www.TheSolutionManual.com www.TheSolutionManual.com www.TheSolutionManual.com Mathematical Physics with Partial Differential Equations www.TheSolutionManual.com This page intentionally left blank James R Kirkwood Sweet Briar College AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier www.TheSolutionManual.com Mathematical Physics with Partial Differential Equations Academic Press is an imprint of Elsevier 225 Wyman Street, Waltham, MA 02451, USA The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK © 2013 Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the Publisher Details on how to seek permission, further information about the Publisher’s permissions policies, and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher, nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein Library of Congress Cataloging-in-Publication Data James R Kirkwood Mathematical physics with partial differential equations / James Kirkwood p cm Includes bibliographical references and index ISBN 978-0-12-386911-1 (hardback) Mathematical physics Differential equations, Partial I Title QC20.7.D5K57 2013 530.14 dc23 2011028883 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library For information on all Academic Press publications visit our website at www.elsevierdirect.com Printed in the United States of America 12 13 14 15 10 www.TheSolutionManual.com This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein) Contents Preliminaries 1-1 Self-Adjoint Operators Fourier Coefficients Exercises 1-2 Curvilinear Coordinates Scaling Factors Volume Integrals The Gradient The Laplacian Spherical Coordinates Other Curvilinear Systems Applications An Alternate Approach (Optional) Exercises 1-3 Approximate Identities and the Dirac-δ Function Approximate Identities The Dirac-δ Function in Physics Some Calculus for the Dirac-δ Function The Dirac-δ Function in Curvilinear Coordinates Exercises 1-4 The Issue of Convergence Series of Real Numbers Convergence versus Absolute Convergence Series of Functions Power Series Taylor Series Exercises 1-5 Some Important Integration Formulas Other Facts We Will Use Later Another Important Integral Exercises Vector Calculus 2-1 Vector Integration Path Integrals Line Integrals Surfaces Parameterized Surfaces xi 1 11 14 17 18 22 23 25 25 31 33 33 34 35 37 40 42 44 45 45 47 48 54 56 60 64 68 69 70 73 73 74 77 80 82 v www.TheSolutionManual.com Preface vi Contents Green’s Functions 3-1 3-2 3-3 3-4 3-5 3-6 Introduction Construction of Green’s Function Using the Dirac-δ Function Exercises Construction of Green’s Function Using Variation of Parameters Exercises Construction of Green’s Function from Eigenfunctions Exercises More General Boundary Conditions Exercises The Fredholm Alternative (or, What If Is an Eigenvalue?) Exercises Green’s Function for the Laplacian in Higher Dimensions Exercises Fourier Series Introduction 4-1 Basic Definitions Exercises 4-2 Methods of Convergence of Fourier Series Fourier Series on Arbitrary Intervals Exercises 4-3 The Exponential Form of Fourier Series Exercises 83 85 91 93 94 97 100 104 104 109 114 122 122 127 135 140 141 142 148 155 155 156 164 164 168 168 171 171 173 173 180 180 186 187 187 188 191 193 203 204 206 207 www.TheSolutionManual.com Integrals of Scalar Functions Over Surfaces Surface Integrals of Vector Functions Exercises 2-2 Divergence and Curl Cartesian Coordinate Case Cylindrical Coordinate Case Spherical Coordinate Case The Curl The Curl in Cartesian Coordinates The Curl in Cylindrical Coordinates The Curl in Spherical Coordinates Exercises 2-3 Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem The Divergence (Gauss’) Theorem Stokes’ Theorem An Application of Stokes’ Theorem An Application of the Divergence Theorem Conservative Fields Exercises vii 4-4 Fourier Sine and Cosine Series Exercises 4-5 Double Fourier Series Exercise Three Important Equations 5-1 5-2 5-3 5-4 5-5 Introduction Laplace’s Equation Exercises Derivation of the Heat Equation in One Dimension Exercise Derivation of the Wave Equation in One Dimension Exercises An Explicit Solution of the Wave Equation Exercises Converting Second-Order PDEs to Standard Form Exercise Sturm-Liouville Theory Introduction Exercises 6-1 The Self-Adjoint Property of a Sturm-Liouville Equation Exercises 6-2 Completeness of Eigenfunctions for Sturm-Liouville Equations Exercises 6-3 Uniform Convergence of Fourier Series Separation of Variables in Cartesian Coordinates 7-1 7-2 7-3 7-4 7-5 7-6 7-7 Introduction Solving Laplace’s Equation on a Rectangle Exercises Laplace’s Equation on a Cube Exercises Solving the Wave Equation in One Dimension by Separation of Variables Exercises Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables Exercises Solving the Heat Equation in One Dimension Using Separation of Variables The Initial Condition Is the Dirac-δ Function Exercises Steady State of the Heat Equation Exercises Checking the Validity of the Solution 208 210 210 212 213 213 215 216 216 218 218 222 222 227 228 232 233 233 234 234 236 237 245 245 251 251 251 256 258 261 262 267 269 271 271 274 276 277 281 283 www.TheSolutionManual.com Contents 404 Mathematical Physics with Partial Differential Equations @θ 21 2y r sin θ sin θ ; 52 52 ÀyÁ2 Uy 2 y @x x x r r 11 x @z 0: @x Similarly, @r sin θ; @y @θ cos θ ; @y r @z 0; @y @r 0; @z @θ 0; @z @z 1: @z So @f @f @r @f @θ @f @z @f @f sin θ U U U cos θ : @x @r @x @θ @x @z @x @r @θ r Also, @f @f @f cos θ sin θ ; @y @r @θ r @f @f : @z @z @2 f We have @x2 @2 f @ @f @ @f @ @f sin θ 5 cos θ : @x2 @x @x @x @r @x @θ r We next compute Now @ @f @ @f @r @ @f @θ @ @f @z cos θ cos θ cos θ cos θ : @x @r @r @r @x @θ @r @x @z @r @x Consider So @ @f @2 f @f @ @2 f cos θ cos θ U cos θ cos θ: @r @r @r @r @r @r @ @f @r @2 f cos θ cos2 θ: @r @r @x @r www.TheSolutionManual.com and Appendix|Computing the Laplacian with the Chain Rule Now so 405 @ @f @2 f @f cos θ cos θ sin θ @θ @r @r@θ @r @ @f @θ @f @f sin θ cos θ cos θ sin θ @θ @r @x @r@θ @r r @2 f cos θ sin θ @f sin2 θ : @r@θ r @r r Also, @z 0: @x Thus, @ @f @ @f @r @ @f @θ @ @f @z cos θ cos θ cos θ cos θ @x @r @r @r @x @θ @r @x @z @r @x @ @f @r @ @f @θ cos θ cos θ @r @r @x @θ @r @x ! 2 @f @ f cos θ sin θ @f sin2 θ : cos θ @r @r@θ r @r r ð2Þ Similarly, @ @f sin θ @ @f sin θ @r @ @f sin θ @θ @ @f sin θ @z 1 @x @θ r @r @θ r @x @θ @θ r @x @z @θ r @x @ @f sin θ @r @ @f sin θ @θ : @r @θ r @x @θ @θ r @x We have so Likewise, @ @f sin θ @2 f sin θ @f sin θ U @r @θ r @r@θ r @θ r @ @f sin θ @r @2 f sin θ cos θ @f sin θ cos θ : U @r @θ r dx @r@θ r @θ r2 @ @f sin θ @2 f sin θ @f cos θ 2U U @θ @θ r r @θ r @θ www.TheSolutionManual.com 52 406 Mathematical Physics with Partial Differential Equations so @ @f sin θ @θ @ f sin θ @f cos θ sin θ U U @θ @θ r @x r @θ r r @θ2 Thus, @2 f sin2 θ @f sin θ cos θ : @θ r2 @θ2 r @ @f sin θ @2 f sin θ cos θ @f sin θ cos θ U @x @θ r @r@θ r @θ r2 ð3Þ @2 f sin2 θ @f sin θ cos θ 2 2 : @θ r2 @θ r To obtain @2 f , we subtract equation (3) from equation (2) to get @x2 @2 f @ @f @ @f sin θ cos θ @x2 @x @r @x @θ r ! @2 f @2 f cos θ sin θ @f sin2 θ cos θ @r @r@θ r @r r @2 f sin θ cos θ @f sin θ cos θ @2 f sin2 θ @f sin θ cos θ 2 2 U @r@θ r @θ r2 @θ r2 @θ r @2 f @2 f cos θ sin θ @2 f sin2 θ @f sin θ cos θ @f sin2 θ cos θ 22 1 : @r @r@θ r @θ r2 @r r @θ2 r ð4Þ Likewise, we must calculate @2 f We have @y2 @f @f @f cos θ sin θ @y @r @θ r so Now ! @2 f @ @f @ @f cos θ sin θ : @y2 @y @r @y @θ r @ @f @ @f @r @ @f @θ sin θ sin θ sin θ : @y @r @r @r @y @θ @r @y www.TheSolutionManual.com 52 Appendix|Computing the Laplacian with the Chain Rule 407 We have @ @f @r @2 f sin θ sin θUsin θ @r @r @y @r and ! @ @f @θ @2 f @f cos θ sin θ sin θ cos θ : @θ @r @y @θ@r @r r @ @f @2 f @2 f sin θ cos θ @f cos2 θ sin θ sin2 θ 1 : @y @r @r @θ@r r @r r ð5Þ Also, @ @f cos θ @ @f cos θ @r @ @f cos θ @θ : @y @θ r @r @θ r @y @θ @θ r @y Now ! @ @f cos θ @r @2 f cos θ @f cos θ sin θ @r @θ r @y @r@θ r @θ r @2 f cos θ sin θ @f cos θ sin θ @r@θ r @θ r2 ð6Þ and ! @ @f cos θ @θ @2 f cos θ @f sin θ cos θ @θ @θ r @y @θ r r @θ2 r 2 @ f cos θ @f cos θ sin θ 2 : @θ r2 @θ r Adding equations (6) and (7) gives ! @ @f cos θ @2 f cos θ sin θ @f cos θ sin θ @y @θ r @r@θ r @θ r2 ! @2 f cos2 θ @f cos θ sin θ @θ r2 @θ2 r @2 f cos θ sin θ @f cos θ sin θ @2 f cos2 θ 22 : @r@θ r @θ r2 @θ2 r ð7Þ ð8Þ www.TheSolutionManual.com So 408 Mathematical Physics with Partial Differential Equations Adding (5) and (8) yields @2 f @ @f @ @f cos θ sin θ @y2 @y @r @y @θ r @2 f @2 f sin θ cos θ @f cos2 θ sin θ 1 @r @θ@r r @r r ! Adding the expressions for @2 f @2 f and gives @x2 @y2 & @2 f @2 f @ f @2 f cos θ sin θ @2 f sin2 θ 1 cos θ 2 @x2 @y2 @r @r@θ r @θ2 r ' @f sin θ cos θ @f sin2 θ 12 @θ r2 @r r & ! @f @2 f sin θ cos θ @f cos2 θ sin θ 1 @r @θ@r r @r r @2 f cos θ sin θ @f cos θ sin θ @2 f cos2 θ 22 @r@θ r @θ r2 @θ2 r !' @2 f @2 f @f sin2 θ cos2 θ sin2 θ @r @r @r r 5 @f cos2 θ @2 f sin2 θ @2 f cos2 θ 2 2 @r r @θ r @θ r @2 f @f @2 f 2: @r r @r r @θ Thus, the Laplacian in cylindrical coordinates is given by Δf @2 f @f @2 f @2 f 1 2 2: @r r @r r @θ @z ð9Þ THE LAPLACIAN IN SPHERICAL COORDINATES To compute the Laplacian in spherical coordinates, we use the variables r, θ, φ, where θ and φ are as shown if Figure 1, and r2 x2 y2 z2 We have www.TheSolutionManual.com ! @2 f cos θ sin θ @f cos θ sin θ @2 f cos2 θ 22 1 2 : @r@θ r @θ r2 @θ r 409 Appendix|Computing the Laplacian with the Chain Rule z (r, θ, φ) θ r y x FIGURE 1-A x r sin θ cos φ; y r sin θ sin φ; z r cos θ: We will adapt some of our computations from cylindrical coordinates In doing so we must be careful, because the standard representation of r in spherical coordinates is not the same r as in cylindrical coordinates If we take ρ in spherical coordinates to be ρ x2 y2 ; then ρ corresponds to r in cylindrical coordinates Also, φ in spherical coordinates was θ in cylindrical coordinates We will also use the subscript notation for partial derivatives Adapting what we found in cylindrical coordinates to spherical coordinate notation, and using u to stand for the function instead of f, we have uxx uyy uρρ 1 uρ uφφ : ρ ρ ð1Þ We need to convert from the variable ρ to the standard spherical coordinates of r, θ, and φ A key observation is that z and ρ in spherical coordinates are obtained from r and θ by the same functions that give x and y from ρ and φ Namely, x ρ cos φ and y ρ sin φ; z r cos θ ρ r sin θ: and and www.TheSolutionManual.com φ 410 Mathematical Physics with Partial Differential Equations The last relationship follows from ρ2 x2 y2 ðr sin θ cos φÞ2 ðr sin θ sin φÞ2 ðr sin θÞ2 ðcos2 φ sin2 φÞ ðr sin θÞ2 so ρ r sin θ: uxx uyy uρρ 1 uρ uφφ ; ρ ρ we have in spherical coordinates uzz uρρ urr 1 ur uθθ : r r ð2Þ We want to compute uxx uyy uzz Adding equations (1) and (2) gives uxx uyy uzz uρρ uρρ 1 1 uρ uφφ urr ur uθθ ρ ρ r r so uxx uyy uzz 1 1 uρ uφφ urr ur uθθ : ρ ρ r r ð3Þ To convert equation (3) to the desired form, we must eliminate the variable ρ If we can express uρ in terms of r, θ, and φ, the rest will be easy The chain rule gives @u @u @r @u @θ @u @φ U U U : @ρ @r @ρ @θ @ρ @φ @ρ Now r ðρ2 z2 Þ2 ; θ tan21 ρz ; φ φ, so @r ρ ρ 5 ; @ρ ðρ2 z2 Þ12 r @θ @ρ 11 1 z z r cos θ cos θ 25 ; 2 U 2 z z 1ρ r r2 r ρ z @φ 0: @ρ www.TheSolutionManual.com So if f(a, b) a cos b and g(a, b) a sin b, then x f(ρ, φ), z f(r, θ) and y g(ρ, φ), ρ g(r, θ) This correspondence means that since we have in cylindrical coordinates Appendix|Computing the Laplacian with the Chain Rule 411 Thus, @u @u ρ @u cos θ @u U U U0: @ρ @r r @θ r @φ Or, converting to subscript notation, uρ u r ρ cos θ uθ : r r Substituting into equation (3) gives 1 1 uρ uφφ urr ur uθθ ρ ρ r r ρ cos θ 1 ur uθ uφφ urr ur uθθ ρ r r ρ r r ur cos θ 1 1 uθ uφφ urr ur uθθ r ρ r ρ r r cos θ 1 uφφ urr uθθ : ur u θ r ρ r ρ r Substituting r sin θ for ρ gives uxx uyy uzz cos θ 1 uφφ uθθ urr ; ur u θ r r sin θ r r ðr sin θÞ which may be rearranged to yield Δu urr 1 ur uθθ cot θ uθ u φφ : r r sin2 θ We note that other sources may express this formula in a slightly different way www.TheSolutionManual.com uxx uyy uzz www.TheSolutionManual.com This page intentionally left blank Apostol, T.M., 1974 Mathematical Analysis Second ed Addison-Wesley Publishing Company, Reading, MA Arfken, G., 1970 Mathematical Methods for Physicists Second ed Academic Press, New York Boas, M.L., 1966 Mathematical Methods in the Physical Sciences John Wiley & Sons, Inc., New York Boyce, W., DiPrima, R., 2008 Elementary Differential Equations and Boundary Value Problems John Wiley & Sons, Inc., New York Brown, J.W., Churchill, R.V., 2008 Fourier Series and Boundary Value Problems McGraw-Hill Book Company, New York Courant, R., and Hilbert, D., 1989 Methods of Mathematical Physics, Vol John Wiley & Sons, Inc., New York Edwards, C., Penny, D., 1994 Elementary Differential Equations with Applications Third ed Prentice-Hall, Englewood Cliffs, NJ Gelbaum, B., Olmsted, J., 1964 Counterexamples in Analysis Holden Day, San Francisco Growers, T (Ed.), 2008 The Princeton Companion to Mathematics Princeton University Press, Princeton, NJ Hoskins, R.F., 1979 Generalized Functions Halsted Press, New York Kirkwood, J.R., 1995 An Introduction to Analysis Second ed Waveland Press, Inc., Prospect Heights, IL Kreysig, E., 1967 Advanced Engineering Mathematics Second ed John Wiley & Sons, Inc., New York Marsden, J.E., Tromba, A.J., 1988 Vector Calculus Third ed W.H Freeman and Company, New York McQuarrie, D., 2003 Mathematical Methods for Scientists and Engineers University Science Books, Sausalito, CA Morse, P.M., Feshbach, H., 1953 Methods of Theoretical Physics McGraw-Hill Book Company, New York Park, D., 1964 Introduction to the Quantum Theory McGraw-Hill Book Company, New York Pinsky, M.A., 1998 Partial Differential Equations and Boundary-Value Problems with Applications Third ed Waveland Press, Inc., Prospect Heights, IL Rogawski, J., 2008 Calculus W.H Freeman and Company, New York Rudin, W., 1964 Principles of Mathematical Analysis McGraw-Hill Book Company, New York Rudin, W., 1973 Functional Analysis McGraw-Hill Book Company, New York Rudin, W., 1987 Real and Complex Analysis Third ed McGraw-Hill Book Company, New York Schey, H.M., 1973 Div, Grad, Curl and All That: An Informal Text on Vector Calculus Norton, New York Stakgold, I., 1967 Boundary Problems in Mathematical Physics Macmillan, New York Stakgold, I., 1979 Green’s Functions and Boundary Value Problems John Wiley & Sons, Inc., New York Weinberger, H.F., 1965 A First Course in Partial Differential Equations John Wiley & Sons, Inc., New York 413 www.TheSolutionManual.com References www.TheSolutionManual.com This page intentionally left blank A Absolute convergence, 47À48 Alternating series test, 48 Ampere’s law, 140 Analytic function, 57, 215 Approximate identities, 35À37 and Dirac-δ Function, 34À45 examples of, 44À45 Associated Legendre function, 319À322 B Bessel equation, 161, 166, 289, 291 cylindrical coordinates, 291, 292À295 of half-integer order, 311 order 0, 305 second solution, 291 in spherical coordinates, 310À315 Bessel function, 58, 287À291 cylindrical coordinates, 292À295 of first kind of order, 291 modified (with imaginary argument), 291 Bessel’s inequality, 7, 190À191, 198, 240 Bipolar coordinates, 29 C Calculus for Dirac-δ Function, 40À42 Cartesian coordinates, 14, 23, 73, 127 curl in, 104À109 cylindrical case, 15 general case, 15 separation of variables, 251À286 Cauchy criterion for uniform convergence of sequences of functions, 52 for uniform convergence of series of functions, 53 Cauchy Integral Formula, 66 Cauchy sequence, 46 Cesaro sum, 199 Chain rule, 403 Change of variables (integration), 34 Circulation, 104 Comparison test, 47 Completeness of eigenfunctions, 202, 237À245 Conservative fields, 142À148 exercises, 153 Continuity of limit function, 48 Convergence, 45À63, 187 absolute, 47À48 conditional, 48 functions, 48À54 pointwise, 48, 187 uniform, 49, 187 L2 convergence, 187 numbers, 45À47 Converting second-order PDEs to standard form, 228À232 Convolution of functions, 47 Fourier transform, 332 Laplace transform, 354 Convolution theorem, 332 Coordinates bipolar, 26À27 Cartesian coordinates, 14 curvilinear, 14À34 cylindrical, 14 elliptic, 26 oblate spheroidal, 26À27 parabolic, 26À27 parabolic cylindrical, 28 prolate spheroidal, 26À27 spherical, 25 Cosine series and Fourier series, 208À210 Coulomb’s law, 141 Coupled spring system, Curl, 93À122 Cartesian coordinates, 104À109 cylindrical coordinates, 109À114 spherical coordinates, 114À121 Curvilinear coordinates, 14À34, 104À109 Curvilinear Systems, 25À30 Cylindrical coordinates, 15, 109À114, 292À299, 403À408 D Del operator, 97 Dirac-δ function, 37À45, 274À275 calculus for, 40À42 415 www.TheSolutionManual.com Index 416 E Eigenfunctions, completeness of, 237 Eigenvalue, Eigenvector, Elliptic cylindrical coordinates, 26 Elliptic equation, 213À214 Equilibrium (steady) state of the heat equation, 277À283 F Fejer kernel, 199 Fejer’s Theorem, 200 Flux, 85 Fourier Coefficients, 5À11 Fourier inversion theorem, 332 Fourier series, 187À212, 245À250 on arbitrary intervals, 203À204 convergence methods, 193À205 definitions, 188À192 double, 210À212 eigenfunctions, 188 exponential form of, 206À207 pointwise convergence, 187 sine and cosine series, 208À210 uniform convergence, 187, 193, 245À250 Fourier transform, 327À350 as decomposition, 328À329 from Fourier series, 329À331 fundamental solution of, 336 of Gaussian distribution, 64, 331À334 heat equation, solving, 335 Laplace’s Equation, solving, 339 negative Laplacian spectrum, in one dimension, 343À346 partial differential equations, solving, 335À343 properties of, 331À335 in three dimensions, 346À350 wave equation, solving, 337 Fourier’s law, 217 Fredholm Alternative, 137À140 G Gamma function, 71 Gaussian functions, 331À334 Gauss’s law for inverse square fields, 133, 142 Gauss’ Theorem See Divergence theorem Geometric series, 46 Gradient, 22 cartesian coordinates, 22 cylindrical coordinates, 22 Green’ function, 155À186 for Bessel’s equation, 161, 166 and Dirac-δ function, 156À164 eigenfunction form, 168À171 general boundary conditions, 171À173 for heat equation, 375À379 for the Laplacian, 180À186 for Poisson’s equation, 398À401 for wave equation, 390À397 variation of parameters, 164À168 Green’s identities, 140, 149À150 Green’s theorem, 122À153 H Hamiltonian operator, Heat equation, 213 derivation, in one dimension, 216À218 on disk, 303À306 Fourier transform, 335 fundamental solution of, 336 Green’s function, 375À379 with heat source, 271À277 Laplace transform, 361À367 method of images, 379À389 nonhomogeneous, 377À378 in one dimension, 271À277 steady state of, 277À283 Heaviside function, 37, 371, 396 Hyperbolic equation, 213À214 www.TheSolutionManual.com Dirac-δ function (Continued) in curvilinear coordinates, 42À44 in Fourier series, 274À275 Green’s function, construction, 156À164 Directional derivative, 74 Dirichlet kernel, 193 Divergence, 93À122 Cartesian coordinate case, 94À97 cylindrical coordinate case, 97À100 spherical coordinate case, 100À104 Divergence theorem, 122À153, 180 application of, 141À142 exercises, 152À153 for inverse square fields, 133 three dimensions, 131 in two dimensions, 129 Dot product See inner product Double Fourier series, 210À212 Drum head problem, 299À303 D’Alembert’s formula, 226, 373 Index 417 Index Increments of volume, 18 Inner product, Integrals, 73À93 line, 77À80 parameterized surfaces, 82À83 path, 74À77 of scalar functions over surfaces, 83À85, 93 surface, 80À81, 85À91, 93 Integral test, 47 Integration formulas, 64À71 Inverse square field, 133 J Jacobian, 19 Jordan’s Lemma, 349 Jump condition on Green’s function, 158 K Kernel, 14 L Laplace transform, 351À374 of convolution of functions, 352À356 heat equation, 361À367 properties and formulas for, 355t solving differential equations, 356À361 using Fourier transform to solve, 339 wave equation, 368À373 Laplace’s equation, 31À33, 213, 215À216 on cube, 258À262 in cylindrical coordinates, 295À299 on rectangle, 251À258 spherical coordinates, 307, 322À325 Laplacian, 23À24, 134, 214 computing, 403 cylindrical coordinates, 23 Green’s function, 180À186 spherical coordinates, 25, 408À411 Legendre equation, 307À310, 315À319 associated Legendre function, 319À322 Legendre polynomial, 317 Linear function See Linear operator Linear operator, self-adjoint, 1À14 Line integrals, 77À80 Method of images, 379À389 on bounded interval, 383À389 on semi-infinite interval, 379À383 N Newton’s law, 3, 220 O Oblate spheroidal coordinates, 26À27 Ordinary differential equations (ODEs), 31À33, 187, 233, 287, 356 Orthogonal basis, coordinates, 15 set, transformation, 13À14 vectors, 12 Orthogonal coordinate system, 17 Orthonormal basis, set, P Parabolic coordinates, 28 Parabolic cylindrical coordinates, 28 Parabolic equation, 213À214 Parameterized surfaces, 82À83 Parseval’s formula, 202 Parseval’s Theorem, 333 Path integrals, 74À77 exercises, 92 Poisson’s equation, 398À401 Poisson’s integral formula, 202, 341 Polar coordinates, 303 Power Series, 54À55 Power Series Expansion, 64 Principle of Superposition, 11 Prolate spheroidal coordinates, 26À27 p-Series Test, 47 R Radius of convergence, 55 Ratio test, 47 Real numbers, series of, 45À47 Resolvent, 343 Riemann-Lebesgue Lemma, 191 Root test, 47 M S Maclaurin series, 57 Maximum modulus principle, 215 Scaling factors, 17À18 Schwarz inequality, 197, 198 www.TheSolutionManual.com I 418 T Taylor polynomial, 56 Taylor Series, 56À60 Telegraph equation, 268 Trigonometric polynomials, 189 U Uniform convergence, 49, 52 of Fourier series, 245À250 V Variation of parameters, 164À168 Vector calculus, 73À154 integrals, 73À93 line, 77À80 parameterized surfaces, 82À83 path, 74À77 of scalar functions over surfaces, 83À85 surface, 80À81, 85À91 Vector field, 77 Volume integrals, 18À21 W Wave equation, 213, 299À303 derivation, in one dimension, 218À222 on disk, 299À303 explicit solution of, 222À228 Fourier transform, 337 Green’s function, 390À397 Laplace transform, 339, 368À373 in one dimension, 218À222, 262À268 separation of variables, 262À268 spherical coordinates, 299À303 in two dimension, 269À271 Weierstrass M-test, 54, 284 Weight function, 3, 235, 294 www.TheSolutionManual.com Self-adjoint operator, 1À14, 188À192, 234À236 Separation of variables in Cartesian coordinates, 251À286 heat equation, 271À283 Laplace’s equation, 251À262 wave equation, 262À271 in cylindrical coordinates, 287À306 Bessel functions, 287À291 Bessel’s equation, 292À295 heat equation, on disk, 303À306 Laplace’s equation, 295À299 wave equation, on disk, 299À303 Sequences functions, 48 numbers, 45 Series cosine series, 208À210 of functions, 48À54 power series, 54À55 of real numbers, 45À47 sine series, 208À210 Taylor series, 56À60 Spectrum, 343À346 Spherical coordinates, 25 Bessel’s equation in, 310À315 curl in, 114À121 Laplace’s equation in, 322À325 Laplacian in, 408À411 Steady state (heat equation), 277À283 Stokes’ theorem, 122À153 application of, 140À141 exercises, 149À152 Stone’s Theorem, Sturm-Liouville Theory, 233À250, 294 eigenfunctions completeness for, 237À245 self-adjoint property of, 234À237 uniform convergence of Fourier series, 245À250 Surface, 80À81 parameterized, 17À18, 82À83 Surface integral, 80À81, 85À91, 93 Index ... Data James R Kirkwood Mathematical physics with partial differential equations / James Kirkwood p cm Includes bibliographical references and index ISBN 978-0-12-386911-1 (hardback) Mathematical physics. .. dx2 Mathematical Physics with Partial Differential Equations © 2013 Elsevier Inc All rights reserved www.TheSolutionManual.com Preliminaries Mathematical Physics with Partial Differential Equations. .. www.TheSolutionManual.com www.TheSolutionManual.com Mathematical Physics with Partial Differential Equations www.TheSolutionManual.com This page intentionally left blank James R Kirkwood Sweet Briar College AMSTERDAM