International Student Edition ADVANCED ENGINEERING MATHEMATICS This page intentionally left blank ADVANCED ENGINEERING MATHEMATICS International Student Edition PETER V O’NEIL University of Alabama at Birmingham Australia Canada Mexico Singapore Spain United Kingdom United States Advanced Engineering Mathematics, International Student Edition by Peter V O’Neil Associate Vice-President and Editorial Director: Evelyn Veitch Publisher: Chris Carson Developmental Editor: Kamilah Reid Burrell/ Hilda Gowaus Production Services: RPK Editorial Services Creative Director: Angela Cluer Copy Editor: Shelly Gerger-Knechtl/ Harlan James Interior Design: Terri Wright Proofreader: Erin Wagner/Harlan James Cover Design: Andrew Adams Indexer: RPK Editorial Services Compositor: Integra Permissions Coordinator: Vicki Gould Production Manager: Renate McCloy Printer: Quebecor World COPYRIGHT © 2007 by Nelson, a division of Thomson Canada Limited ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transcribed, or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, or information storage and retrieval systems—without the written permission of the publisher North America Nelson 1120 Birchmount Road Toronto, Ontario M1K 5G4 Canada Printed and bound in the United States 07 06 For more information contact Nelson, 1120 Birchmount Road, Toronto, Ontario, Canada, M1K 5G4 Or you can visit our Internet site at http://www.nelson.com Library of Congress Control Number: 2006900028 ISBN: 0-495-08237-6 If you purchased this book within the United States or Canada you should be aware that it has been wrongfully imported without the approval of the Publisher or the Author For permission to use material from this text or product, submit a request online at www.thomsonrights.com Every effort has been made to trace ownership of all copyright material and to secure permission from copyright holders In the event of any question arising as to the use of any material, we will be pleased to make the necessary corrections in future printings Asia Thomson Learning Shenton Way #01-01 UIC Building Singapore 068808 Australia/New Zealand Thomson Learning 102 Dodds Street Southbank, Victoria Australia 3006 Europe/Middle East/Africa Thomson Learning High Holborn House 50/51 Bedford Row London WC1R 4LR United Kingdom Latin America Thomson Learning Seneca, 53 Colonia Polanco 11560 Mexico D.F Mexico Spain Paraninfo Calle/Magallanes, 25 28015 Madrid, Spain Contents PART Chapter Ordinary Differential Equations First-Order Differential Equations 1.1 Preliminary Concepts 1.1.1 General and Particular Solutions 1.1.2 Implicitly Defined Solutions 1.1.3 Integral Curves 1.1.4 The Initial Value Problem 1.1.5 Direction Fields 1.2 Separable Equations 11 1.2.1 Some Applications of Separable Differential Equations 14 1.3 Linear Differential Equations 22 1.4 Exact Differential Equations 26 1.5 Integrating Factors 33 1.5.1 Separable Equations and Integrating Factors 37 1.5.2 Linear Equations and Integrating Factors 37 1.6 Homogeneous, Bernoulli, and Riccati Equations 38 1.6.1 Homogeneous Differential Equations 38 1.6.2 The Bernoulli Equation 42 1.6.3 The Riccati Equation 43 1.7 Applications to Mechanics, Electrical Circuits, and Orthogonal Trajectories 1.7.1 Mechanics 46 1.7.2 Electrical Circuits 51 1.7.3 Orthogonal Trajectories 53 1.8 Existence and Uniqueness for Solutions of Initial Value Problems 58 Chapter Second-Order Differential Equations 46 61 2.1 Preliminary Concepts 61 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 69 2.4 The Constant Coefficient Homogeneous Linear Equation 73 2.4.1 Case 1: A − 4B > 73 2.4.2 Case 2: A − 4B = 74 68 v vi Contents 2.4.3 Case 3: A − 4B < 74 2.4.4 An Alternative General Solution in the Complex Root Case 75 2.5 Euler’s Equation 78 2.6 The Nonhomogeneous Equation y + p x y + q x y = f x 82 2.6.1 The Method of Variation of Parameters 82 2.6.2 The Method of Undetermined Coefficients 85 2.6.3 The Principle of Superposition 91 2.6.4 Higher-Order Differential Equations 91 2.7 Application of Second-Order Differential Equations to a Mechanical System 2.7.1 Unforced Motion 95 2.7.2 Forced Motion 98 2.7.3 Resonance 100 2.7.4 Beats 102 2.7.5 Analogy with an Electrical Circuit 103 Chapter The Laplace Transform 107 3.1 Definition and Basic Properties 107 3.2 Solution of Initial Value Problems Using the Laplace Transform 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 122 3.3.3 The Second Shifting Theorem 125 3.3.4 Analysis of Electrical Circuits 129 3.4 Convolution 134 3.5 Unit Impulses and the Dirac Delta Function 139 3.6 Laplace Transform Solution of Systems 144 3.7 Differential Equations with Polynomial Coefficients 150 Chapter Series Solutions 4.1 4.2 4.3 4.4 Chapter 155 Power Series Solutions of Initial Value Problems 156 Power Series Solutions Using Recurrence Relations 161 Singular Points and the Method of Frobenius 166 Second Solutions and Logarithm Factors 173 Numerical Approximation of Solutions 181 5.1 Euler’s Method 182 5.1.1 A Problem in Radioactive Waste Disposal 5.2 One-Step Methods 190 5.2.1 The Second-Order Taylor Method 190 5.2.2 The Modified Euler Method 193 5.2.3 Runge-Kutta Methods 195 5.3 Multistep Methods 197 5.3.1 Case r = 198 5.3.2 Case r = 198 5.3.3 Case r = 199 5.3.4 Case r = 199 187 116 93 Contents PART Chapter Vectors and Linear Algebra 201 Vectors and Vector Spaces 6.1 6.2 6.3 6.4 6.5 Chapter 203 The Algebra and Geometry of Vectors 203 The Dot Product 211 The Cross Product 217 The Vector Space R n 223 Linear Independence, Spanning Sets, and Dimension in R n Matrices and Systems of Linear Equations 228 237 7.1 Matrices 238 7.1.1 Matrix Algebra 239 7.1.2 Matrix Notation for Systems of Linear Equations 242 7.1.3 Some Special Matrices 243 7.1.4 Another Rationale for the Definition of Matrix Multiplication 246 7.1.5 Random Walks in Crystals 247 7.2 Elementary Row Operations and Elementary Matrices 251 7.3 The Row Echelon Form of a Matrix 258 7.4 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 = O 280 7.7 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 7.8 Matrix Inverses 293 7.8.1 A Method for Finding A −1 295 Chapter Determinants 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 Chapter 299 Permutations 299 Definition of the Determinant 301 Properties of Determinants 303 Evaluation of Determinants by Elementary Row and Column Operations Cofactor Expansions 311 Determinants of Triangular Matrices 314 A Determinant Formula for a Matrix Inverse 315 Cramer’s Rule 318 The Matrix Tree Theorem 320 Eigenvalues, Diagonalization, and Special Matrices 9.1 Eigenvalues and Eigenvectors 324 9.1.1 Gerschgorin’s Theorem 328 9.2 Diagonalization of Matrices 330 9.3 Orthogonal and Symmetric Matrices 339 323 307 vii viii Contents 9.4 Quadratic Forms 347 9.5 Unitary, Hermitian, and Skew Hermitian Matrices PART Chapter 10 352 Systems of Differential Equations and Qualitative Methods 359 Systems of Linear Differential Equations 361 10.1 Theory of Systems of Linear First-Order Differential Equations 361 10.1.1 Theory of the Homogeneous System X = AX 365 10.1.2 General Solution of the Nonhomogeneous System X = AX + G 372 10.2 Solution of X = AX when A is Constant 374 10.2.1 Solution of X = AX when A has Complex Eigenvalues 377 10.2.2 Solution of X = AX when A does not have n Linearly Independent Eigenvectors 379 10.2.3 Solution of X = AX by Diagonalizing A 384 10.2.4 Exponential Matrix Solutions of X = AX 386 10.3 Solution of X = AX + G 394 10.3.1 Variation of Parameters 394 10.3.2 Solution of X = AX + G by Diagonalizing A 398 Chapter 11 Qualitative Methods and Systems of Nonlinear Differential Equations 11.1 11.2 11.3 11.4 11.5 11.6 11.7 PART Chapter 12 Nonlinear Systems and Existence of Solutions 403 The Phase Plane, Phase Portraits and Direction Fields Phase Portraits of Linear Systems 413 Critical Points and Stability 424 Almost Linear Systems 431 Lyapunov’s Stability Criteria 451 Limit Cycles and Periodic Solutions 461 406 Vector Analysis 473 Vector Differential Calculus 475 12.1 Vector Functions of One Variable 475 12.2 Velocity, Acceleration, Curvature and Torsion 481 12.2.1 Tangential and Normal Components of Acceleration 488 12.2.2 Curvature as a Function of t 491 12.2.3 The Frenet Formulas 492 12.3 Vector Fields and Streamlines 493 12.4 The Gradient Field and Directional Derivatives 499 12.4.1 Level Surfaces, Tangent Planes and Normal Lines 503 12.5 Divergence and Curl 510 12.5.1 A Physical Interpretation of Divergence 512 12.5.2 A Physical Interpretation of Curl 513 403 Contents Chapter 13 Vector Integral Calculus 517 13.1 Line Integrals 517 13.1.1 Line Integral with Respect to Arc Length 525 13.2 Green’s Theorem 528 13.2.1 An Extension of Green’s Theorem 532 13.3 Independence of Path and Potential Theory in the Plane 536 13.3.1 A More Critical Look at Theorem 13.5 539 13.4 Surfaces in 3-Space and Surface Integrals 545 13.4.1 Normal Vector to a Surface 548 13.4.2 The Tangent Plane to a Surface 551 13.4.3 Smooth and Piecewise Smooth Surfaces 552 13.4.4 Surface Integrals 553 13.5 Applications of Surface Integrals 557 13.5.1 Surface Area 557 13.5.2 Mass and Center of Mass of a Shell 557 13.5.3 Flux of a Vector Field Across a Surface 560 13.6 Preparation for the Integral Theorems of Gauss and Stokes 562 13.7 The Divergence Theorem of Gauss 564 13.7.1 Archimedes’s Principle 567 13.7.2 The Heat Equation 568 13.7.3 The Divergence Theorem as a Conservation of Mass Principle 13.8 The Integral Theorem of Stokes 572 13.8.1 An Interpretation of Curl 576 13.8.2 Potential Theory in 3-Space 576 PART Chapter 14 570 Fourier Analysis, Orthogonal Expansions, and Wavelets 581 Fourier Series 583 14.1 Why Fourier Series? 583 14.2 The Fourier Series of a Function 586 14.2.1 Even and Odd Functions 589 14.3 Convergence of Fourier Series 593 14.3.1 Convergence at the End Points 599 14.3.2 A Second Convergence Theorem 601 14.3.3 Partial Sums of Fourier Series 604 14.3.4 The Gibbs Phenomenon 606 14.4 Fourier Cosine and Sine Series 609 14.4.1 The Fourier Cosine Series of a Function 610 14.4.2 The Fourier Sine Series of a Function 612 14.5 Integration and Differentiation of Fourier Series 614 14.6 The Phase Angle Form of a Fourier Series 623 14.7 Complex Fourier Series and the Frequency Spectrum 630 14.7.1 Review of Complex Numbers 630 14.7.2 Complex Fourier Series 631 ix ...International Student Edition ADVANCED ENGINEERING MATHEMATICS This page intentionally left blank ADVANCED ENGINEERING MATHEMATICS International Student Edition PETER V O’NEIL University of Alabama... United States Advanced Engineering Mathematics, International Student Edition by Peter V O’Neil Associate Vice-President and Editorial Director: Evelyn Veitch Publisher: Chris Carson Developmental... from the fifth edition have been moved to a website, located at http:/ /engineering. thomsonlearning.com I hope that this provides convenient accessibility Material selected for this move includes