Bruce R Kusse and Erik A Westwig Mathematical Physics Applied Mathematics for Scientists and Engineers 2nd Edition WILEYVCH WILEY-VCH Verlag GmbH & Co KGaA This Page Intentionally Left Blank Bruce R Kusse and ErikA Westwig Mathematical Physics Related Titles Vaughn, M T Introduction to Mathematical Physics 2006 Approx 650 pages with 50 figures Softcover ISBN 3-527-40627-1 Lambourne, R., Tinker, M Basic Mathematics for the Physical Sciences 2000.688 pages Softcover ISBN 0-47 1-85207-4 Tinker, M., Lambourne, R Further Mathematics for the Physical Sciences 2000.744 pages Softcover ISBN 0-471-86723-3 Courant, R., Hilbert, D Methods of Mathematical Physics Volume 1989 575 pages with 27 figures Softcover ISBN 0-47 1-50447-5 Volume 1989 852 pages with 61 figures Softcover ISBN 0-471-50439-4 Trigg, G L (ed.) Mathematical Tools for Physicists 2005.686 pages with 98 figures and 29 tables Hardcover ISBN 3-527-40548-8 Bruce R Kusse and Erik A Westwig Mathematical Physics Applied Mathematics for Scientists and Engineers 2nd Edition WILEYVCH WILEY-VCH Verlag GmbH & Co KGaA The Authors Bruce R Kusse College of Engineering Cornell University Ithaca, NY brk2@cornell.edu Erik Westwig Palisade Corporation Ithaca, NY ewestwig@palisade.com For a Solution Manual, lecturers should contact the editorial department at physics@wiley-vch.de, stating their affiliation and the course in which they wish to use the book All books published by Wiley-VCH are carefully produced Nevertheless, authors, editors, and publisher not warrant the information contained in these books, including this book, to be free of errors Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate Library of Congress Card No.: applied for British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library Bibliographicinformation published by Die Dentsehe Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at 02006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheirn All rights reserved (including those of translation into other languages) No part of this book may be reproduced in any form by photoprinting, microfilm, or any other means - nor transmitted or translated into a machine language without written permission from the publishers Registered names, trademarks, etc used in this book, even when not specifically marked as such, are not to be considered unprotected by law ~ Printing Strauss GmbH, Morlenbach Binding J Schaffer Buchbinderei GmbH, Griinstadt Printed in the Federal Republic of Germany Printed on acid-free paper ISBN-13: 978-3-52740672-2 ISBN-10: 3-527-40672-7 This book is the result of a sequence of two courses given in the School of Applied and Engineering Physics at Cornell University The intent of these courses has been to cover a number of intermediate and advanced topics in applied mathematics that are needed by science and engineering majors The courses were originally designed for junior level undergraduates enrolled in Applied Physics, but over the years they have attracted students from the other engineering departments, as well as physics, chemistry, astronomy and biophysics students Course enrollment has also expanded to include freshman and sophomores with advanced placement and graduate students whose math background has needed some reinforcement While teaching this course, we discovered a gap in the available textbooks we felt appropriate for Applied Physics undergraduates There are many good introductory calculus books One such example is Calculus andAnalytic Geometry by Thomas and Finney, which we consider to be a prerequisitefor our book There are also many good textbooks covering advanced topics in mathematical physics such as Mathematical Methods for Physicists by Arfken Unfortunately,these advanced books are generally aimed at graduate students and not work well for junior level undergraduates It appeared that there was no intermediate book which could help the typical student make the transition between these two levels Our goal was to create a book to fill this need The material we cover includes intermediate topics in linear algebra, tensors, curvilinearcoordinatesystems,complex variables, Fourier series, Fourier and Laplace transforms, differential equations, Dirac delta-functions, and solutions to Laplace’s equation In addition, we introduce the more advanced topics of contravariance and covariance in nonorthogonal systems, multi-valued complex functions described with branch cuts and Riemann sheets, the method of steepest descent, and group theory These topics are presented in a unique way, with a generous use of illustrations and V vi PREFACE graphs and an informal writing style, so that students at the junior level can grasp and understand them Throughout the text we attempt to strike a healthy balance between mathematical completeness and readability by keeping the number of formal proofs and theorems to a minimum Applications for solving real, physical problems are stressed There are many examples throughout the text and exercises for the students at the end of each chapter Unlike many text books that cover these topics, we have used an organization that is fundamentally pedagogical We consider the book to be primarily a teaching tool, although we have attempted to also make it acceptable as a reference Consistent with this intent, the chapters are arranged as they have been taught in our two course sequence, rather than by topic Consequently, you will find a chapter on tensors and a chapter on complex variables in the first half of the book and two more chapters, covering more advanced details of these same topics, in the second half In our first semester course, we cover chapters one through nine, which we consider more important for the early part of the undergraduate curriculum The last six chapters are taught in the second semester and cover the more advanced material We would like to thank the many Cornell students who have taken the AEP 3211322 course sequence for their assistance in finding errors in the text, examples, and exercises E.A.W would like to thank Ralph Westwig for his research help and the loan of many useful books He is also indebted to his wife Karen and their son John for their infinite patience BRUCE R KUSSE ERIK A WESTWIG Ithaca, New York CONTENTS A Review of Vector and Matrix Algebra Using SubscriptlSummationConventions 1.1 Notation, I 1.2 Vector Operations, Differential and Integral Operations on Vector and Scalar Fields 18 2.1 Plotting Scalar and Vector Fields, 18 2.2 Integral Operators, 20 2.3 Differential Operations, 23 2.4 Integral Definitions of the Differential Operators, 34 2.5 TheTheorems, 35 Curvilinear Coordinate Systems 3.1 The Position Vector, 44 3.2 The Cylindrical System, 45 3.3 The Spherical System, 48 3.4 General Curvilinear Systems, 49 3.5 The Gradient, Divergence, and Curl in Cylindrical and Spherical Systems, 58 44 670 ERRATALIST p 619 Table 15.25, row Cf], column , change “-1” to “4’ p 623 Change the sentence beginning in the 2nd line of the 2nd paragraph to read: “The net result is that there is one D t l representation, no D f l representation, and one D r l representation in the block diagonalization.” p 625 Equation 15.62,change element (1,3) and (3,l) from “1”to “-1” BIBLIOGRAPHY Abramowitz, Milton, and Irene Stegun, Handbook of Mathematical Function Dover Publications, New York, 1964 Arfken, George, Mathematical Methods of Physics Academic Press, San Diego, CA, 1985 Butkov, Eugene, Mathematical Physics Addison-Wesley, Reading, MA, 1968 Churchill, Rue1 V., and James Ward Brown, Complex Variables and Applications McGrawHill, New York, 1990 Griffel, D.H., Applied Functional Analysis John Wiley and Sons, New York, 1981 Hamemesh, Morton, Group Theory and its Applications to Physical Problems AdclisonWesley, Reading, MA, 1962 Hildebrand, F.B., Methods ofApplied Mathematics Prentice-Hall, Englewood Cliffs, NJ, 1958 Jackson, J.D, Classical Electrodynamics John Wiley and Sons, New York, 1962 Kaplan, Wilfred, Advanced Calculus Addison-Wesley, Reading, M A , 1973 Mathews, Jon, and R.L Walker, MathernaticalMethods of Physics Addison-Wesley,Redwood City, CA, 1970 Misner, Charles W., Kip S Thome, and Jon Archibald Wheeler, Gravitation W.H Freeman, San Francisco, CA, 1973 Sokolnikoff, Ivan, and R.M Redheffer, Mathematics of Physics and Modem Engineering McGraw-Hill, New York, 1966 Taylor, Edwin F., and Jon Archibald Wheeler, Spacetime Physics W.H Freeman, San Francisco, CA, 1963 Thomas, George B Jr., and Ross L Finney, Calculus andAnalytic Geometry Addison-Wesley, Reading, MA, 1951 Tolstov, Georgi P., Fourier Series, translated by Richard A Silverman Dover Publications New York, 1962 Tung, Wu-Ki, Gmup Theory in Physics World Scientific Publishing, Philadelphia, PA, 1985 Watson, G.N., A Treatise on the Theory of Bessel Functions Macmillian, New York, 1944 Wigner, Eugene Paul, Group Theory and its Applicatons to the Quantum Mechanics ofAtomic Spectra Academic Press, New York, 1959 671 This Page Intentionally Left Blank INDEX Abelian group, 598,618 Absolutely integrable function, 254 Aliasing, 240-241 Analytic continuation, 175 Analytic function, see Complex functions, analytic Associated Legendre equation, 460 Associated Legendre polynomials, 472 Autocorrelation,266 Axial vector, 89 Basis functions, 234 Basis vectors contravariant, 576-579 covariant, 573,576-579 curvilinear coordinates,49-5 nonorthogonal, 564 orthonormal, 2,44,75 position dependent, 655 position independent, 2,564 Bessel equation, 444 spherical, 354 Bessel functions first kind, 444446 modified, 453 orthogonality,448450 second kind, 444-446 spherical, 357 Bilateral Laplace transform, see Laplace transform, double-sided Block diagonal mamces, 613 Boundary conditions, 342 Dirichlet, 429 for Green’s functions, 384,387-391 homogeneous, 384 Neumann, 429 nonhomogeneous, 387-39 Branch cuts, 19-522 Branchpoints, 516519 Bromwich contour, 13,333 Cartesian coordinates, Cauchy integral formula, 147-150 Cauchy integral theorem, 144147,510-513 Cauchy-Riemannconditions, 141-144,546 Causality, 370 Character, 610-612 Character table, 621-622 Child-Langmuirproblem, 359-366,410-412 Christoffel symbols, 655-659 Class, 610-612.614 Cleverly closed contour, 540 C~OSUIE, 175-189,284-285.310-312 Cofactor, 572 Completeness Fourier series, 234 group representations,622-623 Sturm-Liouvilleeigenfunctions, 437 Completing the square, 27 Complex functions, 138-202.509-542 analytic, 140-150 derivativesof, 140-144 hyperbolic, 140 logarithm, 140 multivalued functions, 509-542 trigonometric, 139-140 visualization of, 138 673 INDEX 674 Complex plane, 135-136 Complex variables, 135-137 conjugates, 136 magnitude, 136 polar representation, 136-137 real and imaginary parts, 135, 136 Conductivity tensor, 67-70,76 Conformal mapping, see Mapping, conformal Conservative field, 38 Continuity equation, 27-29,361 Contour deformation, 146-147,532-539,542 Contravariant components, 568-582 Convergence absolute, 150, 151 complex series, 150-151 Fourier series, 23 1-234 mean-squared, 232-234 pointwise, 232 ratio test, 151 uniform, 155,232 Convolution, 261-265,325 Fourier transform of, 261-280 Laplace transform of, 320-323 Correlation, 265-266 Cosets, 608-610 Covariant components, 568-570,573-582 Covariant derivatives, 655-659 Cross-correlation, 265 Cross product, 8, 10-12,53,86-92 Curl Cartesian coordinates, 24 curvilinear coordinates, 55-58 cylindrical coordinates, 58 integral definition, 34 nonorthonormal coordinates, 659 physical picture of, 29-32 spherical coordinates, 58 Current density, 28,68 Curvilinear coordinates, 49-58 Cylindrical coordinates, 45-47 Decomposition of block matrices, 622 Del operator Cartesian coordinates, 23-24 curvilinearcoordinates, 54-58 identities, 32-34.41 Delta function, Dirac, see Dirac delta function Delta, Kronecker, see Kronecker delta DeMoivre’s formula, 203 Density functions, singular, 114-121 Differential equations, 339-403 boundary conditions, 342 constant coefficients, 347-349 coupled, 341 exact differential, 343-345 first-order, 342-346 Fourier transform solutions, 366-371 Green’s function solutions, 376403 homogeneous, 341-342 integrating factor, 345-346 Laplace transform solutions, 371-376 linear, 340-342 nonhomogeneous, 341-342.35 1-354 nonlinear, 340,358-359 order, 341-342 ordinary, 340 partial, 340,391 second-order, 347-354 separation of variables, 342-343,424-475 series solutions, 35&358 terminology,339-342 Diffusion equation, 391-398 Dipole, 123-125 Dipole moment, 122 Dirac delta function, 100-126 complicated arguments, 108-1 11 derivatives of, 112-1 14 Fourier transform of, 269-270 integral definition, 106-108 integral of, 111-1 12 Laplace transform of, 314-3 16 sequence definition, 104-105 shifted arguments, 102 three-dimensional, 115 use in Green’s function, 378,382 use in orthogonality relations, 253-254 Direct sum of block matrices, 613 Dirichlet boundary conditions, 429 Discrete Fourier series, 234-242 Displacement vector, 52 Divergence Cartesian coordinates, 24 curvilinear coordinates, 54-55 cylindrical coordinates, 58 integral definition, 34 nonorthonormal coordinates, U physical picture of, 27-29 spherical coordinates, 58 Dot product, 7-10,53 nonorthonormal coordinates, 568-570,573 tensors, 70 Double-sided Laplace transform, see Laplace transform, double-sided Doublet, 112-1 14 Dyadic product, Eigenfunction, 433440 Eigenvalue, 79,435-436 INDEX Eigenvector, 79,435 Einstein summation convention, Electric dipole, 123-125 Electric monopole, 122-123 Electric quadrapole, 133 Elliptical coordinates,62 Essential singularity, 170 Euler angles, 632 Euler constant, 445 Euler’s equation, 6I37 Exact differential, 343-345 Factorial function, 553 Fast Fourier transform (FFT), 234 Field lines, 20 Fourier series, 219-242 circuit analogy, 223-224 convergence, 23 1-234 discrete form, 234-242 exponential form, 227-23 orthogonalityconditions, 221-223,228-229 sine-cosine form, 219-223 of a square wave, 225-227 of a triangular wave, 224-225 Fourier transform, 250-295 circuit analogy, 256.257 of a convolution, 261 of a cross-correlation,266 of a damped sinusoid, 287-290 of a decaying exponential, 283-287 of a delayed function, 258 of a delta function, 269-270 of a derivative, 259-260 differential equation solutions, 366-37 of even and odd functions, 259 existence, 254256 of a Gaussian, 270-273 integral equation solutions, 499 limits of, 303-313 orthogonality condition, 253-254 of a periodic function, 273-275.279-280 of a product, 260-261 of pure real functions, 259 relation to Fourier series, 250-253.275 of a square pulse, 267-269 transform pair, 252 Fredholm equation first kind, 492499 second kind, 492,504 Frobenius, method of, 354358,444, 461, 469 Gamma function, 553-554 Gauss’s theorem, 35-36 675 Gaussian Fourier transform of, 270-273 sequence function, 105 General relativity, 565-566 Generalized coordinates,see Curvilinear coordinates Generalized functions, 100, 103 Geodesics, 566 Gibbs phenomenon, 233 Gradient Cartesian coordinates, 24 curvilinear coordinates, 54 cylindrical coordinates, 58 integral definition, 34 nonorthonormal coordinates, 582,656 physical picture of, 24-27 spherical coordinates, 58 Gram-Schmidt orthogonalization,437 Green’s functions, 376403,647-651 boundary conditions, 384,387-391 for diffusion equation, 391-398 for driven wave equation, multiple independentvariables, 391-403 for a stretched string, 385-391 symmetry properties, 398,422423 translational invariance, 378,398 Green’s theorem, Groups, 597-634 Abelian, 598,618 C,, 598-601 character, 610-612 character table, 621-622 class, 610-612, 614 continuous, 630-634 cosets, 608-610 D2.601403 D3.604-605 definition, 597-598 finite, 598-607 multiplication table, 598-599 order, 598 permutation, 607 rearrangement lemma, 599 representations,see Representations,group rotation, 630-633 SU(2) 633-634 subgroups, 607-608 Harmonic oscillator damped, driven, 368-37 simple, 347 undamped, driven, 378-381 Heaviside step function, 111-1 12, 316-317 Heisenberg uncertainty principle, 273 676 INDEX Hemholtz's theorem, 38-40 Hermitian matrix, 80,435 operator, 436-440 Homogeneous boundary conditions, see Boundary conditions homogeneous Homogeneousdifbmtd equation,see Memntid equstions, homogeneous Hyperboliccoordinates, 62 - Impulse,ideal, 101-102 hdicial equations, 355 ImKr product, see Dot product Integral eq~ations,491-506 classification of, 492493 Follrer transfm SoIutiOns, 499 Laplace transform solutions,499-500 relation to diffenatiplM o m 493-498 separable h l soluriom 504-506 series s o l u t i 501-503 ~ Integrals analytic functions 144-150 closed, 21,175-189,510-513 line,7,21,53 multivalued functions, 10-5 13,534-542 operatorform, 20-21 principal part, 184-188 ,&s 22-23,53 volume, 23,54 Integrating factor, 345-346 Irreducible matrix representation,see Representations,group Irrotational vector field, 40 Jordan's inequality, 285 Kernel, 492 causal, 498 separable, 504-506 translationally invariant, 498 Kronecker delta, 8-10,563464,574,581 L'Hopital's rule, 170 Laguerre polynomials, 488 Laplace contour, 313,333 Laplace equation, 424475 Cartesian coordham, 424-433 confonnal map solutions, 1921% cylindrical coordinates 441,457 spherical coordinates, 458-475 Laplace transform, 313-335 circuit analogy, 326-331 of a convolution 320-323 of a delayed function, 319 of a delta function, 314-316 of a de.rivative, 319-320 differentid equation SOJU~~O~S, 371-376 double-sided, 331-335 of a growing sinusoid, 317-3 18 integral equation solutions, 499-500 inversion contour,313,333 orthogonality condition,316 of a product,323-326 relation to Green"sfunction,377-381 of a step function, 316-317 transform pair, 313 Laplacian, 33,34 Laurent series complex functions, 159-171 real functions, 152 Left-hand rule, 10,88 Left-handedCOordiDBtesystem, 10,8692 polynomials,464-466 hge* Levi-Civita symbol, 11-92,639-641 Line distributions, singular, 119-121 Linearly dependent functions, 350,354 Lorentz t r a n s f d o n , 582585.592 Low-passMter, 292 Mapping, 138 conformal,18P-202 Riemann sheets, 532-534 Schwiuiz-CbristoEel 1!?6-202 Matrices, 3-5 array notation, Hemitian, 80,435 multiplication, 3-4 notation,3 trace, 16,610 transpose,5 unimodular,633 unitary, 633-634 Maxwell equations, 367-368 Metric tensol;569470,575,579,581, 588-592 Minkowski space, 591 Mhor test, forpseudovectors,653-654 Moment of inertia tensor, 78 Moments of a distribution, 121-125 Monopole, 122-123 Monopole moment, 122 Multiplication tabie, see Groups, multiplication table Multiple expansion, 121-125 Neighborhood, 142 Neumann boundary conditions, 429 INDEX Neumann functions Bessel equation solution, 444-446 spherical, 357 Neumann series, 501-503 Nonhomogeneousboundary conditions, see Boundary conditions, nonhomogeneous Nonhomogeneousdifferential equation, see Differential equations, nonhomogeneous Nonisotropic materials, 68 Normal modes of vibration, 624630 Notation, overview, 1-5 Nyquist sampling rate, 294 Ohm’s law, 67-70 Operators group representations, 606607 Hermitian, 436-440 linear differential, 341 SNrm-Liouvilk, 438439 Order of a hfferential equation, 341-342 of a group, 598 Orthogonality relations associated Legendre polynomials, 472 Bessel functions, 448450 exponential Fourier series, 228-229 Fourier series, 221-223 Fourier transform, 253-254 Laplace transform, 16 Legendre polynomials, 4655466 Outer product, see dyadic product Parity conservation, 90 Period, 220 Permutation group, 607 Permutations, even and odd, 11 Phasors, 148 Piecewise smooth function, 232,233 Point mass, 102-103, 114-1 16 Polar vector, 89 Pole, 169 Position vector, 44-45 cylindrical coordinates, 46 spherical coordinates, 49 Positive definite function, 436 Principal part, 184-188 Pseudo-objects, 86-92 pseudoscalar,90-9 pseudotensor,91-92 pseudovector, 11,86-90,653-654 Quadrapole, 133 Quadrapole moment, I22 Quadrature, method of, 358-366 677 Ratio test, 151 Rearrangement lemma,599 Reducible matrix representation, see Representations,group Relativity general, 565-566 special, 564-565.583-592 Removable singularity, 169 Representations, group, 600 completeness, 622-623 decomposition of, 622 equhlent vs inequivalent, 614 irreducible, 13-623 matrix notation, 12-6 14 operator, 606-607 orthogonality,6 17-62 reducible, 613-617 Residue theorem 171-175.510-513 Riemann integration, 125-126,256,327 Riemann sheets, 513-516.532-534 Right-hand rule, 10,22,87 Right-handed coordinate system, 10,46,48,53, 82,86-92 Rodrigues’s formula associated Legendre polynomials, 472 Legendre polynomials, 464 Rotation matrix, 6, 12-13 Saddle point, 545-547.554555 Samplmg theorem, 290-295 Scalar fields, 18-1 Scalar potential, 38.40 Scale factors, 49-5 Schwartz-Christoffelmapping, see Mapping, Schwartz-Christoffel Separation of variables first-order differential equations, 342-343 Laplace equation in Cartesian coordinates, 424-433 Laplace equation in cylindrical coordinates, 441457 Laplace equation in spherical coordinates, 458-475 Sequence functions, 104-105,107-108 Sheet distributions, singular, 116-119 Shur’s lemma, 623-630 Sifting intern, 106,112 Similarity aansform, 610 Sinc function, 107-108,268 Singularity circle, 156, 163-167 Skewed coordinates, 565-567,585-588 Solenoidal vector field, 40 Special relativity, 564-565.583-592 Speed of light, invariance, 564,585 678 Spherical coordinates, Spherical harmonics, 474475 steepestdescent,methodof, 542-555 Step function, see Heaviside step function Stirling’sapproximation,554 Stokes’s theorem, 37-38 String problem, 385-391 Stunn-Liouville form, 438-439 Subgroups,607-608 Subscript notation, Subscriphumation notation, 3,12-15.32-34 Subscript/supedpt notation, 573-575,581 Summation convention, Einstein, Superpositionprinciple, 341, 382 Symmetry operations, 599 Taylor series complex functions, 139-140,153-159 real functions, 152 Tensors, 6746,562-592 contravariant vs covariant, 579-581 coordinate transformations,76-78,84-86, 562-564.580 diagonalization, 78-84 metric, 569-570,575-579,581,58&592 in non-orthogonalcoordinates, 562-592 notation, 69-71 rank,71 INDEX Tokamak, 61 Toroidal coordinates, 61,62 Tour, 517 Trace, see Matrices, trace Transfer function, 293 Transformation matrix, 73-75,8546.570-574 Translational invariance, 378,398,498 Transpose, Unimodular, 633 Unitary matrix, 633-634 Vector fields, 18-20 Vector potential, 40 Vector/te.nsor notation, 70,7 I Vectors coordinate transfomations,71-76,90,570-573 identities, 14-15,17 notation, rotation of, 5-7, 12-13 Vierergruppe, see Groups, D2 Volterra equation fmt kind, 492499 second kind 492-493 Wave equation, 398-403 Weighting function, 436 Wronskian, 349-354 This Page Intentionally Left Blank This Page Intentionally Left Blank This Page Intentionally Left Blank This Page Intentionally Left Blank This Page Intentionally Left Blank This Page Intentionally Left Blank