, vibration of, 644 ff mothball, 394 meromorphic function, 681 motion of a particle, (Problem 16), method of Frobenius, 585 ff, 605 286 See also acceleration; force; method of images, 657 Lagrange’s equations method of undetermined coefficients, 421 in a circle, 283 methods of counting, 736 complex notation, 56, 76, 80, 340–341 metric tensor, 524, 532 damped, 413 in orthogonal coordinates, 533 forced, 426 raising and lowering indices, 532 under gravity, 392 minimum distance: problems, 308, 400, 435, 486 calculus methods, 214, 216 in polar coordinates, 487 calculus of variations, 472 ff probability function for, 750–753 Lagrange multipliers, 217–218, 221–222 reduced mass, 197 vector methods, 109–111, 238 (Problem 2) simple harmonic, 80 minimum surface of revolution, 479 problems, 344, 416, 754, 489–490 minor of a determinant, 89 multiple integrals, Chapter mirror, 407 multiple pole, 680 mixed second partial derivatives, 189 multiple-valued function, 667 not necessarily equal, 190 multiplication (mod N ), 178 reciprocity relations, 190, 231, 306 multiplication tables for groups, 173 ff mixed tensor, 531 multiplying See product mobile, 44 music See sound waves mode, 771 mutually exclusive, 723 mode of vibration, 165, 636, 645–646 modern physics See also quantum mechanics nappe of a cone, 263 charge distribution in atoms, 573 natural frequency, 414, 427 hydrogen atom, 614 (Problems 27, natural logarithm, 72 See also Logarithm 28), 652 (Problem 22) n-dimensional space, 143 ff Millikan oil drop experiment, 400 negative of a vector, 97 radioactive decay, 39, 395, 399, Neumann function, 591 402–403, 755, 768 Neumann problem, 631 relativity, 42, 394 Newton’s law of cooling, 400 waves, 342 (see also waves) Newton’s second law, 80, 289, 390, modified Bessel functions, 595 394, 485 mod N , 174, 178 Newton’s third law, 285 modulus of a complex number, 49 nodal line, 646 modulus of an elliptic integral, 555 nonanalytic function, 669 Moebius strip, 327 non-Cartesian tensors, 529 ff, 531 moment, 277 828 INDEX noncommutative operations, 103, 117, 132 nonconservative field, 302, 306 nonlinear differential equation, 392, 396 non-uniform sample space, 725 norm, 96 normal cumulative distribution, 547 normal derivative, 293 normal distribution, 547, 761, 769 normal error curve, 761 normal frequencies, 166 normalization of Bessel functions, 602–603 Hermite polynomials, 608 Laguerre polynomials, 610, 611 Legendre polynomials, 578 ff normal line, 109, 293 normal matrix, 138 normal modes of vibration, 165 normal (perpendicular), 109 normal vector to a plane, 109 normal vector to a surface, 220, 271, 293 null matrix, 117 number of poles and zeros, 694 octopole moment, 573, 575 odd functions, 364 Ohm’s law, 78 one-sided surface, 327 operators See differential operators; linear operators; vector operators optics applications, 79 See also light combining light waves, 79 Fermat’s principle, 473, 484 Fresnel integrals, 37, 549 (Problem 6), 701 (Problem 41) law of reflection, 474 lenses, 198 multiple reflections, 79 scattering, 389 Snell’s law, 474 orbits, 468 order: of a Bessel function, 589 of a determinant, 80 of a differential equation, 391 of eigenvalues in a diagonal matrix, 152 of a group, 173 of placing electric charges, 18–19 of a pole, 680 of a tensor, 496 of terms in a series, 18 ordinary differential equations, Chapters and 12 See also partial differential equations associated Legendre, 583 auxiliary equation, 409 ff Bernoulli, 404, 431 Bessel, 588, 593, 594 boundary conditions, 393 change of variables in, 406 complementary function, 417 damped motion, 413 dependent variable missing, 430 Euler (Cauchy), 434 Euler (Lagrange), 485 ff exact, 405 exponential right-hand side, 419 family of solutions, 395–396 first integrals, 433 forced vibrations, 417 ff, 426 ff Fourier series solutions, 428 free vibrations, 412, 417 Fuchs’s theorem, 605 generalized power series solution, 585 general solution of, 392, 395, 401, 410, 411, 418 Green functions for, 461 Hermite, 607, 608 homogeneous, 406, 408 ff independent variable missing, 431 inhomogeneous, 408 initial conditions, 393 integrating factors, 401, 405 Laguerre, 609 ff Laplace transform solution, 440 ff Legendre, 564 linear, 391 linear first-order, 401 ff linear second-order, 408 ff, 417 ff method of undetermined coefficients, 421 nonhomogeneous, 408 nonlinear, 391, 392, 396 ff, order, 391 particular solution, 392 ff, 397, 415 ff (see also particular solution) INDEX reduction of order, 434, 567 (Problem 4), 606 (Problems 1–4) regular, 605 second solution, 606 separable, 395 ff series solutions, Chapter 12 simultaneous, Laplace transform solution, 441 slope field, 393, 394 solution of, 391, Sturm-Liouville, 617 variation of parameters, 464 y missing, 430 orthogonal curvilinear coordinates, 521 ff, 708 arc length element, 521 basis vectors, 522 scale factors, 522 vector operators in, 525 velocity, 524 volume element, 524 orthogonal functions, 575 orthogonality of Bessel functions, 601 ff of functions in Fourier series, 351, 601 of Hermite polynomials, 608 of Laguerre polynomials, 610, 611 of Legendre polynomials, 577 proof using differential equation, 577 of solutions of a Sturm-Liouville equation, 617 of vectors in dimensional space, 102 of vectors in n dimensional space, 144 with respect to weight function, 602 orthogonal matrix, 126, 127, 138 orthogonal similarity transformation See orthogonal transformation orthogonal trajectories, 396 orthogonal transformation, 126 ff See also rotation; reflection by an analytic function, 708 diagonalizing a symmetric matrix by, 149 ff matrix, 127 in n dimensions, 143 to normal modes of vibration, 165 829 to principal axes, 162 similarity, 150 in dimensions, 155 orthogonal (perpendicular) vectors, 105, 144 orthonormal basis, 145 functions, 579 oscillating series, oscillator, oscillatory motion See vibrations oscillatory (underdamped) motion, 414 outer product, 501 out of phase, 79 overdamped motion, 413 overtones, 345, 373 paddle wheel probe, 325 parabolic coordinates, 525 parabolic cylinder coordinates, 524 parabolic mirror, 407 parallel axis theorem, 255 parallelogram, 186 parallelogram law, 97 parallel vectors, 102 parameters, variation of, 464 parametric equations circle, 301 curve, 301 cycloid, 483 ff ellipse, 556 line, 108 parent population, 770 ff Parseval’s theorem, 375 partial derivatives, Chapter See also partial differentiation cross partials, 190 of functions of (x, y) or (r, θ), 189 in matrix notation, 200 in thermodynamics, 190, 231 partial differential equations, Chapter 13 change of variables in, 228 ff derivations of 620–621 Fourier series solutions, Chapter 13, Sections 1–4 Green function solutions, 657–658 heat flow or diffusion equation, 619, 628 ff, 659 Helmholtz equation, 620, 629, 645 830 INDEX Laplace’s equation, 619, 621, 638, 648 (see also Laplace’s equation) Poisson’s equation, 619, 652 ff, 657 Schr¨ odinger equation, 620, 628, 631– 632, 651 (Problems 18, 20–22) solution by Fourier transforms, 660 ff solution by Laplace transforms, 659 ff solution by separation of variables, 622, 625, 628, 639, 645, 648 wave equation, 297, 620, 633 ff, 644 ff partial differentiation, Chapter See also partial derivatives chain rule, 199 ff, 203 ff change of variables, 228 ff derivatives not reciprocals, 208 differentials, 193 implicit, 202 of integrals, 233 ff Lagrange multipliers, 216 ff maximum and minimum problems, 211 ff mixed second derivatives, 189–190 notation, 189–190 total differential, 193 ff two-variable power series, 191 ff partial fractions, 451 partial sum of a series, particular solution, 392ff, 397 complex exponentials, 420 exponential right-hand side, 419 ff by inspection, 418 principle of superposition, 425, 428 undetermined coefficients, 421 passive transformation, 127 Pauli spin matrices, 122 pendulum(s), 38, 344, 545 coupled, 490 double, 491 energy of, 545 large vibrations of, 557 lengthening, 598 period of, 343, 545–546, 557 seconds, 343 shortening, 600 simple, 38, 344, 545 small vibrations of, 38, 344, 545 spherical, 489 period of a Fourier series, 350 ff, 360 ff of a function, 343 of lengthening pendulum, 599 of simple harmonic motion, 341 of simple pendulum, 38, 344, 545 periodic function, 340, 343, 345 permutation, even, odd, 509 permutations, 737 permutation symbol (Levi-Civita), 508 ff perpendicular axis theorem, 252 perpendicular (orthogonal) vectors, 105 phase of a complex number, 49 phase space, 143 plane complex, 47 ff distance from point to, 110 equations of, 108 ff perpendicular to a vector, 109 tangent to a surface, 293 through points, 136 (Problem 21) through points, 109 planet, 468 plate, conducting, 322 plate, hottest and coldest points of, 224 plate, temperature in, 621, 661 plotting complex numbers, 48 ff, 62 graphs of complex equations, 54 ff roots of complex numbers, 64 ff point boundary, 223 ff at infinity, 702 of inflection, 212, 472 maximum, minimum, 211 ff in n-dimensional space, 143 saddle, 212 of sample space, 724 ff and vector r, 142 Poisson distribution, 767 ff approximated by normal distribution, 769 mean and standard deviation, 768 Poisson’s equation, 619, 652 ff, 657 Poisson’s summation formula, 389 polar coordinates See also orthogonal coordinates acceleration in, 487 arc length element, 259, 266, 521 area element, 258 change of variables to, 229, 258 INDEX complex numbers in, 48, 50 ff div, 298 Euler equation, 478 grad, 294, 525 integrals in, 258 ff Lagrange’s equations in, 486 Laplacian, 298 partial derivatives, 189, 208 scale factors, 522 unit basis vectors, 288, 522 polar form of a complex number, 48, 61 polar vector, 515 in physics formulas, 517 pole(s) on contour, 691 at infinity, 703 number in a region, 694 of order 2, 681 of order n, 680 residues at, 684 ff simple, 680 ff Polya’s urn model, 741 polynomial approximation, 581 polynomials Hermite, 608 Laguerre, 609 ff Legendre, 566 ff (see also Legendre polynomials) Legendre polynomials, as combinations of, 574 population average and variance, 771 position vector, 286 potential complex, 713 by conformal mapping, 712–713 electrostatic, 304, 712, 713 (see also electrostatic potential) gravitational, 290, 303 ff, 484, Legendre expansion of, 571 ff multipole expansion of, 571–573 Poisson’s equation for, 652 ff scalar, 303 ff vector, 332 velocity, 303, 713 zero at infinity, 304–305 potential energy, 303, 433, 485 power of a matrix, 121 power series See also series absolutely convergent, 10, 19, 58 831 adding, 19, 23, 59 alternating series test, 17 in annular ring, 679 binomial, 28 combining, 23, 26 ff, 59 comparison test, 10 complex, 56 ff computation, 3, 36 ff, 41 ff using computer, 31 conditionally convergent, 18 convergence of, ff, 20 ff, 58 differentiating, 23, 59 disk of convergence, 58, 671 divergence of, ff dividing, 27 ff, 59 expanding functions in, 23 ff functions having no power series, 33 generalized, 585 ff general term, integral test, 11 ff integrating, 23, 30, 59 interval of convergence, 20 Maclaurin, 24 ff multiplying, 23, 26, 59 numerical computation using, 36 preliminary test, ratio test, 13, 14 rearranging terms, 18 remainder of, 7, 13 sequence, 1, solution of differential equations, Chapter 12 See also differential equations substitution of one in another, 23, 29 Taylor, 24, 30, 213, 671 theorems about, 23 two-variable, 191 uniqueness of, 23, 59, 191 powers of a matrix, 121 powers of complex numbers, 63 Poynting vector, 535 preliminary test, principal angle of a complex number, 50 principal axes and moments of inertia, 506 ff principal axes of a conic section, 162 principal part of ∆z, 195 principal part of Laurent series, 678 principal value of 832 INDEX arc tan, 50 an integral, 692 ln z, 72 principle of superposition, 425, 428 probability, Chapter 15 applications, 722 of compound events, 730 conditional, 732 cumulative distribution, 748 definition, 723, 725 density, 753 ff (see also probability functions) distribution, 748 distribution function, 748 experimental measurements, 770 ff functions, 745, 748 binomial, 756 ff continuous, 750 ff normal (Gaussian), 761 ff Poisson, 767 ff of independent events, 731 mathematical, 727 mean value, 746,752,763,768 natural or intuitive, 727 random variables, 744, 747 sample space, 724 ff standard deviation, 747,753,763,768 theorems, 729 variance, dispersion, 747, 753 probable error, 774 product of complex numbers, 62 ff cross, 103 ff, 276 derivative of, 567 of determinants, 118 dot, 101, 102, 276 inner, 100 of matrices, 115, 116 scalar, 101, 276 of series, 23, 26, 191 vector, 103, 276 of vectors, 276 progression, geometric, projections of a vector, 101 proper rotation, 514 p-series test, 13 pseudovector, pseudotensor, 514 axial vector, 515 cross product, 516 pup tent, 212 pure imaginary, 47 pure tone, 345 quadratic formula, 46 quadratures, 433 quadric surface, principal axes of, 163–164 quadrilateral, 100 quadrupole moment, 573 quantum mechanics applications complex numbers, 81 correspondence principle, 197 Dirac delta function, 454 Fourier transforms, 386 Hermite and Laguerre functions, 607–614 hydrogen atom, 614 (Problems 27, 28), 652 (Problem 22) Klein-Gordon equation, 665 ladder operators, 607 ff, 614 (Problems 29, 30) Pauli spin matrices, 122 Schr¨ odinger equation, 620, 628, 631– 632, 651 (Problems 18, 20–22) simple harmonic oscillator, 651 (Problem 21) spherical harmonics, 649, 651 (Problem 16) statistics, 739 ff problems, 744 sums of integers, 651 (Problem 21), 744 (Problem 21) quotient See dividing quotient rule for tensors, 504, 532 radian mode, 49 radians and degrees, 49, 52 radioactive decay, 39, 395, 399, 402– 403, 755 and Poisson distribution, 768 radio-frequency, 347 radio waves, 343, 347 radius of convergence, 58 raindrop, 467 raising and lowering operators, ladder operators, 607, 614, 618 raising and lowering tensor indices, 532 ramp function, 374 random variables, 744, 747 random walk, 757 INDEX rank of a matrix, 86, 94 rank (order) of a tensor, 496 rational functions, 60 ratio test, 13–14 real axis, 48 real part of a complex number, 47, 49 rearranging terms of a series, 18 reciprocals in differentiation, 208 reciprocity relations, 190, 231, 306 rectangular form of a complex number, 48, 51 rectified half-wave, 374 recurrence See recursion relations recursion relations: Bessel functions, 592 gamma function, 538–539 Hermite polynomials, 609 Legendre polynomials, 570 reduced mass, 197 reduction of order in a differential equation, 434, 606 reflection matrix, 128 ff, 155 ff reflection of axes, 126, 128 ff, 156 ff reflection of light, 285, 474 refraction, 474 region, 667 simply connected, 330 regular differential equation, 605 regular function, 667 regular point, 680 relative error, 197, 775 relative intensity of harmonics, 373 relativity, 42, 394 relaxation oscillator, 387 remainder, 7, 13 residues at infinity, 703 using Laurent series, 683 at a multiple pole, 685 at a simple pole, 684 residue theorem, 682 evaluating integrals using, 687 ff resistance of a wire, 197 resonance, 427 response to unit impulse, 449, 459 resultant, 446 rhombus, 106 rhumb line, 269 Riccati, 408 833 Riemann surface, 707, 708 right-handed system, 515 right-hand rule, 515 RLC circuit See electric circuits rocket, 467 Rodrigues’ formula, 568 Hermite polynomials, 608 Laguerre polynomials, 609 Legendre polynomials, 568 root-mean-square, 348 roots of auxiliary equation, 409–414 roots of complex numbers, 65, 66 rotation(s) angle, 127, 129, 151 axes rotated, 127 in the complex plane, 77 (Problem 1), 131 (Problem 19) equations, 127, 151 improper, 514 matrix, 120, 127, 129, 155 ff in n dimensions, 143 non-commuting, 132 not a vector, 497 to principal axes, 162–164 proper, 514 of a rigid body, 497, 505 ff tensor, 535 (problem 9) in three dimensions, 129, 155 ff in two dimensions, 127, 151 of a vector, 127 vector rotated, 127 rot v, rotation v, 325 row matrix, 114 row operations, 86 row reduction, 83 ff row vector, 114 saddle point, 212 sample average, 771 sample space, 724 ff sawtooth voltage, 346, 374 scalar, 82, 496 field, 290 operator, 296 potential, 303 ff, 332 (see also potential) product, 101, 276 projection, 101 triple product, 278 834 INDEX scale factors and basis vectors, 522 Schr¨ odinger equation, 620, 628, 631– 632, 651 (Problems 18, 20–22) particle in a box, 632 time independent, 631 Schwarz inequality in complex Euclidean space, 146 in Euclidean space, 145 sec γ method, 271 second derivative tests for maximum, minimum, 213 (Problem 2) second-rank tensor: Cartesian, 500 contravariant, 532 covariant, 531 mixed, 531 skew-symmetric, 503 symmetric, 503 seconds pendulum, 343 secular determinant 148 seesaw, 277 selections (combinations), 737 separable differential equations, 395 separation constant, 622 integral valued, 639, 645, 648 sign of, 624, 629, 634, 639, 648 zero value of, 628, 644 separation of variables: ordinary differential equations, 395 partial differential equations, 622, 633, 639 sequence 1, 5, series, Chapters 1, 2, 7, 11, 12 See also individual series absolutely convergent, 10, 19, 58 adding, 19, 23, 59 alternating, 17, 34 alternating harmonic, 17, 18, 21, 37 approximations, 33 ff, 38 ff, 545, 548 asymptotic, 549 ff, 552 Bessel, 640 ff, 645 ff (see also Bessel functions) binomial, 28 complex, 56 ff, 673, 678 ff computation with, 3, 36, 41 ff, 548– 549, 624 ff conditionally convergent, 18 convergent, 6, 20, 58 (see also convergence) divergent, ff dividing, 23, 27, 59 Fourier, Chapter (see also Fourier series) geometric, ff harmonic, 11, 12 infinite, Chapter Laurent, 678 ff Legendre, 580, 649 (see also Legendre series) Maclaurin, 24 ff multiplying, 23, 26, 191 oscillating, partial sum of, of positive terms, 10 power, Chapter (see also power series) rearranging terms in, 18 remainder of, 7, 13, 33 for alternating series, 34 solutions of differential equations, 428, 562 ff summing, 37 sum of, 2, 7, 18 Taylor, 24, 30, 671 useful facts about, 19 series solutions of differential equations, Chapter 12 Bessel 588, 593, 594 Hermite, 607 ff Laguerre, 609 ff Legendre, 564 ff sgn x, 460 shear forces in stress tensor, 496 sheet of Riemann surface, 708 shifting theorems, 439, 468 SHM See simple harmonic motion, 340 ff shortening pendulum, 600 shortest distance along a surface, see geodesics shortest time See brachistochrone side bands, 347 Sierpi´ nski gasket, similarity transformation, 150 ff orthogonal, 150 ff, 154 unitary, 154 similar matrices, 150 simple curve, 674 INDEX simple harmonic motion, 38, 80, 340 ff, 412 See also pendulum problems, 344, 416, 489–490, 754 simple pendulum See pendulum simple pole, 680 simply connected region, 330 simultaneous diagonalization of matrices, 158 simultaneous differential equations, 441 sine of complex number, 68 Fourier series, 366 Fourier transform, 381 hyperbolic, 70 inverse, 74 ff power series for, 26, 69 single-valued function, 356, 667 singularity, 670 singular matrix, 119 singular point, 670 singular solution, 397 sinh z, 70 sink, sink density, 316, 714 sinusoidal functions, 341 skew lines, 111 skew-symmetric matrix, 138 skew-symmetric tensor, 503 skin effect, 596 slant height, 255 slope of a curve, 202–203 Snell’s law, 474 sn u, 558 soap film, 473, 480 solenoidal field, 332 solid angle, 320 solid of revolution, 253 sound waves, 345, 372, 378 air pressure in, 345, 372 energy in, 373 frequency: continuous, 378; fundamental, 345, 373; overtones, 345, 373 harmonics, 345, 373 intensity of, 373 pure tone, 345 vibrating drum, 644 vibrating string, 633 ff source, source density, 714 space, 142 ff See linear vector space span, 143 835 special comparison test, 15 specific heats, 211 (Problem 28) spectrum, continuous, 378 sphere and complex plane, 703 spherical Bessel functions, 596 spherical coordinates, 261 See also orthogonal coordinates acceleration, 523 arc length element, 266 curl, 527 div, 298, 526 grad, 294, 525 Laplace’s equation, 648 Laplacian, 298 scale factors and unit basis vectors, 523 velocity, 266, 523 volume element, 261, 263, 524 spherical harmonics, 649 spherical pendulum, 489 spring constant, 80 square of a vector, 101 square wave, 353 stability of a vertical wire, 600 standard deviation for binomial and normal distributions, 763 of the mean, 772 for Poisson distribution, 768 of a random variable, 747, 753 of a single measurement, 771 of sum, product, etc., 772 ff standard error, 722 standard normal distribution, 763, 764 star ∗ meaning complex conjugate, 50 star ∗ meaning convolution, 446 statics problems, 42–43 stationary point, 472 ff statistical mechanics applications, 554, 739 ff Bose-Einstein, Fermi-Dirac, MaxwellBoltzmann statistics, 742 problems, 743–744 statistics, 770 ff central limit theorem, 774 combination of measurements, 772 ff confidence interval, probable error, 774 836 INDEX and experimental measurements, 770 ff population mean and variance, 771 standard deviation of the mean 772 steady state, 426 steady state temperature in a cylinder, 638 ff in a finite plate, 625 insulated boundaries, 631 insulated edges, 627 (Problem 14) problems, 626 ff in a semi-infinite rectangular plate, 621 in a sphere, 648 ff stereographic projection, 703 Stirling’s formula, 552 error in, 553 in statistical mechanics, 554 stochastic process, 746 Stokes’ theorem, 313, 324 ff, 327 ff and Amp`ere’s law, 329 strain, strain tensor, 519 stream function, 713 streamlines, 325, 714 See also flow of water stress tensor, 496, 519, 520 stretch, 152 string, vibrating See vibrating string Sturm-Liouville equation, 617 subgroup, 173 subscripts on D, 568 on ∆, 298 in partial differentiation, 189–190, 205 in tensor notation, 502 ff, 530 on vectors, 97 ff subspace, 143 ff substituting one series in another, 29 subtraction of vectors, 98 summation convention, 502 ff summation sign, summing numerical series: by computer, 41, 44, 45 using Fourier series, 358, 377 power series of a function, 37 sum of alternating harmonic series, 37 complex numbers, 51 conditionally convergent series, 18 geometric progression, infinite series, 2, 7, 10 ff, 37 matrices, 115 power series, 23 two series, 18, 23 vectors, 97 superposition principle, 425 superscripts, 530 surface area, 271 (see also area) distance to, 216 integral, 270 ff level, 290, 293 of minimum area, 479 normal to, 220, 271, 293, 317, 327 one-sided and two-sided, 327 of revolution, 254, 479 tangent plane to, 194, 293 symmetric equations of line, 107–108 matrix, 138 tensor, 503 symmetry groups, 174, 178 systems of masses and springs, 165–172 tables approximate formulas for Bessel functions, 604 Laplace transforms, 469–471 matrices, 137–138 orthogonal polynomials: Hermite and Laguerre, 804 Legendre, 803 vector identities, 339 tangent approximation, 193 tangent line, 203 tangent plane, 293 tangent, series for, 28 Taylor series, 24, 30, 671 See also power series temperature: in a bar, 629 ff, 659 on boundary, 720 in a cylinder, 638 ff gradient, 292–293 in a half cylinder, 712 in a hemisphere, 650 maximum and minimum in a plate, 224–225 INDEX in a plate, 621, 625 scale, 650 in a semicircular plate, 711 in a sphere, 648 ff tensor product See direct product tensor(s), Chapter 10 angular momentum, 505 ff antisymmetric, 503 applications of, 505, 518 ff associated, 533 Cartesian, 498 ff combining, 504 components and basis vectors, 530 contraction, 502 ff contravariant, 530 covariant, 530 cross product, 516 definition, 499 direct product, 501 dual, 512 electric dipole, quadrupole, 574 (Problem 15) field, 520 in general coordinates, 531 inertia, 505 ff isotropic, 509 Kronecker δ, 508 ff Levi-Civita symbol, 508 ff and matrices, 503 metric, 524, 532 non-Cartesian, 529 ff notation, 502 order (see rank) pseudo-vectors, pseudo-tensors, 514 ff quotient rule, 504, 532 raising and lowering tensor indices, 532 rank, 496 rotation, 535 (Problem 9) 2nd -rank, 496, 501, 506 strain, 519 stress, 496, 519 summation convention, 502 symmetric, 503 vector identities, 511 vector operators, 525 and vectors, 496 ff tent, 212 term-by-term addition of series, 19, 23 terminal speed, 400 837 thermodynamics, 190, 211 See also heat flow Legendre transformation, 231, 233 reciprocity relations, 190, 231, 306 specific heats, 211 time constant, 399 torque about a line, 277 torque about a point, 281 torus, 257 total differentials, 193 ff tower of books, 44 circular, 43 trace, 140 and group character, 176 of a product of matrices, 140 of a rotation matrix, 157 transfer function, 444 transformations See also linear transformations; rotation active, 127 conformal mapping, 705 ff general, 520 Jacobian of, 261 ff, 283, 536 (Problem 12), 710 (Problems 12, 13) Legendre, 231, 233 linear, 125 orthogonal, 126 ff passive, 127 similarity, 150 transforms, integral, 378 ff, 437 ff Fourier, 378 ff (see also Fourier transforms) Hilbert, 437 Laplace, 437 ff (see also Laplace transforms) transients, 426 translation or shifting theorems, 439 transpose of a matrix, 84, 137 transpose of product of matrices, 139 trapezoid, 100 triangle inequality, 148 trigonometric functions of complex numbers, 67 ff identities, 69, 70 integrals of, 349, 351 inverse of, 74 series for, 23 ff triple integrals, 242 ff 838 INDEX triple scalar product, 278 triple vector product, 280 trivial solution, 134 true vector (polar vector), 515 tuning fork, 345 two-sided surface, 327 two-variable power series, 191 unbounded region, 659 underdamped motion, 413, 414 undetermined coefficients, 421 undetermined multipliers, 216 uniformly distributed, 751 uniform sample space, 725 unitary matrix, 138 similarity transformation, 154 transformation, 154 unit basis vectors, 100, 144, 288, 522 See also basis vectors cross product, 104, in curvilinear coordinates, 522 in cylindrical coordinates, 522 derivatives, 288, 523 dot product, 102 in general curvilinear coordinates, 523 in polar coordinates, 288 unit circle, 687 unit eigenvectors, 149, 164 unit element of a group, 172 unit impulse, 449 unit matrix, 118 unit step, 470 unit vector, 98 See also unit basis vectors in polar coordinates, 287, 288 universe, 770 variables See also change of variables; coordinates; dependent variable; independent variable complex, Chapters and 14 dummy, 380, 541 random, 744, 747 variance, 747, 753, 763, 771 See also standard deviation variational notation, 493 variation of parameters, 464 varied curves, 475 vector coordinate, 286 vector field, 289 ff, 332, 393, 394 vector identities, table 339 polar, cylindrical, spherical, 298 proved in tensor form, 511 vector integral theorems, 325 vector operators, 296 ff, 525 ff curl, 296, 324, 511, 527 in curvilinear coordinates, 525 ff in cylindrical coordinates, 525 ff divergence, 296, 314, 526 gradient, 290 ff, 525 Laplacian, 297, 298, 527 in tensor notation, 533 vector potential, 332, 336 (Problem 3) vector product, 276 not commutative, 103 of parallel vectors, 103 triple, 280 vectors, Chapters 3, and 10 acceleration, 286 complex, 56 addition of, 97 angle between, 102 associated, 533 associative law for addition, 97 axial, 515 basis, 287 (see also basis vectors) Cartesian, 498 ff characteristic, 148 ff column, 114 commutative law for addition, 97 components, 82, 97, 100, 114, 497, 530 covariant and contravariant, 530 cross product, 103 ff, 276, 516 definition, 82, 497 derivatives of, 285 ff displacement, 286 dot product, 101, 102, 276 length, 96 linearly independent, 132 magnitude, 96 in matrix notation, 114 multiplication of, 100 ff in n dimensions, 143 negative of, 97 norm of, 96 notation, 96 INDEX orthogonal, 144 parallel, 102 perpendicular, 102 polar, 515 in polar coordinates, 287, 288 position, 286 products of, 276 pseudo-, 514 row, 114 scalar product, 276 subtraction of, 98 transformation law, 498 triple products of, 278 ff, 511, 516 unit, 100 (see also unit basis vectors) zero, 98 vector space See linear vector space velocity amplitude, 343 angular, 278, 324, 340 complex, 56, 717 in cylindrical coordinates, 523 of electrons, 42 of escape, 435 of light, 474 in orthogonal coordinates, 524 potential, 303, 620, 713 in spherical coordinates, 266 vector, 286 wave, 342, 620, 634 vibrating string, 633 ff characteristic frequencies, 636 in elastic medium, 665 with free end, 637 Green function solution, 462 plucked, 634 standing waves, 636 struck, 635 of variable density, 600 vibrations See also pendulum; sound waves; waves; vibrating string characteristic frequencies, 636 of circular membrane (drum), 644 ff damped, 413–414 electrical, 340, 426 ff, 444 forced, 417 ff, 426 ff free, 80, 417 due to impulse, 449 of lengthening pendulum, 598 839 normal modes, 645, 646 of pendulum, 38, 344, 545 ff, 557 period of, 341 resonance, 427 simple harmonic, 38, 341, 412 problems, 344, 416 in a sound wave, 345, 373 steady state, 426 of string, 633 tuning fork, 345 violin, 645 voltage, 346, 353, 378 volume, 242 ff element, 253, 261, 524 finding by integration, 242, 253 integral, 253 ff, 318 of revolution, 253 vortex, 714 water flow, 325, 713-714 See also flow of water water waves, 342 wave equation, 297, 620, 633 ff, 644 ff for vibrating drum, 644 for vibrating string, 633 ff wavelength, 342 wave number, 634 waves, 342, 620, 634 light, 346, 378 notation, 344 (Problem 17) radio, 343 sound, 345, 372 velocity of, 342, 620, 634 Weber function, 591 weighted coin, 727, 734 weight function, 602 wire, vertical stability of, 600 work, 277 independent of the path, 303 Wronskian, 133, 136 (Problem 16) yo-yo, 489 zero factorial, zero matrix, 117 zero-rank tensor, 496 zeros of Bessel functions, 591 zeros of f (z), 694 zeta (ζ) function, 41 z-score, 764