E. Kreyszig / E. J. Norminton

MAPLE COMPUTER GUIDE

A Self-Contained Introduction

For

Erwin Kreyszig

ADVANCED ENGINEERING MATHEMATICS

Ninth Edition

JOHN WILEY & SONS, INC.

Contents

PART A. ORDINARY DIFFERENTIAL EQUATIONS (ODEs) 7 Chapter 1 First-Order ODEs 8 Ex. 1.1 General Solutions, 8 Ex. 1.2 Direction Fields, 9 Ex. 1.3 Mixing Problems, 10 Ex. 1.4 Integrating Factors, 11 Ex. 1.5 Bernoulli's Equation, 12 Ex. 1.6 ^ C i r c u i t , 13 Problems for Chapter 1, 14

Chapters 2 and 3 Linear ODEs of Second and Higher Order 16 Ex. 2.1 General Solution. Initial Value Problem, 16 Ex. 2.2 Mass-Spring System. Complex Roots. Damped Oscillations, 18 Ex. 2.3 The three Cases of Damping, 18 Ex. 2.4 The three Cases for an Euler-Cauchy Equation, 20 Ex. 2.5 Wronskian, 21 Ex. 2.6 Nonhomogeneous Linear ODE's, 22 Ex. 2.7 Solution by Undetermined Coefficients, 23 Ex. 2.8 Solution by Variation of Parameters, 24 Ex. 2.9 Forced Vibrations. Resonance. Beats, 25 Ex. 2.10 ALC-Circuit, 27 Problems for Chapters 2 and 3, 28

Chapter 4 Systems of ODEs. Phase Plane, Qualitative Methods 31

Ex. 4.1 How to Write a System of ODEs? Initial Value Problem, 31 Ex. 4.2 Use of Matrices in Solving Systems of ODEs, 32 Ex. 4.3 Critical Points. Node, 33 Ex. 4.4 Proper Node, Saddle Point, Center, Spiral Point, 35 Ex. 4.5 Pendulum Equation, 36 Ex. 4.6 Nonhomogeneous System, 37 Ex. 4.7 Method of Undetermined Coefficients, 38 Ex. 4.8 Van der Pol Equation. Limit Cycle, 39 Problems for Chapter 4, 40

Chapter 5 Series Solutions of ODEs. Special Functions 42 Ex. 5.1 Power Series Solutions. Plots from them. Numeric Values, 42 Ex. 5.2 Legendre Polynomials. Orthopoly Package. Procedures, 44 Ex. 5.3 Legendre's Differential Equation, 46 Ex. 5.4 Orthogonality. Fourier-Legendre Series, 47 Ex. 5.5 Frobenius Method, 49 Ex. 5.6 Bessel's Equation. Bessel Functions, 50 Problems for Chapter 5, 52

VI Contents

Ex. Ex. Ex. Ex. Ex.

7.2 7.3 7.4 7.5 7.6

Chapter 6 Laplace Transforms 55 Ex. 6.1 Further Transforms and Inverse Transforms, 55 Ex. 6.2 Differential Equations, 56 Ex. 6.3 Forced Vibrations. Resonance, 58 Ex. 6.4 Unit Step Function (Heaviside Function), Dirac's Delta, 59 Ex. 6.5 Solution of Systems by Laplace Transform, 62 Ex. 6.6 Formulas on General Properties of the Laplace Transform, 63 Problems for Chapter 6, 65

PART B. LINEAR ALGEBRA, VECTOR CALCULUS 67 Chapter 7 Matrices, Vectors, Determinants. Linear Systems

of Equations 67 Ex. 7.1 Matrix Addition, Scalar Multiplication, Matrix Multiplication.

Vectors, 67 Special Matrices, 70 Changing and Composing Matrices, Accessing Entries. Submatrices, 70 Solution of a Linear System, 72 Gauss Elimination; Further Cases, 73 Rank. Row Space. Linear Independence, 74

Problems for Chapter 7, 76

Chapter 8 Linear Algebra: Matrix Eigenvalue Problems 78 Ex. 8.1 Eigenvalues, Eigenvectors, Accessing Spectrum, 78 Ex. 8.2 Real Matrices with Complex Eigenvalues, 79 Ex. 8.3 Orthogonal Matrices and Transformations, 80 Ex. 8.4 Complex Matrices, 81 Ex. 8.5 Similarity of Matrices. Diagonalization, 83 Problems for Chapter 8, 85

Chapter 9 Vector Differential Calculus. Grad, Div, Curl 87

Ex. 9.1 Vectors, Length, Addition, Scalar Multiplication, 87 Ex. 9.2 Inner Product. Cross Product, 88 Ex. 9.3 Differentiation of Vectors. Curves and their Properties, 89 Ex. 9.4 Gradient. Directional Derivative. Potential, 91 Ex. 9.5 Divergence, Laplacian, Curl, 93 Problems for Chapter 9, 94

Chapter 10 Vector Integral Calculus. Integral Theorems 96 Ex. 10.1 Line Integrals, 96

Independence of Path, 97 Double Integrals. Moments of Inertia, 98 Green's Theorem in the Plane, 99 Surface Integrals. Flux, 101 Divergence Theorem of Gauss, 102 Stokes's Theorem, 104

Problems for Chapter 10, 105

Ex. Ex. Ex. Ex. Ex. Ex.

10.2 10.3 10.4 10.5 10.6 10.7

Contents V l l

PART С FOURIER ANALYSIS. PARTIAL DIFFERENTIAL EQUATIONS (PDEs) 107

Chapter 11 Fourier Series, Integrals, and Transforms 107 Ex. 11.1 Functions of Period 2тт. Even Functions. Gibbs Phenomenon, 108 Ex. 11.2 Functions of Arbitrary Period. Odd Functions, 109 Ex. 11.3 Half-Range Expansions, 111 Ex. 11.4 Rectifier, 113 Ex. 11.5 Trigonometric Approximation. Minimum Square Error, 114 Ex. 11.6 Fourier Integral, Fourier Transform, 115 Problems for Chapter 11, 116

Chapter 12 Partial Differential Equations (PDEs) 118 Ex. 12.1 Wave Equation. Separation of Variables. Animation, 118 Ex. 12.2 Wave Equation: D'Alembert's Solution Method., 120 Ex. 12.3 One-Dimensional Heat Equation, 121 Ex. 12.4 Heat Equation, Laplace Equation 122 Ex. 12.5 Rectangular Membrane. Double Fourier Series, 124 Ex. 12.6 Laplacian. Circular Membrane. Bessel Equation, 125 Problems for Chapter 12, 128

PART D. COMPLEX ANALYSIS 130 Chapter 13/17 Complex Numbers and Functions.

Conformal Mapping 130 Ex. 13/17.1 Complex Numbers. Polar form. Plotting, 130 Ex. 13/17.2 Equations. Roots. Sets in the Complex Plane, 132 Ex. 13/17.3 Cauchy-Riemann Equations. Harmonic Functions, 134 Ex. 13/17.4 Conformal Mapping, 136 Ex. 13/17.5 Exponential, Trigonometric, and Hyperbolic Functions, 137 Ex. 13/17.6 Complex Logarithm, 140 Problems for Chapters 13 and 17, 141

Chapter 14 Complex Integration 143 Ex. 14.1 Indefinite Integration of Analytic Functions, 143 Ex. 14.2 Integration: Use of Path. Path Dependence, 143 Ex. 14.3 Contour Integration by Cauchy's Integral Theorem and Formula, 145 Problems for Chapter 14, 146

Chapter 15 Power Series, Taylor Series 148 Ex. 15.1 Sequences and their Plots, 148 Ex. 15.2 Convergence Tests for Complex Series, 149 Ex. 15.3 Power Series. Radius of Convergence, 150 Ex. 15.4 Taylor Series, 151 Ex. 15.5 Uniform Convergence, 153 Problems for Chapter 15, 155

Chapter 16 Laurent Series. Residue Integration 157 Ex. 16.1 Laurent Series, 157 Ex. 16.2 Singularities and Zeros, 158 Ex. 16.3 Residue Integration, 159

V l l l Contents

Ex. 16.4 Real Integrals of Rational Functions of cos and sin, 161 Ex. 16.5 Improper Real Integrals of Rational Functions, 162 Problems for Chapter 16, 164

Chapter 17 See before

Chapter 18 Complex Analysis and Potential Theory 165 Ex. 18.1 Complex Potential. Related Plots, 165 Ex. 18.2 Use of Conformal Mapping, 166 Ex. 18.3 Fluid Flow, 167 Ex. 18.4 Series Representation of Potential, 168 Ex. 18.5 Mean Value Theorem for Analytic Functions, 169 Problems for Chapter 18, 170

PART E. NUMERIC ANALYSIS 171 Chapter 19 Numerics in General 171 Ex. 19.1 Loss of Significant Digits. Quadratic Equation, 171 Ex. 19.2 Fixed-Point Iteration, 172 Ex. 19.3 Solving Equations by Newton's Method, 174 Ex. 19.4 Solving Equations by the Secant Method, 174 Ex. 19.5 Solving Equations by the Bisection Method. Procedure, 175 Ex. 19.6 Polynomial Interpolation, 176 Ex. 19.7 Spline Interpolation, 177 Ex. 19.8 Numeric Integration, 179 Problems for Chapter 19, 180

Chapter 20 Numeric Linear Algebra 182 Ex. 20.1 Gauss Elimination. Pivoting, 182 Ex. 20.2 Doolittle LU-Factorization, 185 Ex. 20.3 Cholesky Factorization, 186 Ex. 20.4 Gauss-Jordan Elimination. Matrix Inversion, 186 Ex. 20.5 Gauss-Seidel Iteration for Linear Systems, 188 Ex. 20.6 Vector and Matrix Norms. Condition Numbers, 189 Ex. 20.7 Fitting Data by Least Squares, 190 Ex. 20.8 Approximation of Eigenvalues: Collatz Method, 192 Ex. 20.9 Approximation of Eigenvalues: Power Method, 193 Ex. 20.10 Approximation of Eigenvalues: QR-Factorization, 194 Problems for Chapter 20, 196

Chapter 21 Numerics for ODEs and PDEs 198 Ex. 21.1 Two Ways of Writing an ODE, 198 Ex. 21.2 Euler Method, 198 Ex. 21.3 Improved Euler Method, 200 Ex. 21.4 Classical Runge-Kutta Method (RK). Procedure, 201 Ex. 21.5 Adams-Moulton Multistep Method, 202 Ex. 21.6 Classical Runge-Kutta Method for Systems (RKS), 204 Ex. 21.7 Classical Runge-Kutta-Nystroem Method (RKN), 205 Ex. 21.8 Laplace Equation. Boundary Value Problem, 206 Ex. 21.9 Heat Equation. Crank-Nicolson Method, 210 Problems for Chapter 21, 212

Contents ix

PART F. OPTIMIZATION, GRAPHS 214 Chapter 22 Unconstrained Optimization. Linear Programming 214 Ex. 22.1 Method of Steepest Descent, 214 Ex. 22.2 Simplex Method of Constrained Optimization, 216 Problems for Chapter 22, 218

Chapter 23 Graphs and Combinatorial Optimization 219 Ex. 23.1 Graphs and Digraphs. Their Matrices, 219 Ex. 23.2 Shortest Spanning Trees, 221 Ex. 23.3 Flow in Networks, 222 Problems for Chapter 23, 224

PART G. PROBABILITY, STATISTICS 225 Chapter 24 Data Analysis. Probability Theory 225 Ex. 24.1 Data Analysis: Mean, Variance, Standard Deviation, 225 Ex. 24.2 Data Analysis: Histograms, Boxplots, 226 Ex. 24.3 Discrete Probability Distributions, 227 Ex. 24.4 Normal Distribution, 230 Problems for Chapter 24, 232

Chapter 25 Mathematical Statistics 234 Ex. 25.1 Random Numbers, 234 Ex. 25.2 Confidence Interval for the Mean of the Normal Distribution

With Known Variance, 235 Ex. 25.3 Confidence Interval for the Mean of the Normal Distribution

With Unknown Variance. ^-Distribution, 236 Ex. 25.4 Confidence Interval for the Variance of the Normal Distribution.

^-Distribution, 237 Test for the Mean of the Normal Distribution, 238 Test for the Mean: Power Function, 238 Test for the Variance of the Normal Distribution, 240 Comparison of Means, 240 Comparison of Variances. F-Distribution, 241 Chi-Square Test for Goodness of Fit, 242 Regression, 242

Problems for Chapter 25, 244

Appendix 1 References A l

Appendix 2 Answers to Odd-Numbered Problems A2

Ex. Ex. Ex. Ex. Ex. Ex. Ex.

25.5 25.6 25.7 25.8 25.9 25.10 25.11

Index J l