Part A Ordinary Differential Equations (ODEs) 1

Chapter 1 First-Order ODEs 2

1.1 Basic Concepts. Modeling 2

1.2 Geometric Meaning of y’= ƒ(x, y). Direction Fields, Euler’s Method 9

1.3 Separable ODEs. Modeling 12

1.4 Exact ODEs. Integrating Factors 20

1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 27

1.6 Orthogonal Trajectories. Optional 36

1.7 Existence and Uniqueness of Solutions for Initial Value Problems 38

Chapter 1 Review Questions and Problems 43

Summary of Chapter 1 44

Chapter 2 Second-Order Linear ODEs 46

2.1 Homogeneous Linear ODEs of Second Order 46

2.2 Homogeneous Linear ODEs with Constant Coefficients 53

2.3 Differential Operators. Optional 60

2.4 Modeling of Free Oscillations of a Mass–Spring System 62

2.5 Euler–Cauchy Equations 71

2.6 Existence and Uniqueness of Solutions. Wronskian 74

2.7 Nonhomogeneous ODEs 79

2.8 Modeling: Forced Oscillations. Resonance 85

2.9 Modeling: Electric Circuits 93

2.10 Solution by Variation of Parameters 99

Chapter 2 Review Questions and Problems 102

Summary of Chapter 2 103

Chapter 3 Higher Order Linear ODEs 105

3.1 Homogeneous Linear ODEs 105

3.2 Homogeneous Linear ODEs with Constant Coefficients 111

3.3 Nonhomogeneous Linear ODEs 116

Chapter 3 Review Questions and Problems 122

Summary of Chapter 3 123

Chapter 4 Systems of ODEs. Phase Plane. Qualitative Methods 124

4.0 For Reference: Basics of Matrices and Vectors 124

4.1 Systems of ODEs as Models in Engineering Applications 130

4.2 Basic Theory of Systems of ODEs. Wronskian 137

4.3 Constant-Coefficient Systems. Phase Plane Method 140

4.4 Criteria for Critical Points. Stability 148

4.5 Qualitative Methods for Nonlinear Systems 152

4.6 Nonhomogeneous Linear Systems of ODEs 160

Chapter 4 Review Questions and Problems 164

Summary of Chapter 4 165

Chapter 5 Series Solutions of ODEs. Special Functions 167

5.1 Power Series Method 167

5.2 Legendre’s Equation. Legendre Polynomials P_n(x) 175

5.3 Extended Power Series Method: Frobenius Method 180

5.4 Bessel’s Equation. Bessel Functions J_v(x) 187

5.5 Bessel Functions of the Y_v(x). General Solution 196

Chapter 5 Review Questions and Problems 200

Summary of Chapter 5 201

Chapter 6 Laplace Transforms 203

6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting) 204

6.2 Transforms of Derivatives and Integrals. ODEs 211

6.3 Unit Step Function (Heaviside Function).

Second Shifting Theorem (t-Shifting) 217

6.4 Short Impulses. Dirac’s Delta Function. Partial Fractions 225

6.5 Convolution. Integral Equations 232

6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients 238

6.7 Systems of ODEs 242

6.8 Laplace Transform: General Formulas 248

6.9 Table of Laplace Transforms 249

Chapter 6 Review Questions and Problems 251

Summary of Chapter 6 253

Part B Linear Algebra. Vector Calculus 255

Chapter 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems 256

7.1 Matrices, Vectors: Addition and Scalar Multiplication 257

7.2 Matrix Multiplication 263

7.3 Linear Systems of Equations. Gauss Elimination 272

7.4 Linear Independence. Rank of a Matrix. Vector Space 282

7.5 Solutions of Linear Systems: Existence, Uniqueness 288

7.6 For Reference: Second- and Third-Order Determinants 291

7.7 Determinants. Cramer’s Rule 293

7.8 Inverse of a Matrix. Gauss–Jordan Elimination 301

7.9 Vector Spaces, Inner Product Spaces. Linear Transformations. Optional 309

Chapter 7 Review Questions and Problems 318

Summary of Chapter 7 320

Chapter 8 Linear Algebra: Matrix Eigenvalue Problems 322

8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors 323

8.2 Some Applications of Eigenvalue Problems 329

8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices 334

8.4 Eigenbases. Diagonalization. Quadratic Forms 339

8.5 Complex Matrices and Forms. Optional 346

Chapter 8 Review Questions and Problems 352

Summary of Chapter 8 353

Chapter 9 Vector Differential Calculus. Grad, Div, Curl 354

9.1 Vectors in 2-Space and 3-Space 354

9.2 Inner Product (Dot Product) 361

9.3 Vector Product (Cross Product) 368

9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives 375

9.5 Curves. Arc Length. Curvature. Torsion 381

9.6 Calculus Review: Functions of Several Variables. Optional 392

9.7 Gradient of a Scalar Field. Directional Derivative 395

9.8 Divergence of a Vector Field 403

9.9 Curl of a Vector Field 406

Chapter 9 Review Questions and Problems 409

Summary of Chapter 9 410

Chapter 10 Vector Integral Calculus. Integral Theorems 413

10.1 Line Integrals 413

10.2 Path Independence of Line Integrals 419

10.3 Calculus Review: Double Integrals. Optional 426

10.4 Green’s Theorem in the Plane 433

10.5 Surfaces for Surface Integrals 439

10.6 Surface Integrals 443

10.7 Triple Integrals. Divergence Theorem of Gauss 452

10.8 Further Applications of the Divergence Theorem 458

10.9 Stokes’s Theorem 463

Chapter 10 Review Questions and Problems 469

Summary of Chapter 10 470

Part C Fourier Analysis. Partial Differential Equations (PDEs) 473

Chapter 11 Fourier Analysis 474

11.1 Fourier Series 474

11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions 483

11.3 Forced Oscillations 492

11.4 Approximation by Trigonometric Polynomials 495

11.5 Sturm–Liouville Problems. Orthogonal Functions 498

11.6 Orthogonal Series. Generalized Fourier Series 504

11.7 Fourier Integral 510

11.8 Fourier Cosine and Sine Transforms 518

11.9 Fourier Transform. Discrete and Fast Fourier Transforms 522

11.10 Tables of Transforms 534

Chapter 11 Review Questions and Problems 537

Summary of Chapter 11 538

Chapter 12 Partial Differential Equations (PDEs) 540

12.1 Basic Concepts of PDEs 540

12.2 Modeling: Vibrating String, Wave Equation 543

12.3 Solution by Separating Variables. Use of Fourier Series 545

12.4 D’Alembert’s Solution of the Wave Equation. Characteristics 553

12.5 Modeling: Heat Flow from a Body in Space. Heat Equation 557

12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem 558

12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms 568

12.8 Modeling: Membrane, Two-Dimensional Wave Equation 575

12.9 Rectangular Membrane. Double Fourier Series 577

12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series 585

12.11 Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential 593

12.12 Solution of PDEs by Laplace Transforms 600

Chapter 12 Review Questions and Problems 603

Summary of Chapter 12 604

Part D Complex Analysis 607

Chapter 13 Complex Numbers and Functions. Complex Differentiation 608

13.1 Complex Numbers and Their Geometric Representation 608

13.2 Polar Form of Complex Numbers. Powers and Roots 613

13.3 Derivative. Analytic Function 619

13.4 Cauchy–Riemann Equations. Laplace’s Equation 625

13.5 Exponential Function 630

13.6 Trigonometric and Hyperbolic Functions. Euler’s Formula 633

13.7 Logarithm. General Power. Principal Value 636

Chapter 13 Review Questions and Problems 641

Summary of Chapter 13 641

Chapter 14 Complex Integration 643

14.1 Line Integral in the Complex Plane 643

14.2 Cauchy’s Integral Theorem 652

14.3 Cauchy’s Integral Formula 660

14.4 Derivatives of Analytic Functions 664

Chapter 14 Review Questions and Problems 668

Summary of Chapter 14 669

Chapter 15 Power Series, Taylor Series 671

15.1 Sequences, Series, Convergence Tests 671

15.2 Power Series 680

15.3 Functions Given by Power Series 685

15.4 Taylor and Maclaurin Series 690

15.5 Uniform Convergence. Optional 698

Chapter 15 Review Questions and Problems 706

Summary of Chapter 15 706

Chapter 16 Laurent Series. Residue Integration 708

16.1 Laurent Series 708

16.2 Singularities and Zeros. Infinity 715

16.3 Residue Integration Method 719

16.4 Residue Integration of Real Integrals 725

Chapter 16 Review Questions and Problems 733

Summary of Chapter 16 734

Chapter 17 Conformal Mapping 736

17.1 Geometry of Analytic Functions: Conformal Mapping 737

17.2 Linear Fractional Transformations (Möbius Transformations) 742

17.3 Special Linear Fractional Transformations 746

17.4 Conformal Mapping by Other Functions 750

17.5 Riemann Surfaces. Optional 754

Chapter 17 Review Questions and Problems 756

Summary of Chapter 17 757

Chapter 18 Complex Analysis and Potential Theory 758

18.1 Electrostatic Fields 759

18.2 Use of Conformal Mapping. Modeling 763

18.3 Heat Problems 767

18.4 Fluid Flow 771

18.5 Poisson’s Integral Formula for Potentials 777

18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem 781

Chapter 18 Review Questions and Problems 785

Summary of Chapter 18 786

Part E Numeric Analysis 787

Software 788

Chapter 19 Numerics in General 790

19.1 Introduction 790

19.2 Solution of Equations by Iteration 798

19.3 Interpolation 808

19.4 Spline Interpolation 820

19.5 Numeric Integration and Differentiation 827

Chapter 19 Review Questions and Problems 841

Summary of Chapter 19 842

Chapter 20 Numeric Linear Algebra 844

20.1 Linear Systems: Gauss Elimination 844

20.2 Linear Systems: LU-Factorization, Matrix Inversion 852

20.3 Linear Systems: Solution by Iteration 858

20.4 Linear Systems: Ill-Conditioning, Norms 864

20.5 Least Squares Method 872

20.6 Matrix Eigenvalue Problems: Introduction 876

20.7 Inclusion of Matrix Eigenvalues 879

20.8 Power Method for Eigenvalues 885

20.9 Tridiagonalization and QR-Factorization 888

Chapter 20 Review Questions and Problems 896

Summary of Chapter 20 898

Chapter 21 Numerics for ODEs and PDEs 900

21.1 Methods for First-Order ODEs 901

21.2 Multistep Methods 911

21.3 Methods for Systems and Higher Order ODEs 915

21.4 Methods for Elliptic PDEs 922

21.5 Neumann and Mixed Problems. Irregular Boundary 931

21.6 Methods for Parabolic PDEs 936

21.7 Method for Hyperbolic PDEs 942

Chapter 21 Review Questions and Problems 945

Summary of Chapter 21 946

Part F Optimization, Graphs 949

Chapter 22 Unconstrained Optimization. Linear Programming 950

22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent 951

22.2 Linear Programming 954

22.3 Simplex Method 958

22.4 Simplex Method: Difficulties 962

Chapter 22 Review Questions and Problems 968

Summary of Chapter 22 969

Chapter 23 Graphs. Combinatorial Optimization 970

23.1 Graphs and Digraphs 970

23.2 Shortest Path Problems. Complexity 975

23.3 Bellman’s Principle. Dijkstra’s Algorithm 980

23.4 Shortest Spanning Trees: Greedy Algorithm 984

23.5 Shortest Spanning Trees: Prim’s Algorithm 988

23.6 Flows in Networks 991

23.7 Maximum Flow: Ford–Fulkerson Algorithm 998

23.8 Bipartite Graphs. Assignment Problems 1001

Chapter 23 Review Questions and Problems 1006

Summary of Chapter 23 1007

Part G Probability, Statistics 1009

Software 1009

Chapter 24 Data Analysis. Probability Theory 1011

24.1 Data Representation. Average. Spread 1011

24.2 Experiments, Outcomes, Events 1015

24.3 Probability 1018

24.4 Permutations and Combinations 1024

24.5 Random Variables. Probability Distributions 1029

24.6 Mean and Variance of a Distribution 1035

24.7 Binomial, Poisson, and Hypergeometric Distributions 1039

24.8 Normal Distribution 1045

24.9 Distributions of Several Random Variables 1051

Chapter 24 Review Questions and Problems 1060

Summary of Chapter 24 1060

Chapter 25 Mathematical Statistics 1063

25.1 Introduction. Random Sampling 1063

25.2 Point Estimation of Parameters 1065

25.3 Confidence Intervals 1068

25.4 Testing Hypotheses. Decisions 1077

25.5 Quality Control 1087

25.6 Acceptance Sampling 1092

25.7 Goodness of Fit. χ ²-Test 1096

25.8 Nonparametric Tests 1100

25.9 Regression. Fitting Straight Lines. Correlation 1103

Chapter 25 Review Questions and Problems 1111

Summary of Chapter 25 1112

Appendix 1 References A1

Appendix 2 Answers to Odd-Numbered Problems A4

Appendix 3 Auxiliary Material A63

A3.1 Formulas for Special Functions A63

A3.2 Partial Derivatives A69

A3.3 Sequences and Series A72

A3.4 Grad, Div, Curl, ∇² in Curvilinear Coordinates A74

Appendix 4 Additional Proofs A77

Appendix 5 Tables A97

Index I1

Photo Credits P1

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