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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 f(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 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 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 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 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 (x) 187
5.5 Bessel Functions of the Y (x). General Solution 196
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
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 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 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 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
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 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
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 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 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 16 Laurent Series. Residue Integration 708
16.1 Laurent Series 708
16.2 Singularities and Zeros. Infinity 714
16.3 Residue Integration Method 719
16.4 Residue Integration of Real Integrals 725
CHAPTER 17 Conformal Mapping 735
17.1 Geometry of Analytic Functions: Conformal Mapping 736
17.2 Linear Fractional Transformations (Möbius Transformations) 741
17.3 Special Linear Fractional Transformations 745
17.4 Conformal Mapping by Other Functions 749
17.5 Riemann Surfaces. Optional 753
CHAPTER 18 Complex Analysis and Potential Theory 756
18.1 Electrostatic Fields 757
18.2 Use of Conformal Mapping. Modeling 761
18.3 Heat Problems 765
18.4 Fluid Flow 768
18.5 Poisson''s Integral Formula for Potentials 774
18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem 778
PART E Numeric Analysis 785 Software 786
CHAPTER 19 Numerics in General 788
19.1 Introduction 788
19.2 Solution of Equations by Iteration 795
19.3 Interpolation 805
19.4 Spline Interpolation 817
19.5 Numeric Integration and Differentiation 824
CHAPTER 20 Numeric Linear Algebra 841
20.1 Linear Systems: Gauss Elimination 841
20.2 Linear Systems: LU-Factorization, Matrix Inversion 849
20.3 Linear Systems: Solution by Iteration 855
20.4 Linear Systems: Ill-Conditioning, Norms 861
20.5 Least Squares Method 869
20.6 Matrix Eigenvalue Problems: Introduction 873
20.7 Inclusion of Matrix Eigenvalues 876
20.8 Power Method for Eigenvalues 882
20.9 Tridiagonalization and QR-Factorization 885
CHAPTER 21 Numerics for ODEs and PDEs 897
21.1 Methods for First-Order ODEs 898
21.2 Multistep Methods 908
21.3 Methods for Systems and Higher Order ODEs 912
21.4 Methods for Elliptic PDEs 919
21.5 Neumann and Mixed Problems. Irregular Boundary 928
21.6 Methods for Parabolic PDEs 933
21.7 Method for Hyperbolic PDEs 939
PART F Optimization, Graphs 947
CHAPTER 22 Unconstrained Optimization. Linear Programming 948
22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent 949
22.2 Linear Programming 952