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 Pn(x) 175
5.3 Extended Power Series Method: Frobenius Method 180
5.4 Bessel's Equation. Bessel Functions Jv(x) 187
5.5 Bessel Functions of the Yv(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 (Mobius 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. ?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
