Part I: Ordinary Differential Equations
1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1
1.1
Introduction 1
1.2
Definitions 2
1.3
Introduction to Modeling 9
2 EQUATIONS OF FIRST ORDER 18
2.1
Introduction 18
2.2
The Linear Equation 19
2.2.1
Homogeneous case 19
2.2.2
Integrating factor method 22
2.2.3
Existence and uniqueness for the linear equation 25
2.2.4
Variation-of-parameter method 27
2.3
Applications of the Linear Equation 34
2.3.1
Electrical circuits 34
2.3.2
Radioactive decay; carbon dating 39
2.3.3
Population dynamics 41
2.3.4
Mixing problems 42
2.4
Separable Equations 46
2.4.1
Separable equations 46
2.4.2
Existence and uniqueness optional
48
2.4.3
Applications 53
2.4.4
Nondimensionalization optional
56
2.5
Exact Equations and Integrating Factors 62
2.5.1
Exact differential equations 62
2.5.2
Integrating factors 66
Chapter 2 Review 71
3 LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER AND HIGHER 73
3.1
Introduction 73
3.2
Linear Dependence and Linear Independence 76
3.3
Homogeneous Equation: General Solution 83
3.3.1
General solution 83
3.3.2
Boundary-value problems 88
3.4
Solution of Homogeneous Equation: Constant Coefficients 91
3.4.1
Euler''s formula and review of the circular and hyperbolic functions 91
3.4.2
Exponential solutions 95
3.4.3
Higher-order equations n>2
99
3.4.4
Repeated roots 102
3.4.5
Stability 105
3.5
Application to Harmonic Oscillator: Free Oscillation 110
3.6
Solution of Homogeneous Equation: Nonconstant Coefficients 117
3.6.1
Cauchy-Euler equation 118
3.6.2
Reduction of order optional
123
3.6.3
Factoring the operator optional
126
3.7
Solution of Nonhomogeneous Equation 133
3.7.1
General solution 134
3.7.2
Undetermined coefficients 136
3.7.3
Variation of parameters 141
3.7.4
Variation of parameters for higher-order equations optional
144
3.8
Application to Harmonic Oscillator: Forced Oscillation 149
3.8.1
Undamped case 149
3.8.2
Damped case 152
3.9
Systems of Linear Differential Equations 156
3.9.1
Examples 157
3.9.2
Existence and uniqueness 160
3.9.3
Solution by elimination 162
Chapter 3 Review 171
POWER SERIES SOLUTIONS 173
4.1
Introduction 173
4.2
Power Series Solutions 176
4.2.1
Review of power series 176
4.2.2
Power series solution of differential equations 182
4.3
The Method of Frobenius 193
4.3.1
Singular points 193
4.3.2
Method of Frobenius 195
4.4
Legendre Functions 212
4.4.1
Legendre polynomials 212
4.4.2
Onhogonality of the Pn''s 214
4.4.3
Generating functions and properties 215
4.5
Singular Integrals; Gamma Function 218
4.5.1
Singular integrals 218
4.5.2
Gamma function 223
4.5.3
Order of magnitude 225
4.6
Bessel Functions 230
4.6.1
v integer 231
4.6.2
v=integer 233
4.6.3
General solution of Bessel equation 235
4.6.4
Hankel functions optional
236
4.6.5
Modified Bessel equation 236
4.6.6
Equations reducible to Bessel equations 238
Chapter 4 Review 245
5 LAPLACE TRANSFORM 247
5.1
Introduction 247
5.2
Calculation of the Transform 248
5.3
Properties of the Transform 254
5.4
Application to the Solution of Differential Equations 261
5.5
Discontinuous Forcing Functions; Heaviside Step Function 269
5.6
Impulsive Forcing Functions; Dirac Impulse Function Optional
275
5.7
Additional Properties 281
Chapter 5 Review 290
6 QUANTITATIVE METHODS: NUMERICAL SOLUTION
OF DIFFERENTIAL EQUATIONS 292
6. l
Introduction 292
6.2
Euler''s Method 293
6.3
Improvements: Midpoint Rule and Runge-Kutta 299
6.3.1
Midpoint rule 299
6.3.2
Second-order Runge-Kutta 302
6.3.3
Fourth-order Runge-Kutta 304
6.3.4
Empirical estimate of the order optional
307
6.3.5
Multi-step and predictor-col-rector methods optional
308
6.4
Application to Systems and Boundary-Value Problems 313
6.4.1
Systems and higher-order equations 313
6.4.2
Linear boundary-value problems 317
6.5
Stability and Difference Equations 323
6.5.1
Introduction 323
6.5.2 Stability 324
6.5.3 Difference equations optional
328
Chapter 6 Review 335
7 QUALITATIVE METHODS: PHASE PLANE AND NONLINEAR
DIFFERENTIAL EQUATIONS 337
7.1
Introduction 337
7.2
The Phase Plane 338
7.3
Singular Points and Stability 348
7.3.1
Existence and uniqueness 348
7.3.2
Singular points 350
7.3.3
The elementary singularities and their stability 352
7.3.4
Nonelementary singularities 357
7.4
Applications 359
7.4.1
Singularities of nonlinear systems 360
7.4.2
Applications 363
7.4.3
Bifurcations 368
7.5
Limit Cycles, van der Pol Equation, and the Nerve Impulse 372
7.5.1
Limit cycles and the van der Pol equation 372
7.5.2
Application to the nerve impulse and visual perception 375
7.6
The Duffing Equation: Jumps and Chaos 380
7.6.1
Duffing equation and the jump phenomenon 380
7.6.2
Chaos. 383
Chapter 7 Review 389
Part II: Linear Algebra
8 SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS; GAUSS ELIMINATION 391
8.1
Introduction 391
8.2
Preliminary Ideas and Geometrical Approach 392
8.3
Solution by Gauss Elimination 396
8.3.1
Motivation 396
8.3.2
Gauss elimination 401
8.3.3
Matrix notation 402
8.3.4
Gauss-Jordan reduction 404
8.3.5
Pivoting 405
Chapter 8 Review 410
9 VECTOR SPACE 412
9.1
Introduction 412
9.2
Vectors; Geometrical Representation 412
9.3
Introduction of Angle and Dot Product 416
9.4
n-Space 418
9.5
Dot Product, Norm, and Angle for n-Space 421
9.5.1
Dot product, norm, and angle 421
9.5.2
Properties of the dot product 423
9.5.3
Properties of the norm 425
9.5.4
Orthogonality 426
9.5.5
Normalization 427
9.6
Generalized Vector Space 430
9.6.1
Vector space 430
9.6.2
Inclusion of inner product and/or norm 433
9.7
Span and Subspace 439
9.8
Linear Dependence 444
9.9
Bases, Expansions, Dimension 448
9.9.1
Bases and expansions 448
9.9.2
Dimension 450
9.9.3
Orthogonal bases 453
9.10 Best Approximation 457
9.10.1 Best approximation and orthogonal projection 458
9.10.2 Kronecker delta 461
Chapter 9 Review 462
10 MATRICES AND LINEAR EQUATIONS 465
10.1 Introduction 465
10.2 Matrices and Matrix Algebra 465
10.3 The Transpose Matrix 481
10.4 Determinants 486
10.5 Rank; Application to Linear Dependence and to Existence
and Uniqueness for Ax=c 495
10.5.1 Rank 495
10.5.2 Application of rank to the system Ax =c 500
10.6 Inverse Matrix, Cramer''s Rule, Factorization 508
10.6.1 Inverse matrix 508
10.6.2 Application to a mass-spring system 514
10.6.3 Cramer''s rule 517
10.6.4 Evaluation of A- 1 by elementary row operations 518
10.6.5 LU-factorization 520
10.7 Change of Basis Optional
526
10.8 Vector Transformation Optional
530
Chapter 10 Review 539
11 THE EIGENVALUE PROBLEM 541
11.1 Introduction 541
11.2 Solution Procedure and Applications 542
11.2.1 Solution and applications 542
11.2.2 Application to elementary singularities in the phase plane 549
11.3 Symmetric Matrices 554
11.3.1 Eigenvalue problem Ax =x 554
11.3.2 Nonhomogeneous problem Ax = x c optional
561
11.4 Diagonalization 569
11.5 Application to First-Order Systems with Constant Coefficients optional
583
I 1.6 Quadratic Forms Optional
589
Chapter 11 Review 596
12 EXTENSION TO COMPLEX CASE OPTIONAL
599
12.1 Introduction 599
12.2 Complex n-Space 599
12.3 Complex Matrices 603
Chapter 12 Review 611
Part III: Scalar and Vector Field Theory
13 DIFFERENTIAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES 613
13.1 Introduction 613
13.2 Preliminaries 614
13.2.1 Functions 614
13.2.2 Point set theory definitions 614
13.3 Partial Derivatives 620
13.4 Composite Functions and Chain Differentiation 625
13.5 Taylor''s Formula and Mean Value Theorem 629
13.5.1 Taylor''s formula and Taylor series for f x
630
13.5.2 Extension to functions of more than one variable 636
13.6 Implicit Functions and Jacobians 642
13.6.1 Implicit function theorem 642
13.6.2 Extension to multivariable case 645
13.6.3 Jacobians 649
13.6.4 Applications to change of variables 652
13.7 Maxima and Minima 656
13.7.1 Single variable case 656
13.7.2 Multivariable case 658
13.7.3 Constrained extrema and Lagrange multipliers 665
13.8 Leibniz Rule 675
Chapter 13 Review 681
14 VECTORS IN 3-SPACE 683
14.1 Introduction 683
14.2 Dot and Cross Product 683
14.3 Cartesian Coordinates 687
14.4 Multiple Products 692
14.4.1 Scalar triple product 692
14.4.2 Vector triple product 693
14.5 Differentiation of a Vector Function of a Single Variable 695
14.6 Non-Cartesian Coordinates Optional
699
14.6.1 Plane polar coordinates 700
14.6.2 Cylindrical coordinates 704
14.6.3 Spherical coordinates 705
14.6.4 Omega method 707
Chapter 14 Review 712
15 CURVES, SURFACES, AND VOLUMES 714
15.1 Introduction 714
15.2 Carves and Line Integrals 714
15.2.1 Curves 714
15.2.2 Arc length 716
15.2.3 Line integrals 718
15.3 Double and Triple Integrals 723
15.3.1 Double integrals 723
15.3.2 Triple integrals 727
15.4 Surfaces 733
15.4.1 Parametric representation of surfaces 733
15.4.2 Tangent plane and normal 734
15.5 Surface Integrals 739
15.5.1 Area element dA 739
15.5.2 Surface integrals 743
15.6 Volumes and Volume Integrals 748
15.6.1 Volume element dV 749
15.6.2 Volume integrals 752
Chapter 15 Review 755
16 SCALAR AND VECTOR FIELD THEORY 757
16.1 Introduction 757
16.2 Preliminaries 758
16.2.1 Topological considerations 758
16.2.2 Scalar and vector fields 758
16.3 Divergence 761
16.4 Gradient 766
16.5 Curl 774
16.6 Combinations; Laplacian 778
16.7 Non-Cartesian Systems; Div, Grad, Curl, and Laplacian Optional
782
16.7.1 Cylindrical coordinates 783
16.7.2 Spherical coordinates 786
16.8 Divergence Theorem 792
16.8.1 Divergence theorem 792
16.8.2 Two-dimensional case 802
16.8.3 Non-Cartesian coordinates optional
803
16.9 Stokes''s Theorem 810
16.9.1 Line integrals 814
16.9.2 Stokes''s theorem 814
16.9.3 Green''s theorem 818
16.9.4 Non-Cartesian coordinates optional
820
16.10 lrrotational Fields 826
16.10.1 Irrotational fields 826
16.10.2 Non-Cartesian coordinates 835
Chapter 16 Review 841
Part IV: Fourier Methods and Partial Differential Equations
17 FOURIER SERIES, FOURIER INTEGRAL, FOURIER TRANSFORM 844
17.1 Introduction 844
17.2 Even, Odd, and Periodic Functions 846
17.3 Fourier Series of a Periodic Function 850
17.3.1 Fourier series 850
17.3.2 Euler''s formulas 857
17.3.3 Applications 859
17.3.4 Complex exponential form for Fourier series 864
17.4 Half- and Quarter-Range Expansions 869
17.5 Manipulation of Fourier Series Optional
873
17.6 Vector Space Approach 881
17.7 The Sturm-Liouville Theory 887
17.7.1 Sturm-Liouville problem 887
17.7.2 Lagrange identity and proofs optional
897
17.8 Periodic and Singular Sturm-Liouville Problems 905
17.9 Fourier Integral 913
17.10 Fourier Transform 919
17.10.1 Transition from Fourier integral to Fourier transform 920
17.10.2 Properties and applications 922
17.11 Fourier Cosine and Sine Transforms, and Passage
from Fourier Integral to Laplace Transform Optional
934
17.11.1 Cosine and sine transforms 934
17.11.2 Passage from Fourier integral to Laplace transform 937
Chapter 17 Review 940
18 DIFFUSION EQUATION 943
18.1 Introduction 943
18.2 Preliminary Concepts 944
18.2.1 Definitions 944
18.2.2 Second-order linear equations and their classification 946
18.2.3 Diffusion equation and modeling ''948
18.3 Separation of Variables 954
18.3.1 The method of separation of variables 954
18.3.2 Verification of solution optional
964
18.3.3 Use of Sturm-Liouville theory optional
965
18.4 Fourier and Laplace Transforms Optional
981
18.5 The Method of Images Optional
992
18.5.1 Illustration of the method 992
18.5.2 Mathematical basis for the method 994
18.6 Numerical Solution 998
18.6.1 The finite-difference method 998
18.6.2 Implicit methods: Crank-Nicolson, with iterative solution optional
1005
Chapter 18 Review 1015
19 WAVE EQUATION 1017
19.1 Introduction 1017
19.2 Separation of Variables; Vibrating String 1023
19.2.1 Solution by separation of variables 1023
19.2.2 Traveling wave interpretation 1027
19.2.3 Using Sturm-Liouville theory optional
1029
19.3 Separation of Variables; Vibrating Membrane 1035
19.4 Vibrating String; d''Alembert''s Solution 1043
19.4.1 d''Alembert''s solution 1043
19.4.2 Use of images 1049
19.4.3 Solution by integral transforms optional
1051
Chapter 19 Review 1055
20 LAPLACE EQUATION 1058
20.1 Introduction 1058
20.2 Separation of Variables; Cartesian Coordinates 1059
20.3 Separation of Variables; Non-Cartesian Coordinates 1070
20.3.1 Plane polar coordinates 1070
20.3.2 Cylindrical coordinates optional
1077
20.3.3 Spherical coordinates optional
1081
20.4 Fourier Transform Optional
1088
20.5 Numerical Solution 1092
20.5.1 Rectangular domains 1092
20.5.2 Nonrectangular domains 1097
20.5.3 Iterative algorithms optional
1100
Chapter 20 Review 1106
Part V: Complex Variable Theory
21 FUNCTIONS OF A COMPLEX VARIABLE 1108
21.1 Introduction 1108
21.2 Complex Numbers and the Complex Plane 1109
21.3 Elementary Functions 1114
21.3.1 Preliminary ideas 1114
21.3.2 Exponential function 1116
21.3.3 Trigonometric and hyperbolic functions 1118
21.3.4 Application of complex numbers to integration and the
solution of differential equations 1120
21.4 Polar Form, Additional Elementary Functions, and Multi-valuedness 1125
21.4.1 Polar form 1125
21.4.2 Integral powers of z and de Moivre''S formula 1127
21.4.3 Fractional powers 1128
21.4.4 The logarithm ofz 1129
21.4.5 General powers ofz 1130
21.4.6 Obtaining single-valued functions by branch cuts 1131
21.4.7 More about branch cuts optional
1132
21.5 The Differential Calculus and Analyticity 1136
Chapter 21 Review 1148
22 CONFORMAL MAPPING 1150
22.1 Introduction 1150
22.2 The Idea Behind Conformal Mapping 1150
22.3 The Bilinear Transformation 1158
22.4 Additional Mappings and Applications 1166
22.5 More General Boundary Conditions 1170
22.6 Applications to Fluid Mechanics 1174
Chapter 22 Review 1180
23 THE COMPLEX INTEGRAL CALCULUS 1182
23.1 Introduction 1182
23.2 Complex Integration 1182
23.2.1 Definition and properties 1182
23.2.2 Bounds 1186
23.3 Cauchy''s Theorem 1189
23.4 Fundamental Theorem of the Complex Integral Calculus 1195
23.5 Cauchy Integral Formula 1199
Chapter 23 Review 1207
24 TAYLOR SERIES, LAURENT SERIES, AND THE RESIDUE THEOREM 1209
24.1 Introduction 1209
24.2 Complex Series and Taylor Series 1209
24.2.1 Complex series 1209
24.2.2 Taylor series 1214
24.3 Laurent Series 1225
24.4 Classification of Singularities 1234
24.5 Residue Theorem 1240
24.5.1 Residue theorem 1240
24.5.2 Calculating residues 1242
24.5.3 Applications of the residue theorem 1243
Chapter 24 Review 1258
REFERENCES 1260
APPENDICES
A Review of Partial Fraction Expansions 1263
B Existence and Uniqueness of Solutions of Systems of
Linear Algebraic Equations 1267
C Table of Laplace Transforms 1271
D Table of Fourier Transforms 1274
E Table of Fourier Cosine and Sine Transforms 1276
F Table of Conformal Maps 1278
ANSWERS TO SELECTED EXERCISES 1282
INDEX 1315
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