ADVANCE D ENGINEERIN G MATHEMATIC S International Student Editio n PETER V O'NEIL University of Alabam a at Birmingham Al* Ea -14gg Rift Wt * 47 THOMSON TTY tt n.:0935-33786 mail;hhchouCtunghuaconitw Australia Canada Mexico Singapore Spain United Kingdom United States THOMSO N Advanced Engineering Mathematics, International Student Editio n by Peter V O'Neil Associate Vice-President and Editorial Director: Evelyn Veitch Publisher: Chris Carson Developmental Editor : Kamilah Reid Bur ell/ Hilda Gowaus Production Services: RPK Editorial Services Creative Director: Angela Cluer Copy Editor : Shelly Gerger-Knechtl / Harlan James Interior Design: Terri Wrigh t Proofreader: Erin Wagner/Harlan James Cover Design : Andrew Adams Indexer: RPK Editorial Services Compositor: Integra Permissions Coordinator: Vicki Goul d Production Manager: Renate McCloy Printer: Pu Hsin Wang COPYRIGHT © 2007 by Nelson , a division of Thomson Canad a Limited ALL RIGHTS RESERVED No part of this work covered by the copyright herein may b e reproduced, transcribed, or use d in any form or by any means-graphic, electronic, o r mechanical, including photocopying, recording, taping , Web distribution, or informatio n storage and retrieval systems-without the written permission of the publisher North America Nelso n 1120 Birchmount Road Toronto, Ontario M1K 5G4 Canad a Printed and bound in Taiwan 07 06 For more information contac t Nelson, 1120 Birchmount Road, Toronto, Ontario, Canada, M1K 5G4 Or you can visit our Internet site at http ://www.nelson com Library of Congress Contro l Number: 2006900028 ISBN : 495.08237.6 If you purchased this book within the United States or Canada yo u should be aware that it has been wrongfully imported without the approval of the Publisher or th e Author For permission to use material from this text or product, submit a request online at www.thomsonrights.com Every effort has been made t o trace ownership of all copyright material and to secure permission from copyright holders In th e event of any question arising as to the use of any material, we will be pleased to make the necessary corrections in future printings Asi a Thomson Learnin g Shenton Way #01-0 UIC Building Singapore 06880 Australia/New Zealand Thomson Learning 102 Dodds Street Southbank, Victori a Australia 300 Europe/Middle East/Africa Thomson Learning High Holborn House 50/51 Bedford Row London WC1R 4LR United Kingdom Latin Americ a Thomson Learning Seneca, 53 Colonia Polanco 11560 Mexico D.F Mexico Spain Paraninfo Calle/Magallanes, 28015 Madrid, Spain Contents PART Chapter 1 Ordinary Differential Equations First-Order Differential Equations 1.1 Preliminary Concepts 1 General and Particular Solutions 1 Implicitly Defined Solutions 1 Integral Curves 1 The Initial Value Problem 1.1 Direction Fields Separable Equations 1 2.1 Some Applications of Separable Differential Equations Linear Differential Equations 2 Exact Differential Equations 1.5 Integrating Factors 3 Separable Equations and Integrating Factors Linear Equations and Integrating Factors 1.6 Homogeneous, Bernoulli, and Riccati Equations 1.6.1 Homogeneous Differential Equations 6.2 The Bernoulli Equation 6.3 The Riccati Equation Applications to Mechanics, Electrical Circuits, and Orthogonal Trajectories 46 7.1 Mechanics 7.2 Electrical Circuits 1 7.3 Orthogonal Trajectories 1.8 Existence and Uniqueness for Solutions of Initial Value Problems Chapter Second-Order Differential Equations 2.1 Preliminary Concepts 2.2 Theory of Solutions of y" + p(x)y' + q(x)y = f(x) 62 2.2.1 The Homogeneous Equation y" + p(x)y' + q(x) = 64 2.2.2 The Nonhomogeneous Equation y" + p(x)y' + q(x)y = f(x) 2.3 Reduction of Order The Constant Coefficient Homogeneous Linear Equation 2.4.1 Case : A2 - 4B > 2.4.2 Case : Az - 4B = 74 68 Contents 2.4.3 Case : A2 - 4B < 74 2.4.4 An Alternative General Solution in the Complex Root Case 2.5 Euler' s Equation 2.6 The Nonhomogeneous Equation y" + p(x)y' + q(x)y = f(x) 82 2.6.1 The Method of Variation of Parameters 2.6.2 The Method of Undetermined Coefficients 2.6.3 The Principle of Superposition 2.6.4 Higher-Order Differential Equations 2.7 Application of Second-Order Differential Equations to a Mechanical System 2.7.1 Unforced Motion 2.7.2 Forced Motion 98 2.7.3 Resonance 10 Beats 10 2 Analogy with an Electrical Circuit 10 Chapter The Laplace Transform 107 3.1 Definition and Basic Properties 10 3.2 Solution of Initial Value Problems Using the Laplace Transform 11 3.3 Shifting Theorems and the Heaviside Function 120 3.3.1 The First Shifting Theorem 120 3.3.2 The Heaviside Function and Pulses 12 3 The Second Shifting Theorem 125 3 Analysis of Electrical Circuits 12 3.4 Convolution 13 Unit Impulses and the Dirac Delta Function 13 Laplace Transform Solution of Systems 144 Differential Equations with Polynomial Coefficients 150 Chapter Series Solutions 155 4.1 Power Series Solutions of Initial Value Problems 15 Power Series Solutions Using Recurrence Relations 16 4.3 Singular Points and the Method of Frobenius 16 4.4 Second Solutions and Logarithm Factors 17 Chapter Numerical Approximation of Solutions 18 Euler's Method 18 5.1 A Problem in Radioactive Waste Disposal 187 One-Step Methods 190 5.2 The Second-Order Taylor Method 19 5.2 The Modified Euler Method 19 5.2 Runge-Kutta Methods 195 Multistep Methods 197 5.3 Case r = 19 5.3 Case r = 19 5.3 Case r = 19 5.3 Case r = 199 Contents PART Chapter Vectors and Linear Algebra 20 Vectors and Vector Spaces 203 6.1 The Algebra and Geometry of Vectors 203 6.2 The Dot Product 21 6.3 The Cross Product 21 6.4 The Vector Space R" 223 6.5 Linear Independence, Spanning Sets, and Dimension in R" 22 Chapter Matrices and Systems of Linear Equations 23 7.1 Matrices 23 1 Matrix Algebra 239 7.1 Matrix Notation for Systems of Linear Equations 242 Some Special Matrices 243 7.1 Another Rationale for the Definition of Matrix Multiplicatio n 24 7.1 Random Walks in Crystals 247 7.2 Elementary Row Operations and Elementary Matrices 25 7.3 The Row Echelon Form of a Matrix 258 The Row and Column Spaces of a Matrix and Rank of a Matrix 266 7.5 Solution of Homogeneous Systems of Linear Equations 272 7.6 The Solution Space of AX = 280 7 Nonhomogeneous Systems of Linear Equations 283 7.7.1 The Structure of Solutions of AX = B 284 7.7.2 Existence and Uniqueness of Solutions of AX = B 285 Matrix Inverses 293 7.8 A Method for Finding A- 295 Chapter Determinants 29 8.1 Permutations 29 8.2 Definition of the Determinant 30 8.3 Properties of Determinants 30 Evaluation of Determinants by Elementary Row and Column Operations 30 8.5 Cofactor Expansions 31 8.6 Determinants of Triangular Matrices 31 8.7 A Determinant Formula for a Matrix Inverse 31 8.8 Cramer' s Rule 31 8.9 The Matrix Tree Theorem 320 Chapter Eigenvalues, Diagonalization, and Special Matrices 32 Eigenvalues and Eigenvectors 324 9.1 Gerschgorin's Theorem 328 ' 9.2 Diagonalization of Matrices 33 Orthogonal and Symmetric Matrices 33 viii Contents Quadratic Forms 347 Unitary, Hermitian, and Skew Hermitian Matrices 35 PART Chapter 10 Systems of Differential Equations and Qualitative Methods 35 Systems of Linear Differential Equations 36 10 Theory of Systems of Linear First-Order Differential Equations 36 10.1 Theory of the Homogeneous System X' = AX 365 10.1 General Solution of the Nonhomogeneous System X' = AX + G 37 10 Solution of X' = AX when A is Constant 374 10.2.1 Solution of X' = AX when A has Complex Eigenvalues 37 10.2.2 Solution of X' = AX when A does not have n Linearly Independent Eigenvectors 379 10.2 Solution of X' = AX by Diagonalizing A 384 10.2 Exponential Matrix Solutions of X' = AX 38 10.3 Solution of X' = AX + G 39 10.3 Variation of Parameters 39 10.3 Solution of X' = AX + G by Diagonalizing A 398 Chapter 11 Qualitative Methods and Systems of Nonlinear Differential Equations 40 11 Nonlinear Systems and Existence of Solutions 40 11 The Phase Plane, Phase Portraits and Direction Fields 40 11 Phase Portraits of Linear Systems 41 11 Critical Points and Stability 42 11 Almost Linear Systems 43 11 Lyapunov ' s Stability Criteria 45 11 Limit Cycles and Periodic Solutions 46 PART Chapter 12 Vector Analysis 47 Vector Differential Calculus 47 12 Vector Functions of One Variable 47 12 Velocity, Acceleration, Curvature and Torsion 48 12 2.1 Tangential and Normal Components of Acceleration 48 12 2.2 Curvature as a Function of t 49 12.2 The Frenet Formulas 492 12 Vector Fields and Streamlines 49 12 The Gradient Field and Directional Derivatives 49 12.4 Level Surfaces, Tangent Planes and Normal Lines 50 12 Divergence and Curl 51 12.5 A Physical Interpretation of Divergence 51 12.5 A Physical Interpretation of Curl 513 Contents Chapter 13 Vector Integral Calculus 51 13 Line Integrals 51 13.1 Line Integral with Respect to Arc Length 525 13 Green' s Theorem 52 13.2 An Extension of Green's Theorem 53 13 Independence of Path and Potential Theory in the Plane 53 13.3 A More,Critical Look at Theorem 13 539 13 Surfaces in 3-Space and Surface Integrals 54 13.4.1 Normal Vector to a Surface 54 13 4.2 The Tangent Plane to a Surface 55 13 Smooth and Piecewise Smooth Surfaces 55 13.4 Surface Integrals 55 13 Applications of Surface Integrals 55 13.5 Surface Area 557 13.5 Mass and Center of Mass of a Shell 55 13 5.3 Flux of a Vector Field Across a Surface 56 13.6 Preparation for the Integral Theorems of Gauss and Stokes 56 13.7 The Divergence Theorem of Gauss 564 13.7 Archimedes 's Principle 567 13.7 The Heat Equation 568 13.7 The Divergence Theorem as a Conservation of Mass Principle 570 13.8 The Integral Theorem of Stokes 572 13 8.1 An Interpretation of Curl 57 13 Potential Theory in 3-Space 57 PART Chapter 14 Fourier Analysis, Orthogonal Expansions, and Wavelets 58 Fourier Series 583 14.1 Why Fourier Series? 58 14.2 The Fourier Series of a Function 58 14.2.1 Even and Odd Functions 589 14.3 Convergence of Fourier Series 59 14.3.1 Convergence at the End Points 59 14.3 A Second Convergence Theorem 60 14.3 Partial Sums of Fourier Series 604 14.3 The Gibbs Phenomenon 606 14.4 Fourier Cosine and Sine Series 609 14.4 The Fourier Cosine Series of a Function 61 14.4 The Fourier Sine Series of a Function 61 14.5 Integration and Differentiation of Fourier Series 61 14.6 The Phase Angle Form of a Fourier Series 62 14.7 Complex Fourier Series and the Frequency Spectrum 63 14.7.1 Review of Complex Numbers 63 14.7.2 Complex Fourier Series 631 x Contents Chapter 15 The Fourier Integral and Fourier Transforms 63 15 The Fourier Integral 637 15 Fourier Cosine and Sine Integrals 64 15 The Complex Fourier Integral and the Fourier Transform 64 15 Additional Properties and Applications of the Fourier Transform 65 15.4.1 The Fourier Transform of a Derivative 65 15.4.2 Frequency Differentiation 65 15.4 The Fourier Transform of an Integral 65 15.4 Convolution 65 15.4 Filtering and the Dirac Delta Function 66 15.4 The Windowed Fourier Transform 66 15.4 The Shannon Sampling Theorem 66 15.4 Lowpass and Bandpass Filters 66 15 The Fourier Cosine and Sine Transforms 67 15 The Finite Fourier Cosine and Sine Transforms 67 15.7 The Discrete Fourier Transform 67 15 7.1 Linearity and Periodicity 67 15 The Inverse N-Point DFT 67 15 DFT Approximation of Fourier Coefficients 67 15.8 Sampled Fourier Series 68 15 8.1 Approximation of a Fourier Transform by an N-Point DFT 685 15 8.2 Filtering 68 15 The Fast Fourier Transform 69 15 Use of the FFT in Analyzing Power Spectral Densities of Signals 69 15.9.2 Filtering Noise From a Signal 69 15.9.3 Analysis of the Tides in Morro Bay 69 Chapter 16 Special Functions, Orthogonal Expansions, and Wavelets 70 16.1 Legendre Polynomials 70 16.1 A Generating Function for the Legendre Polynomials 704 16.1 A Recurrence Relation for the Legendre Polynomials 70 16.1.3 Orthogonality of the Legendre Polynomials 70 16.1 Fourier-Legendre Series 709 16.1 Computation of Fourier-Legendre Coefficients 71 16.1 Zeros of the Legendre Polynomials 71 16.1 Derivative and Integral Formulas for Pn(x) 715 16.2 Bessel Functions 71 16.2.1 The Gamma Function 71 16.2.2 Bessel Functions of the First Kind and Solutions of Bessel 's Equation 72 16.2.3 Bessel Functions of the Second Kind 72 16.2.4 Modified Bessel Functions 72 16.2.5 Some Applications of Bessel Functions 72 16.2.6 A Generating Function for L(x) 732 16.2.7 An Integral Formula for L(x) 733 16.2.8 A Recurrence Relation for Jv (x) 735 16.2.9 Zeros of Jv (x) 737 Contents 16.2 10 Fourier-Bessel Expansions 73 16.2 11 Fourier-Bessel Coefficients 74 16.3 Sturm-Liouville Theory and Eigenfunction Expansions 74 16.3.1 The Sturm-Liouville Problem 74 16.3 The Sturm-Liouville Theorem 75 16.3 Eigenfunction Expansions 75 16.3 Approximation in the Mean and Bessel ' s Inequality 75 16.3 Convergence in the Mean and Parseval 's Theorem 76 16.3.6 Completeness of the Eigenfunctions 76 16 Wavelets 765 16.4.1 The Idea Behind Wavelets 765 16.4 The Haar Wavelets 76 16.4 A Wavelet Expansion 77 16.4 Multiresolution Analysis with Haar Wavelets 77 16.4 General Construction of Wavelets and Multiresolution Analysis 77 16.4.6 Shannon Wavelets 77 PART Chapter 17 Partial Differential Equations 779 The Wave Equation 78 17 The Wave Equation and Initial and Boundary Conditions 78 17.2 Fourier Series Solutions of the Wave Equation 78 17.2.1 Vibrating String with Zero Initial Velocity 78 17 2 Vibrating String with Given Initial Velocity and Zero Initial Displacement 79 17.2 Vibrating String with Initial Displacement and Velocity 79 17.2 Verification of Solutions 79 17.2 Transformation of Boundary Value Problems Involving the Wave Equation 79 17.2 Effects of Initial Conditions and Constants on the Motion 79 17.2 Numerical Solution of the Wave Equation 80 17 Wave Motion Along Infinite and Semi-Infinite Strings 80 17.3 Wave Motion Along an Infinite String 80 17.3 Wave Motion Along a Semi-Infinite String 81 17.3 Fourier Transform Solution of Problems on Unbounded Domains 81 17.4 Characteristics and d'Alembert' s Solution 822 17 4.1 A Nonhomogeneous Wave Equation 82 17 4.2 Forward and Backward Waves 82 17.5 Normal Modes of Vibration of a Circular Elastic Membrane 83 17.6 Vibrations of a Circular Elastic Membrane, Revisited 83 17.7 Vibrations of a Rectangular Membrane 83 Chapter 18 The Heat Equation 84 18 The Heat Equation and Initial and Boundary Conditions 84 18.2 Fourier Series Solutions of the Heat Equation 844 Contents 18.2 Ends of the Bar Kept at Temperature Zero 844 18.2 Temperature in a Bar with Insulated Ends 84 18.2 Temperature Distribution in a Bar with Radiating End 84 18.2 Transformations of Boundary Value Problems Involving the Heat Equation 85 18.2 A Nonhomogeneous Heat Equation 85 18.2 Effects of Boundary Conditions and Constants on Heat Conduction 85 18.2 Numerical Approximation of Solutions 85 18.3 Heat Conduction in Infinite Media 86 18.3.1 Heat Conduction in an Infinite Bar 86 18.3.2 Heat Conduction in a Semi-Infinite Bar 86 18.3.3 Integral Transform Methods for the Heat Equation in an Infinite Medium 86 18.4 Heat Conduction in an Infinite Cylinder 87 18.5 Heat Conduction in a Rectangular Plate 877 Chapter 19 The Potential Equation 879 19 Harmonic Functions and the Dirichlet Problem 87 19 Dirichlet Problem for a Rectangle 88 19.3 Dirichlet Problem for a Disk 88 19 Poisson' s Integral Formula for the Disk 886 19 Dirichlet Problems in Unbounded Regions 88 19.5 Dirichlet Problem for the Upper Half Plane 88 19.5 Dirichlet Problem for the Right Quarter Plane 89 19.5 An Electrostatic Potential Problem 89 19 A Dirichlet Problem for a Cube 89 19 The Steady-State Heat Equation for a Solid Sphere 89 19.8 The Neumann Problem 90 19 8.1 A Neumann Problem for a Rectangle 90 19 8.2 A Neumann Problem for a Disk 90 19.8.3 A Neumann Problem for the Upper Half Plane 90 PART Chapter 20 Complex Analysis 91 Geometry and Arithmetic of Complex Numbers 91 20.1 Complex Numbers 91 20.1.1 The Complex Plane 91 20.1 Magnitude and Conjugate 91 20.1 Complex Division 91 20.1.4 Inequalities 91 20.1.5 Argument and Polar Form of a Complex Number 91 20.1.6 Ordering 920 20.2 Loci and Sets of Points in the Complex Plane 92 20.2.1 Distance 922 20.2.2 Circles and Disks 92 20.2.3 The Equation lz -al = Iz - bI 923 20.2.4 Other Loci 925 20.2.5 Interior Points, Boundary Points, and Open and Closed Sets 925 12 Inde x Bounded sequences, points, 93 Bounded set, points, 93 C Calculus, see Differential calculus ; Integral calculus Cauchy principal value, 105 Cauchy's theorem, 990-1005 consequences of, 994-100 deformation theorem, 995-997, 1002-1005 higher derivatives, 1000-100 independence of path, 994-99 integral formula, 997-1000 Liouville's theorem, and, 1001-100 proof of, 99 terminology for, 990-99 Cauchy-Riemann equations, 945-95 Cauchy-Schwarz inequality, 216-217, 22 inequality in R", 225 theorem of, 216-21 Center, measures of, 1143-114 mean, 1143-1145 median, 1145-1146 Central limit theorem, 1181-118 Cesit o filter, 690-69 Characteristic equation, 73-7 Characteristic polynomial, 326-32 Characteristic vectors, see Eigenvectors Characteristics, wave equation, 822-83 Circulation, fluid flow, 108 Closed set, points, 92 Coefficients, 587-588, 679-68 DFT approximation of, 679-68 Fourier, 587-58 Cofactor expansion, 311-31 column, by a, 31 defined, 31 row, by a, 31 Column operations, 307-31 Column spaces, matrices, 266-26 Compact set, points, 935-93 Complementary events, 112 Complex analysis, 911-109 conformal mappings, 1055-1095 functions, 939-973 integration, 975-1005 numbers, geometry and arithmetic of, 913-93 residue theorem, 1030-105 series representations of functions, 1007-102 singularities, 1023-103 Complex Fourier integral, 642-65 Complex Fourier series, 630-63 Complex numbers, 630-631, 913-937, 99 argument, 918-920 conjugate, 915-91 defined, 91 division, 91 geometry and arithmetic of, 913-93 inequalities, 91 loci, 921-936 magnitude, 915-91 ordering, 920-921 planes, 914, 921-93 points, sets of, 921-93 polar form, 918-92 review of, 630-63 rules of, 91 simply connected, 991-992 Complex sequences, points, 931-93 Conditional probability, 1122-1125 Conditions, 781-785, 798-801, 834-837, 841-84 boundary, 781-785, 841-844 heat equation, 841-84 initial, 781-785, 798-801, 841-84 periodicity, vibration, 834-83 wave equation, 781-785, 798-80 Confidence intervals, 1185-118 Conformal mappings, 1062-109 angle preserving, 106 complex function models, 1087-109 construction of between domains, 1072-108 defined, 106 Dirichlet problem, 1080-108 fluid flow, 1087-1094 harmonic functions, 1080-1086 Joukowski transformation, 1093-109 linear fractional transformations, 1064-107 mean value property, 108 Mobius transformation, 1064 orientation preserving, 1062 Riemann mapping theorem, 1072-107 Schwartz-Christoffel transformation, 1077-108 three point theorem, 1069-107 Conjugate, complex numbers, 915-91 Connected set, defined, 99 Conservative field, test for, 539-543 Conservative vector field, 53 Consistent system of equations, 285-28 Continuity adjustment, 1168-1172 Continuity function, 941-942 Continuous random variable, 115 Convergence, 593-609, 611, 613, 620-622, 757-759, 762-763, 932-934, 952-95 cosine series, 61 eigenfunction expansions, 757-75 end points, at, 599-60 Fourier series, 593-609, 611, 613, 620-62 Gibbs phenomenon, 606-609 mean, 762-763 open disk of, 954 partial sums, 604-606 points, 932-934 power series, 952-95 radius of, 95 sine series, 61 theorems, 596-599, 601-60 uniform and absolute, 620-622 Convolution, 134-139, 657-66 commutativity of, 137, 65 defined, 134 Fourier transform, 657-66 frequency, 658-65 inverse version, 136 Index linearity, 65 theorem, 135-13 time, 658 Correlation, 1194-119 Cosine, 609-612, 640-642, 670-671, 67 Fourier series, 609-61 Fourier transform, 670-671, 67 integral, 640-642 Counting, see Probability Cramer's rule, 318-32 Critical damping, Critical points, 413, 424-431, 46 asymptomatically stable, 42 defined, 425 enclosure of, 46 linear systems, 41 nonlinear systems, 424-43 stability of, 427 Critically damped forced motion, 9 Cross product, 217-22 defined, 21 parallelogram, 220-22 properties of, 218-21 rectangular parallelopiped, 221-222 right-hand rule, 21 Crystals, random walks in, 247-25 Curl, 510-512, 513-514, 57 defined, 51 physical interpretation of, 513-514, 57 Stoke's theorem, 576 Curvature, 483-488, 491-49 defined, 48 function of t, 491-49 unit normal vector, 486-48 Curves, 517-525, 975-979, 108 complex planes, in, 975-97 continuous, 977 coordinate functions, 975-976 differentiable, 977 equipotential, 108 equivalent, 977-979 initial point, 97 line integrals of, 517-525 simple, 977 terminal point, 97 D d'Alembert's solution, 822-83 Damping constant, Data, statistical, 114 Deformation theorem, 995-997, 1002-100 Derivatives, 601, 652-655, 943-945, 1000-100 bounds on, 1001-1002 Cauchy's integral formula for, 1000-100 complex functions, of, 943-945 Fourier transform of a, 652-65 higher, 1000-100 Liouville's theorem, 1001-1002 right and left, 601 Determinants, 299-32 cofactor expansions, 311-31 column operations, 307-31 Cramer's rule, 318-32 defined, 301-302 evaluation of, 307-31 matrix inverse, 315-31 matrix tree theorem, 320-322 minor, 31 permutations, 299-30 properties of, 303-30 row operations, 307-31 triangular matrices, 314-31 Diagonalization, 330-339, 384-386, 398-40 matrices, 330-33 solution of X' =AX+G, 398-40 solution of X' = AX, 384-38 Differential calculus, 473, 475-51 acceleration, 481-483, 488-49 curl, 510-512, 513-51 curvature, 483-488, 491-49 directional derivatives, 501-504 divergence, 510-511, 512-51 Frenet formulas, 492-49 gradient field, 499-501, 504-50 level surfaces, 503-507 normal components, 488-49 normal lines, 508-50 one variable, 475-48 streamlines, 495-49 tangent planes, 507-50 tangential components, 488-49 torsion, 49 unit normal vector, 486-488 vector fields, 493-495 vector functions, 475-48 velocity, 481-48 Differential equations, 1-200, 359, 361-402, 403-472, 779 , 781-840, 841-878, 879-909 Bernoulli equation, 42-43, 45-4 Bessel's equation, 171, 17 characteristic, 73-7 defined, Euler's equation, 78-8 exact, 26-3 first order, 3-6 heat equation, 841-878 higher-order, 91-9 homogeneous, 38-42, 45-46, 64-6 introduction to, 1-2 Laplace transform, 107-154 linear, 22-26, 37, 61-62, 361-402 nonhomogeneous, 68, 82-93 nonlinear, 403-47 numerical approximations of solutions, 181-20 order, defined, partial, 779, 781-840, 841-878, 879-90 polynomial coefficients, with, 150-15 potential equation, 879-90 second order, 61-106 13 14 Index Differential equations (continued ) separable, 11-21, 37 series solutions, 155-18 solution, defined, systems of, 359, 361-402, 403-472 wave equation, 781-840 Diffusivity, defined, 842 Dirac delta function, 139-144, 660-66 filtering property, 140-14 Fourier transform, 660-66 unit impulses, 139-144 Direct fields, first order differential equations, 7-1 Direction fields, nonlinear systems, 406-41 Directional derivatives, 501-504 Dirichlet problem, 879-886, 888-898, 1080-108 conformal mappings, 1080-108 cube, for a, 896-89 defined, 88 disk, for a, 883-88 electrostatic potential, 893-89 harmonic functions and, 879-880, 1080-108 planes, 889-893 rectangle, for a, 881-883 solution of by conformal mapping, 1083-108 unbounded regions, 888-89 Discrete Fourier transform (DFT), 675-681, 685-68 approximation of, 685-68 coefficients, approximation of, 679-68 inverse N-point, 678-67 linearity, 678 N-point, 676-678, 685-68 periodicity, 67 Discrete random variable, 115 Displacement, 791-79 initial, 793-79 wave equation, 791-794 zero initial, 791-793 Distributions, sampling of, 1178-118 Divergence, 510-511, 512-513, 564-567, 570-57 conservation of mass principle, 570-57 defined, 51 Gauss theorem, 564-567 physical interpretation of, 512-51 theorem, 564-567, 570-57 Division, complex, 91 Domain, 539-545, 578-579, 815-821, 908, 991 , 1072-1080 Cauchy's theorem, 99 conditions of, 54 conformal mappings, construction of between, 1072-108 conservative field, test for, 539-54 defined, 99 plane, in a, 540-542 Riemann mapping theorem, 1072-107 simply connected, 54 3-space, in surfaces, 578-57 unbounded, 815-821, 908 wave equation, solution on unbounded, 815-82 Dot product, 211-21 Cauchy-Schwarz inequality, 216-21 defined, 211 orthogonal vectors, 215-21 properties of, 211-21 E Eigenfunction, 749-752, 755-759, 763-765 completeness of, 763-76 expansions, 755-759 Sturm-Liouville theory, 745-75 Eigenvalues, 323-33 defined, 323-324 characteristic polynomial, 326-32 Gerschgorin ' s theorem, 328-330 Eigenvectors, 323-33 Electrical circuits, 51-53, 57, 103-105, 106, 129-13 first order differential equation applications to, 51-53, Kirchoff's current and voltage law, second order differential equations, analogy with, 103-105, 10 shifting theorems, analysis of by, 129-13 Electrostatic potential, 893-89 Elementary matrix, 253-25 Enclosure of critical points, 46 End points, convergence at, 599-600 Equality matrices, 23 Equilibrium point, 41 Equipotential curves, 1089 Euler's equation, 78-8 Euler's formula, 75 Euler's method, 182-190, 193-194 defined, 18 modified, 193-194 numerical approximation solutions, 182-19 Events, 1112-1122, 1126-1129 complementary, 1121-112 defined, 111 experiment, 1112-111 independent, 1126-1129 principle of complementarity, 112 probability of, 1116-1120 product rule, 1128-112 Exact differential equations, 26-32 defined, potential function, test for exactness, 30-3 Existence and uniqueness theorems, 59-6 Expansions, 581-582, 708-711, 739-741, 745-765 , 1019-1022 eigenfunction, 582, 745-755, 755-75 Fourier-Bessel, 739-74 Fourier-Legendre series, 709-71 Laurent, 1019-102 orthogonality, 708-709, 740-74 Sturm-Liouville theory, 745-75 wavelets, 582, 765-77 Exponential functions, 957-96 Exponential matrix solutions, 386-39 F Fast Fourier transform (FFT), 694-69 analysis of tides in Morro Bay, 697-69 analyzing power spectral densities in signals, 695 filtering noise from a signal, 69 procedure of, 694 Index Filtering, 140-142, 660-661, 667-669, 689-692, 69 bandpass, 667-66 Cesi ro filter, 690-692 Dirac delta function, 140-14 Fourier transformations, 660-661, 667-66 lowpass, 667-669 noise, using FFT, 696 property, 140-142 sampled Fourier series, 689-69 First order differential equations, 3-6 Bernoulli, 42-43, 45-46 direct fields, 7-1 electrical circuits, applications to, 51-53, exact, 26-32 general solutions, 3-4 homogeneous, 38-42, 45-46 implicitly defined solutions, initial value problem, 6-7, 58-60 integral curves, 5-6 integrating factors, 33-3 linear, 22-2 mechanics, applications to, 46-51, 55-5 orthogonal trajectories, applications to, 53-55, particular solutions, 3- preliminary concepts, 3-1 Riccati, 43-4 separable equations, 11-2 theorems for, 30, Fluid flow, 495-499, 1087-1094 circulation, 1087 complex function models for, 1087-109 equipotential curves, 108 flux, 1087-108 lines, 495-49 plane-parallel, 108 potential, 108 solenoidal, 108 stagnation point, 108 stationary, 1087 streamlines, 495-499, 108 vector fields, 495-49 vortex, strength of, 108 Flux, fluids, 1087-108 Forced motion, 98-10 Forth order (RK4) Runge-Kutta method, 195-19 Fourier analysis, 581-582, 583-635, 637-699, 701-74 Fourier integral, 637-65 Fourier series, 583-635, 681-693 Fourier transform, 642-681, 685-689, 694-69 importance of, 581-582 special functions, 581-582, 701-745 Fourier integral, 637-65 complex, 642-65 cosine, 640-64 formulas, 637-640 Laplace's, 641-642 sine, 640-642 Fourier series, 583-635, 681-693, 786-808 , 844-864 amplitude spectrum, 633-635 Bessel's inequalities, 618-620 coefficients, 587-58 complex, 630-63 convergence of, 593-60 cosine, 609-612 differentiation, 641-615, 617-61 endpoints, 599-60 even and odd functions, 589-59 frequency spectrum, 633-63 function, of a, 586-59 Gibbs phenomenon, 606-60 heat equation, solutions, 844-864 integration, 614-61 left derivative, 60 Parseval's theorem, 622-623 partial sums of, 604-60 phase angle form, 623-63 piecewise continuous function, 593-595 piecewise smooth function, 595-59 right derivative, 60 sampled, 681-693 sine, 612-61 special functions, 581-58 wave equation, solutions of, 786-80 Fourier transform, 642-681, 685-689, 694-699, 816-821 , 890-873 applications of, 652-670 approximation of, 685-68 complex Fourier integral and, 642-65 convolution, 657-660 cosine, 670-671, 67 defined, 64 derivative, of a, 652-65 Dirac delta function, 660-66 discrete (DFT), 675-681, 685-68 fast (FFT), 694-69 filtering, 660-661, 667-669, 69 finite, 673-67 frequency differentiation, 655-65 frequency shifting, 64 heat equation, integral transform methods for, 869-87 integral, of a, 65 modulation, 65 operational formula, 67 scaling, 649-65 Shannon sampling theorem, 665-66 sine, 671-672, 673-67 symmetry, 650-65 time reversal, 650 time shifting, 647-64 wave equation, solution on unbounded domains, 815-82 windowed, 661-66 Frenet formulas, 492-19 Frequency differentiation, 655-65 Frequency of outcomes, 1159-116 Frequency shifting, Fourier transform theorem of, 64 Frequency spectrum, complex Fourier series, 633-63 Frobenius, method of, 166-171, 173-17 linear independent solutions, 173-17 series, 168 singular points, 167-17 theorem, 170 I5 Inde x Functions, 122-125, 475-481, 589-596, 600-661, 701-745, 879-880, 939-973, 980-990, 1007-1022 See also Bessel functions; Eigenfunction s analytic, 1008 bounded, 942-943 Cauchy-Riemann equations, 945-95 complex, 939-973, 980-990 continuity, 941-942 derivative of complex, 943-945 Dirac delta, 660-66 even and odd, 589-592 exponential, 957-96 generating, 704-706, 73 harmonic, 879-88 heaviside, 122-12 integrals of complex, 980-99 integrals of series of, 988-98 limit, 939-94 logarithm, 966-969 piecewise continuous, 593-595 piecewise smooth, 595-596 power series, 950-957 powers, 969-972 pulse, 124-12 series representations of, 1007-1022 special, 701-745 trigonometric, 957-966 vector, 475-48 G Gamma function, 719-72 Gauss's divergence theorem, 564-567 Gauss-Jordan method, 276-279 General solutions, first order differential equations, 3- Generating function, 704-706, 73 Bessel functions, 73 Legendre polynomials, 701-71 Gerschgorin's theorem, 328-33 Gibbs phenomenon, 606-609 Gradient field, 499-501, 504-50 defined, 50 normal vector, as, 504-50 Graph, defined, 274 Green's theorem, 528-535, 563-564, 903 extension of, 532-535 first identity, 90 rewritten, 563-56 uses of, 530-53 vector integral calculus, 528-535 H Haar wavelets, 767-774, 774-775 Harmonic functions, 879-880, 1080-108 conformal mappings, 1080-1086 Dirichlet problems and, 879-880, 1080-1086 Heat conduction, 857-859, 865-878 See also Steady-state heat equation effects of boundary conditions and constants on, 857-85 infinite cylinder, in, 873-877 infinite media, in, 865-873 integral transform methods, 869-87 Laplace transform solution, 871-87 rectangular plate, in, 877-87 Heat equation, 568-570, 841-87 boundary conditions, 841-844 determination of, 568-57 Fourier series solutions, 844-86 heat conduction, 857-859, 865-87 initial conditions, 841-844 integral transform methods, 869-87 nonhomogeneous, 854-85 numerical approximation of, 859-86 temperature, 843, 844-848 transfer coefficient, 849 vector integral calculus, as, 568-570 Heaviside functions, 122-12 Hermitian matrices, 355-357 Homogeneous equations, 38-42, 45-46, 64-68, 272-280 See also Nonhomogeneous equation s complete pivoting, 276 first order differential, 38-42, 45-46 Gauss-Jordan method, 276-279 linear dependence and independence, 65-66 second order differential, 64-68 solutions of, 272-28 systems of linear equations, 272-28 Wronskian test, 66-67 Hooke's law, I Identity matrix, 244 Identity theorem, 1015-101 Implicitly defined solutions, Improper node, 420-421, 429 Inconsistent system of equations, 285, 289-292 Independence of path, 536-545, 987, 994-99 Cauchy's theorem, 994-99 complex integrals, 987 defined, 53 potential theory in the plane, and, 536-545 test for a conservative field, 539-543 Independent events, 1126-112 Inequalities, complex numbers, 91 Initial conditions, 6, 781-785, 798-801, 841-84 defined, heat equation, 841-844 motion, effects on, 798-80 temperature, 843 wave equation, 781-785, 798-80 Initial value problem, 6-7, 58-60, 116-120, 156-16 defined, existence and uniqueness for solutions of, 58-60 first order differential equations, 6-7 Laplace transform, solutions of using, 116-12 power series solutions of, 156-16 theorems for, 59, 116-117, 156, 15 Input frequency, 10 Instability, Lyapunov's direct method for, 45 Integral calculus, 473, 517-58 applications of surface integrals, 557-562 Archimedes' principle, 567-568 Index conservative vector field, 536 curves, 517-525 divergence theorem, 564-567, 570-57 domain, 540-542 Gauss's divergence theorem, 564-56 Green's theorem, 528-535, 563-56 heat equation, 568-570 independence of path, 536-545 lines, 517-52 potential theory, 536-545, 576-57 Stoke's theorem, 572-57 surfaces, 545-562 Integral curves, 5-6 Integrals, 517-528, 545-562, 637-652, 656, 980-990, 997-1000, 1040-1052 arc length, with respect to, 525-52 Cauchy principal value, 1052 Cauchy's formula, 997-100 complex functions, of, 980-990 complex in terms of real, 983-98 evaluation of real, 1040-105 Fourier, 637-652 Fourier transform of, 65 independence of path, 98 line, 517-52 linearity, 98 properties of complex, 985-98 residue theorem, 1040-105 reversal of orientation, 98 series of functions, of, 988-98 surface, 545-56 term-by-term integration, 988-98 vector calculus, 517-52 Integrating factors, 22, 33-3 defined, 22, 3 linear equations and, 22, nonzero function, 33-34 separable equations and, 37 Integration, 975-100 Cauchy's theorem, 990-1005 complex functions, 975-1005 curves, complex plane, 975-97 integrals, 980-98 term-by-term, 988-989 Interior points, 925 Inverses, 136, 293-298, 315-318, 678-67 convolution, 136 determinant formula for matrix, 315-31 matrix, 293-298, 315-31 method of finding A -1 , 295-29 N-point, DFT, 678-679 nonsingular matrix, 29 singular matrix, 294 systems of linear equations, relation to, 296-297 uniqueness of, 29 J Jordan curve theorem, 52 Joukowski transformation, 1093-109 Jump discontinuities, 593 K Kirchoff's current and voltage law, L Laplace transform, 107-1.54, 379-398, 641-642, 87]-873 , 1039-1040 boundary condition values, solution for, 871-87 convolution, 134-139 defined, 10 derivative, 116-11 differential equations, solving, 150-15 Dirac delta function, 139-14 functions, table of, 109-11 ] higher derivative, 117 initial value problems, solution of using, 116-12 integrals, 641-642 inversion formula for, 1039-104 Lerch's theorem, 11 linearity of, 11 piecewise continuity, 112-11 polynomial coefficients, solving equation s with, 150-1.5 residue theorem, 1039-1040 shifting theorems, 120-13 systems, solution of, 144-15 theorems for, 112, 113, 114, 115, 116-117, 120, 126 , 135-136, 137, 140, 150, 15 unit impulses, 139-144 variation of parameters method and, 397-39 Laplace's equation, see Steady-state heat equatio n Lattice points, 80 Laurent expansion, 1019-1022 Leading entry, 25 Legendre polynomials, 701-719, 750-752 approaches to, 701-70 derivative formula, 715-71 eigenfunctions, as, 749-750 Fourier-Legendre coefficients, 711-71 Fourier-Legendre series, 709-71 generating function, 704-70 integral formula, 716-71 orthogonality, 708-709 recurrence relation, 706-70 Rodrigues' formula, 715-71 zeros, 713-71 Lerch's theorem, 11 Level surfaces, 503-50 Lienard theorem, 468-47 Limit cycles, 461-46 Bendixson theorem, 466-46 enclosure of critical points, 46 Lienard theorem, 468-470 Poincare-Bendixson theorem, 467-46 van der Pol equation, 468-47 Limit function, 939-94 Limit points, 929-93 Line integrals, 517-528 Lineal elements, Linear combinations in R", 22 Linear correlation, 1194-1198 17 18 Index Linear dependence and independence, 65-68, 228-235, 365-37 defined, 65, 230, 36 linear differential equations for, 365-37 vectors, 230-23 Wronskian test, 6 Linear differential equations, 22-26, 37, 61-62, 361-402 , 413-424 defined, 2 dependence, 365-37 diagonalization, 384-386, 398-40 exponential matrix solutions, 386-39 first order, 26-27, 37, 361-37 homogeneous system X' = AX, 365-372 independence, 365-37 integrating factors, and, 37 nonhomogeneous system X' = AX + G, 372-373 phase portrait, 413-424 second order, 61-62 solution of X' = AX+G, 394-40 solution of X' = AX when A is constant, 374-39 systems of, 361-402 theory of systems, 361-347 variation of parameters method, 394-39 Linear equations, 237, 242-243, 272-293, 296-29 Gauss-Jordan method, 276-27 homogeneous systems of, solution of, 272-28 inverses, relation to, 296-29 nonhomogeneous systems, 283-293 notation for systems of, 242-24 solution space, 280-28 use of matrices and, 23 Linear fractional transformations, 1064-107 defined, 106 three point theorem, 1069-107 Linear independence, 365-37 Linearity, complex integrals, 98 Liouville's theorem, 1001-1002 Loci, see Points Logarithm, 966-969 complex, 966-96 natural, 966 Lyapunov's stability criteria, 451-46 direct method, 45 positive definite, 45 semidefinite, 45 M Magnitude, complex numbers, 915-91 Mappings, 1055-109 angle preserving, 1062 conformal, 1062-109 defined, 1055 functions as, 1055-106 inversion, 1066-106 orientation preserving, 1062 stereographic projection, 107 Mass, surface, 557-55 Mathematical modeling, 20 Matrices, 237-298, 299-322, 330-339, 339-347, 352-35 addition of, 23 adjacency, 248 algebra, 239-242 augmented, 285-28 cofactor, 31 column operations, 307-31 column spaces, 266-269 crystals, random walks in, 247-25 defined, 23 determinants, 299-32 diagonalization of, 330-33 elementary, 253-25 equality, 23 Hermitian, 355-35 homogeneous systems of linear equations, 272-28 identity, 24 inverses, 293-298 linear equations, systems of, 237, 272-29 matrix inverse, 293-298, 315-31 minor, 31 multiplication of, 239-241, 246-247 nonhomogeneous systems of linear equations, 283-29 nonsingular, 29 notation for systems of linear equations, 242-24 orthogonal, 339-343 product of, with a scalar, 23 rank, 269-27 reduced row echelon, 259-26 row echelon form of, 258-26 row equivalence, 25 row operations, 251-258, 261-265, 307-31 row spaces, 266-269 singular, 294 Skew-Hermitian, 355-35 solution space, 280-28 symmetric, 343-34 transpose, 24 tree theorem, 320-322 triangular, 314-31 unitary, 352-35 zero, 243-244 Maximum modulus theorem, 1016-101 Mean, 759-761, 762-763, 1082, 1143-1145, 1152-1153 , 1190-119 approximation in, 759-76 Bessel inequality, 759-76 convergence in, 762-763 defined, 1143 Parseval's theorem, 762-763 population, estimating, 1190-119 random variables, 1152-115 statistical, 1143-1145, 1152-115 value property, 108 Mechanical systems, 46-51, 55-57, 93-10 applications to, 46-51, 55-57, 93-10 beats, 102-10 critical damping, 96 electrical circuits, analogy with, 103-105, 10 first order differential equations, 46-51, 55-5 forced motion, 98-100 motion, 48-49, 50-5 Newton's laws, 46-4 overdamping, 95-96 Index resonance, 100-102 second order differential equations, 93-10 terminal velocity, 47-48 underdamping, 96-97 unforced motion, 95-9 velocity, 49-5 Median, statistical, 1145-114 Minor, matrices, 31 Mixing problems, 24-25 Mbbius transformation, see Linear fractional transformation s Modulation, Fourier transform theorem of, 65 Motion, 48-49, 50-51, 93-106, 98-1.00, 798-801, 808-82 beats, 102-103 critically damped forced, 9 damping constant, 94 effects of initial conditions and constants on, 798-80 electrical circuit, analogy with, 103-10 forced, 98-10 Fourier transform solution on unbounded domains, 815-82 Hooke's law, 94 infinite string, along an, 808-81 mathematical models of, 48-49, 50-5 mechanical systems, 48-49, 50-51, 98-10 Newton's laws, 46-47, overdamped forced, 9 overdamping, 95-97 resonance, 100-102 semi-infinite string, along an, 813-81 spring equation, 94-95 underdamped forced, 10 unforced, 95-9 wave, 808-822 Multiplication principle, 1099-110 Multiresolution analysis, 774-775, 775-776 defined, 774 Haar wavelets, 774-77 scaling function, 77 wavelets, general construction of, 775-77 Multistep methods, numerical approximation, 197-200 N Natural frequency, 100 Neumann problem, 902-90 disk, for a, 906-90 Green's first identity, 90 rectangle, for a, 904-906 unbounded domain, 90 Newton's laws, 46-47, 94, 842 cooling, 84 motion, 46-47, 94 Nodal source, 415-418, 429 linear systems, 415-41 nonlinear systems, 42 Noise, FFT filtering, 696 Nonhomogeneous equations, 68, 82-93, 283-293, 372-373, 825-828, 854-85 consistent, 285-28 existence and uniqueness of solutions, 285-29 heat equation, 854-85 higher-order differential equations, 91-92 inconsistent, 285, 289-29 linear, systems of, 283-293 solution of X' = AX+G, 372-37 structure of solutions, 284-28 superposition, principle of, systems of linear, 283-293, 372-37 theorem for, 68 undetermined coefficients, method of, 85-9 variation of parameters, method of, 82-8 wave equation, 825-828 Nonlinear differential equations, 403-47 almost linear systems, 431-45 critical points, 424-431, 466 direction fields, 406-41 limit cycles, 461-46 Lyapunov's stability criteria, 451-46 periodic solutions, 466-47 phase plane, 406-40 phase portrait, 406-41 solutions, existence of, 403-406 systems, 403-40 trajectories, 407-43 translation, 409-41 uniqueness, 40 Nonsingular matrix, 294 Normal distribution, 1162, 1167-116 Normal lines, 508-509 Normal vector, 486-488, 504-509, 545-56 defined, 504 gradient field, as, 504-50 surfaces, 548-55 unit, 486-48 Numbers, see Complex numbers Numerical approximations, 181-200, 859-862 Adams-Bashforth method, 199 Adams-Moulton method, 19 Euler's method, 182-190, 193-19 features of, 181-182 heat equation solutions, 859-862 multistep methods, 197-200 one-step methods, 190-197 Runge-Kutta methods, 195-197 Taylor method, second order, 190-192 One-step methods, numerical approximation, 190-19 Open set, points, 926-92 Operational formula, Fourier transform, 67 Order p method, 19 Ordering, complex numbers, 920-92 Orientation preserving mapping, 1062 Origin (center), 423-424, 429-43 linear systems, 423-42 nonlinear systems, 429-43 Orthogonal expansions, see Expansion s Orthogonal matrices, 339-343 Orthogonal trajectories, 53-55, Orthogonal vectors, 21 Orthogonality, 708-709, 740-74 Bessel functions, 740-74 Legendre polynomials, 708-709 19 11 Index Overdamped forced motion, 9 Overdamping, 95-9 P Parallelogram, 206-207, 220-22 cross product, 220-22 law, 206-20 Parseval's theorem, 622-62 Partial differential equations, 779, 781-840, 841-878, 879-909 heat equation, 841-87 potential equation, 879-90 use of, 779 wave equation, 781-840 Particular solutions, first order differential equations, 3- Path, defined, 99 Periodicity conditions, vibration, 834-837 Permutations, 299-301, 1102-110 defined, 299, 1102 determinants, 299-30 probability, 1102-1103 Phase plane, nonlinear systems, 406-408 Phase portrait, 406-413, 4131124 linear systems, 413-424 nonlinear systems, 406-41 Piecewise, 112-113, 552-553, 593-595, 595-596 continuity, 112-11 continuous function, 593-59 functions, 593-595, 595-59 smooth function, 595-59 surfaces, smooth, 552-553 Plane-parallel flow, 108 Planes, 507-508, 889-893, 914, 921-936, 975-979, 1070 , 1087-1094 complex, 914, 921-936, 975-97 curves in, 975-979 Dirichlet problem, 889-893 extended complex, 107 fluid flow, complex function models of, 1087-109 tangent, 507-50 Poincare-Bendixson theorem, 467-46 Points, 166-173, 921-936, 975 Bessel functions, 171-17 Bolzano-Weierstrass theorem, 936 boundary, 927-928 bounded sequences, 935 bounded set, 93 circles, in, 922-92 closed set, 92 compact set, 935-93 complex pane, in the, 921-93 complex sequences, 931-934 convergence, 932-93 disks, in, 923 distance between, 922, 923-92 equation Iz - a* = 923-924 Frobenius, method of, 166-17 initial, 975 interior, 92 irregular singular, 167 limit, 929-93 loci, 921-936 open set, 926-927 ordinary, 166 regular singular, 167 sets of, 921-93 singular, 166-173 subsequences, 934-93 terminal, 97 Poisson distribution, 1157-115 Poisson's integral formula, 886-88 Polar form, complex numbers, 918-92 Poles, 1024, 1026-1029, 1031-103 condition for, 1026-102 double, 1026 order m, of, 1024, 1026-1028, 1033-103 products, of, 102 quotients, of, 1028-1029 residue at, 1031-103 simple, 1031-103 single, 102 Polynomial coefficients, solving equations with, 150-154 Population, 1178, 1185-119 defined, 117 distributions, sampling of, 1178-118 mean, estimating, 1190-119 student t distributions, 1191-119 Potential equation, 879-909 Dirichlet problems, 879-88 electrostatic, 893-895 harmonic functions, 879-88 Laplace's equation, 879-880 Neumann problem, 902-90 Poisson's integral formula, 886-88 steady-state heat equation, 879-880, 898-90 Potential function, 28 Potential theory, 536-545, 576-57 independence of path, 536-54 Stokes theorem, 576-57 3-space, 576-579 Power series, 156-161, 161-166, 950-957, 1007-101 complex numbers, 951-95 complex polynomial, 95 convergence, 953-95 defined, 952 identity theorem, 1015-101 initial value problems, 156-16 isolated zeros, 1012-101 maximum modulus theorem, 1016-101 recurrence relations, 161-16 representations of functions, 1007-101 solutions, 156-161, 161-16 Taylor coefficients, 95 Taylor series, 1007-101 Power spectral densities, 1'1'1 analysis of, 695 Powers, 969-972 complex numbers, 969-97 integer, 969 positive integer n, 969-97 rational, 971-97 Prediction interval, 1198-120 Principal axis theorem, 350-351 Index Probability, 1097, 1099-1141, 1150-115 Bayes' theorem, 1134-113 choosing r objects from is objects, 1104-111 complementary events, 112 conditional, 1122-1125 counting and, 1099-114 distributions, 1150-115 events, 1112-1122, 1126-1129 expected value, 1139-1140 experiment, 1112-111 independent events, 1126-1129 multiplication principle, 1099-110 permutations, 1102-1103 sample space, 1112-111 statistics and, 1097 tree diagrams, 1107-1111, 1130-113 Probability distribution, defined, 115 Product rule, 1128-1129 Products, poles of, 102 Proper node, 419, 429 Pulse functions, 124-12 Pure imaginary number, 91 Q Quadratic forms, 347-352 Quotients, poles of, 1028-1029 R Random variables, 1150-115 continuous, 115 countable, 115 defined, 1153-1154 discrete, 115 experiment of, 115 mean, 1152-115 probability distributions, and, 1150-115 standard deviation, 1152-1153 uncountable, 1154 Random walk, defined, 24 Range, statistical, 1146-1147 Rank, matrices, 269-27 Rectangular parallelopiped, cross product, 221-22 Recurrence relations, 161-166, 706-708, 735-73 Bessel functions, 735-73 Legendre polynomials, 701-71 power series solutions using, 161-166 Reduced matrix, 258-26 Reduction of order, 69-72 Regression, 1198-120 Residue, 1030-105 applications of, 1037-1054 argument principle, 1037-103 Cauchy principal value, 105 defined, 1030 Laplace transform, inversion formul a for, 1039-1040 pole of order m, 1033-103 real integrals, evaluation of, 1040-105 simple pole, 1031-103 theorem, 1030-105 Resonance, 100-102 Reversal of orientation, complex integrals, 98 Riccati equation, 43-4 Riemann mapping theorem, 1072-107 Right-hand rule, 21 Rodrigues' formula, 715-71 Row echelon form, 258-26 elementary row operations, 261-265 reduced matrix, 258-26 reducing, 26 Row equivalence, 25 Row operations, 251-258, 261-265, 307-31 determinants, evaluation of by, 307-31 elementary matrices, 251-25 reduced matrices, 261-265 Row spaces, matrices, 266-26 Runge-Kutta methods, 195-197 S Saddle point, 418, 429 Sample space, 1112-111 Scalar field, 49 Scalar, 203-205, 23 defined, 20 product of, with a vector, 204-205 product of, with a matrix, 23 Scaling, 649-650, 77 Fourier transform theorem of, 649-65 function, 776 Schwartz-Christoffel transformation, 1077-108 Second order differential equations, 61-10 characteristic equation, 73-74 constant coefficient homogeneous linear equation, 72-7 defined, electrical circuits, analogy with, 103-105, 10 Euler's equation, 78-8 Euler's formula, 75 homogeneous equations, 64-6 linear dependence and independence, 65-6 linear equations, 61-62 mechanical systems, applications for, 93-10 nonhomogeneous equations, 68-69, 82-9 preliminary concepts, 61-6 reduction of order, 69-7 superposition, principle of, theory of solutions, 62-6 Separable differential equations, 11-21, applications of, 14-2 defined, 1 integrating factors, and, mathematical modeling, 20 Torricelli's law, 18-1 Separation constant, 78 Series representations of functions, 1007-1022 functions, of, 1007-102 identity theorem, 1015-101 isolated zeros, 1012-101 Laurent expansion, 1019-1022 maximum modulus theorem, 1016-101 power, 1007-101 Taylor, 1007-1012 I11 112 Index Series solutions, 155-180 See also Points analytic function, 15 Bessel functions, 171-172, 178-18 Frobenius, method of, 166-171, 173-17 initial value problems, 156-16 linear independent, 173-178, 178-180 logarithm factors, 173-18 power series solutions, 156-161, 161-16 recurrence relations, 161-16 second solutions, 173-18 singular points, 166-17 Shannon sampling theorem, 665-66 Shannon wavelets, 776-77 Shifting theorems, 120-134 electrical circuits, analysis of, 129-13 first, 120-122 heaviside functions, 122-125 pulse functions, 124-125 second, 125-129 Simply connected, complex numbers, 991-99 Sine, 612-614, 640-642, 671-672, 673-674 Fourier series, 612-61 Fourier transform, 671-672, 673-674 integral, 640-642 Singular matrix, 294 Singularities, 1023-103 classification of, 1025 essential, 1024 isolated, 102 pole of order m, 1024, 1026-102 poles, 1026-1029 removable, 1024-1025 68, 95, 99.7 rule, 1176-1177 Skew-Hermitian matrices, 355-35 Solenoidal fluid, 108 Solution space, 280-28 Spanning set, 229-23 Special functions, 581-582, 701-745 Bessel functions, 719-745 Legendre polynomials, 701-71 Sturm-Liouville theory, 745-75 Speed, see Velocity Spheres, 107 Spiral point, 421-424, 42 Spring equation, 94-9 Stability, 424-431, 451-46 critical points, 424-43 direct method, 45 Lyapunov's criteria, 451-46 Stagnation point, 1089 Standard deviation, 1147-1149, 1152-1153 defined, 1147 random variables, 1152-115 Standard representation, 207 Stationary flow, 108 Statistics, 1097, 1143-1204 bell curve, 1165-1176 binomial distribution, 1154-115 center, measures of, 1143-114 central limit theorem, 1181-1184 confidence intervals, 1185-1189 continuity adjustment, 1168-117 correlation, 1194-119 data, 114 defined, 1143 frequency of outcomes, 1159-1164 mean, 1143-1145, 1152-115 median, 1145-1146 normal distribution, 1162, 1167-1168 Poisson distribution, 1157-115 population, 1178, 1185-1193 prediction interval, 1198-120 probability and, 109 probability distributions, 1150-115 random variables, 1150-115 range, 1146-114 regression, 1198-120 sampling distributions, 1178-118 68, 95, 99 rule, 1176-1177 standard deviation, 1147-1149, 1152-115 variation, measures of, 1146-114 Steady-state heat equation, 879-880, 898-90 harmonic function, as, 878-88 solid sphere, for a, 898-90 Stereographic projection, 107 Stoke's theorem, 572-579 curl, physical interpretation of, 576 integral calculus, 572-57 potential theory in 3-space, 576-579 Streamlines, 495-499, 108 defined, 495 fluid flow, 1089 vector fields, 495-49 Sturm-Liouville theory, 745-75 eigenfunctions, 749-752 periodic problem, 747, 748-74 regular problem, 745-746, 747-74 singular problem, 747, 749-752 theorem, 752-75 Subsequences, points, 934-93 Subspace of vectors, 226-227, 227-228 , 229-23 basis, 23 defined, 22 dimensions, 23 R2 , theorem of, 227-22 spanning set, 229-230 Superposition, principle of, Surfaces, 503-507, 545-56 area, 55 center of mass, 559 flux of a vector field across, 560-56 integrals, 553-562, 557-56 level, 503-507 mass, 557-559 normal vector to, 548-55 piecewise smooth, 552-553 tangent plane to, 551-55 3-space, 545, 576-57 Symmetric matrices, 343-347 Symmetry, Fourier transform theorem of, 650-651 Index T Tangent planes, 503-509, 551-55 defined, 504 differential calculus, 503-509 equation of, 505-50 surfaces, to, 551-55 Taylor method, second order, 190-19 Taylor series, 956, 1007-1 01 coefficients, 95 theorems for, 1007-101 Temperature, 843, 844-84 conditions of, 843 distribution, 848-85 insulated boundary conditions, 843, 847-848 zero, 844-84 Term-by-term integration, 988-98 Terminal velocity, 47-4 Test for exactness, 30-32 Three point theorem, 1069-107 3-space, 545-554, 576-579 domain, 578-57 potential theory in, 576-57 surfaces in, 545-556 Tides, FFT analysis of, 697-69 Time reversal, Fourier transform theorem of, 65 Time shifting, Fourier transform theorem of, 647-64 Torricelli's law, 18-1 Torsion, 493 Trajectories, 407-413, 413-42 linear systems, 413-424 nonlinear systems, 407-41 translation, 409-41 Transfer coefficient, 849 Transformations, 660-661, 667-669, 851-853, 1064-1071, 1077-1080, 1093-1094 See also Mapping s boundary values, 851-85 Fourier, 660-661, 667-66 Joukowski, 1093-1094 linear fractional, 1064-107 Schwartz-Christoffel, 1077-1080 Transpose, 245 Tree diagrams, 1107-1111, 1130-113 computing probabilities, 1130-113 configuration of, 1107-111 Trigonometric functions, 957-966 U Unbounded domains, 815-821, 90 Underdamped forced motion, 100 Underdamping, Undetermined coefficients, method of, 85-9 Unforced motion, 95-97 Uniform and absolute convergence, 620-62 Unique solutions, 58-60, 285-292, 29 existence and, theorem for, 59-6 initial value problems, 58-6 inverses, 294 nonhomogeneous systems of linear equations, 285-292 Uniqueness, nonlinear systems, 40 Unit impulses, 139-14 Unit normal vector, 486-488 Unit vectors, 20 Unitary matrices, 352-355 V van der Pol equation, 468-470 Variation of parameters method, 82-84, 394-39 Laplace transform, 379-39 nonhomogeneous equations, solution for, 82-8 solution of X' = AX +G, 394-39 Variation, measures of, 1146-114 range, 1146-1147 standard deviation, 1147-1149 Vector analysis, 473, 475-515, 517-58 differential calculus, 473, 475-51 integral calculus, 473, 517-580 Vector fields, 493-499, 560-56 defined, 493 differential calculus, 493-49 flux across surfaces, 560-56 streamlines (flow lines), 495-49 Vector functions, 475-48 differential calculus, 475-48 one variable, 475-48 Vector space R", 223-22 algebra of, 223-22 Cauchy-Schwarz inequality in, 22 dot product of, 224-22 n-vector, defined, 223 subspace, 226-227, 227-228 Vectors, 201-35 algebra and geometry of, 203-21 basis, 254 Cauchy-Schwarz inequality, 216-21 components, 204 cross product, 217-22 defined, 20 determinants, 299-322 diagonalization, 330-33 dimension, 23 dot product, 211-21 eigenvalues, 323-33 linear equations, 237-29 linear independence, 228-23 matrices, 237-298, 330-339, 339-347, 352-35 norm of, 20 orthogonal, 21 parallel, 205 parallelogram law, 206-20 product of a scalar and, 20 quadratic forms, 347-35 space R",223-22 spanning set, 229-23 subspace, 226-227, 227-22 sum of, 206 unit, 207 Velocity, 47-48, 481-483, 786-79 initial, 791-79 terminal, 47-4 vector, analysis, 481-48 wave equation, 786-794 zero initial, 786-791 113 114 Index Vibration, 831-84 circular membrane, 831-83 normal modes, 831-83 periodicity conditions, 834-837 rectangular membrane, 837-840 wave equation, 831-84 Vortex, strength of, 108 W Walk, defined, 247 Wave equation, 781-84 approximations, 801-804 boundary conditions, 781-78 boundary value problems, 786-79 characteristics, 822-83 d'Alemnbert's solution, 822-830 displacement, 791-794 forward and backward, 828-83 Fourier series solutions, 786-808 Fourier transform solution on unbounded domains, 815-82 initial conditions, 781-785, 798-80 lattice points, 803 motion, 798-801, 808-822 nonhomogeneous, 825-82 numerical solution, 801-805 velocity, 786-794 vibration, 831-840 Wavelets, 581-582, 765-77 construction of, 775-77 expansion, 774 Haar, 767-774, 774-775 idea behind, 765-767 multiresolution analysis, 774-775, 775-776 Shannon, 776-77 use of, 582 Windowed Fourier transform, 661-66 Wronskian test, 66-67 Z Zero matrix, 243-244 Zero row, 25 Zeros, 713-714, 737-739, 1012-101 Bessel functions, 737-73 isolated, 1012-101 Legendre polynomials, 701-71 order of, 1013-1014 Guide to Notation The following symbols and notation are used throughout this text Each symbol is paired with a section in which it is defined or used Standard symbols, such as notation for integrals an d sums, are not included W[f,g] Wronskian off and g (2.2) G[j] Laplace transform off (3 ) L[f](s) Laplace transform off evaluated at s (3 ) [F] G H(t) inverse Laplace transform of F Heaviside function (3 ) 8(t) Dirac delta function (3 5) (a, b, c) vector with three components (6 ) norm (magnitude) of a vector v (6 ) Il v il F G dot product of F and G (6 2) F x G cross product of F and G (6 ) R" n-space ; set of all n-vectors (6 ) [au] matrix whose i, j element is a, (7 ) n x rn zero matrix (7 ) I,, n x n identity matrix (7 3) A' transpose of A (7 ) AR reduced row echelon form of A (7 ) rank(A) rank of A (7 4) [A:B] augmented matrix (7 ) A -' V del operator (12 ) Vcp or grad(co) gradient of cp (12 4) D„cp(P) directional derivative of cp in the directio n of u, evaluated at P (12 4) fc fdx+gdy+hdz line integral over C (13 ) F fc dR another notation for fc fdx+gdy+hdz , with F= fi+gj+hk (13 ) fc f(x, y, z)ds line integral off with respect to arc length (13 1 ) 8(f,g) Jacobian of f and g with respect to u and 8(u, v) v (13 ) f f* f(x, y, z)du surface integral off over a surface E (13 4) f(xo -), f(xo+) left and right limits, respectively , off at xo (14 ) f',(xo), ff(xo) left and right derivatives (respectively) off at xo (14 2) f], or f inverse of A (7 ) [AI or det(A) determinant of A (8 2) Au often denotes the minor of the i, j element of A (9 ) pA (A) characteristic polynomial of A (9 ) ,fl In the context of a system X ' = AX, denotes a fundamental matrix (10.1) ; in the context of th e fast Fourier transform, denotes the set of n" roots of unity (15 ) T often denotes a unit tangent vector to a curve (12.] ) K N often denotes a normal (or unit normal) to a curve (12 2) curvature (12 2) [f] s w, [f] sw Fourier transform off (15 ) inverse Fourier transform off (15 ) windowed Fourier transform off (15 4.6) [f] windowed Fourier transform of shifted f (15 4.6) 2i c[f], or fc((0 ) (15 ) or fs( w ) (15 ) ^SS[ ,