اسم المؤلف
Michael D. Greenberg
التاريخ
18 سبتمبر 2017
التصنيف
المشاهدات
827
التقييم

Michael D. Greenberg
Contents
Part I: Ordinary Differential Equations
1 INTRODUCTION TO DIFFERENTIAL EQUATIONS I
Introduction 1
Definitions 2
Introduction to Modeling
2 EQUATIONS OF FIRST ORDER
2.1 Introduction 18
The Linear Equation
Homogeneous case 19
Integrating factor method 22
Existence and uniqueness for the linear equation 25
Variation-of-parameter method 27
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
2.4 Separable Equations 46
2.4.1 Separable equations 46
Existence and uniqueness (optional) 48
Applications 53
Nondimensionalization (optional)
Exact Equations and Integrating Factors 62
2.5. 1 Exact differential equations 62
2.5.2 Integrating factors 66
Chapter 2 Review
3 LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER AND HIGHER 73
3.1 Introduction 73
3.2 Linear Dependence and Linear Independence 76
vvi Contents
Homogeneous Equation: General Solution 83
3.3.1 General solution 83
3.3.2 Boundary-value problems
Solution of Homogeneous Equation: Constant Coefficients
Euler’s formula and review of the circular and hyperbolic functions 91
Exponential solutions 95
3.4.3 Higher-order equations (« > 2) 99
Repeated roots 102
Stability
Application to Harmonic Oscillator: Free Oscillation 110
Solution of Homogeneous Equation: Nonconstant Coefficients
Cauchy-Euler equation 118
Reduction of order (optional) 123
Factoring the operator (optional)
Solution of Nonhomogeneous Equation 133
3.7.1 General solution
3.7.2 Undetermined coefficients
Application to Harmonic Oscillator: Forced Oscillation 149
3.8. 1 Undamped case 149
3.8.2 Damped case 152
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
Variation of parameters 141
Variation of parameters for higher-order equations (optional)
4 POWER SERIES SOLUTIONS
Introduction 173
Power Series Solutions 176
4.2.1 Review of power series 176
4.2.2 Power series solution of differential equations 182
The Method of Frobenius
Singular points 193 i
Method of Frobenius 195
Legendre Functions
Singular Integrals; Gamma Function 218
4.5.1 Singular integrals 218
4.5.2 Gamma function 223
4.5.3 Order of magnitude 225
Bessel Functions 230
4.6. 1 v ^ integer
Legendre polynomials 212
Orthogonality of the Pn ’s 214
Generating functions and properties
4.6.2 v = integer 233
General solution of Bessel equation 235
Hankel functions (optional) 236
Modified Bessel equation 236
Equations reducible to Bessel equations 238
Chapter 4 Review
5 LAPLACE TRANSFORM
5.1 Introduction 247
Calculation of the Transform 248
Properties of the Transform 254
Application to the Solution of Differential Equations 261
Discontinuous Forcing Functions; Heaviside Step Function 269
Impulsive Forcing Functions; Dirac Impulse Function (Optional) 275
Chapter 5 Review
6 QUANTITATIVE METHODS: NUMERICAL SOLUTION
OF DIFFERENTIAL EQUATIONS 292
/ ) 6.1 Introduction 292
Euler’s Method 293
Improvements: Midpoint Rule and Runge-Kutta
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
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
Midpoint rule 299
Second-order Runge-Kutta 302
Fourth-order Runge-Kutta 304
Empirical estimate of the order (optional) 307
Multi-step and predictor-corrector methods (optional)
7 QUALITATIVE METHODS: PHASE PLANE AND NONLINEAR
DIFFERENTIAL EQUATIONS 337
Introduction 337
The Phase Plane 338
Singular Points and Stability
Applications
Existence and uniqueness 348
Singular points 350
The elementary singularities and their stability 352
Nonelementary singularities
Singularities of nonlinear systems 360
Applications 363
Bifurcations
Limit Cycles, van der Pol Equation, and the Nerve Impulse
Limit cycles and the van der Pol equation 372
Application to the nerve impulse and visual perception 375
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
Part II: Linear Algebra
8 SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS; GAUSS ELIMINATION 391
Introduction 391
Preliminary Ideas and Geometrical Approach 392
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
9 VECTOR SPACE 412
Introduction 412
Vectors; Geometrical Representation 412
Introduction of Angle and Dot Product 416
n-Space 418
Dot Product, Norm, and Angle for /?-Space
Dot product, norm , and angle 421
Properties of the dot product 423
Properties of the norm 425
Orthogonality 426
Normalization 427
9.6 Generalized Vector Space
Vector space 430
Inclusion of inner product and/or norm 433
9.7 Span and Subspace 439
Linear Dependence 444
Bases, Expansions, Dimension
Bases and expansions 448
Dimension 450
Orthogonal bases 453
9.10 Best Approximation 457Contents i x
Best approximation and orthogonal projection 458
Kronecker delta
Chapter 9 Review 462
( •) 10 MATRICES AND LINEAR EQUATIONS 465
i \ Introduction 465
Matrices and Matrix Algebra 465
The Transpose Matrix 481
Determinants 486
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
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
Change of Basis (Optional) 526
Vector Transformation (Optional ) 530
Chapter 10 Review
11 THE EIGENVALUE PROBLEM 541
1 1 . 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
Eigenvalue problem Ax = Xx 554
Nonhomogeneous problem Ax = Ax + c (optional)
11.4 Diagonalization 569
11.5 Application to First-Order Systems with Constant Coefficients (optional) 583
Chapter ! I Review
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 61 l
Part III: Scalar and Vector Field Theory
13 DIFFERENTIAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES 613x Contents
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 i 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
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 Curves 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/
Contents xi
15.4. 1 Parametric representation of surfaces 733
15.4.2 Tangent plane and normal 734
Surface Integrals 739
15.5. 1 Area element dA 739
15.5.2 Surface integrals 743
Volumes and Volume Integrals 748
15.6.1 Volume element clV 749
15.6.2 Volume integrals 752
Chapter 15 Review
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
16.5 Curl 774
16.6 Combinations; Lapiacian 778
16.7 Non-Cartesian Systems; Div, Grad, Curl, and Lapiacian (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 Irrotational Fields 826
16.10.1 Irrotational fields 826
16.10.2 Non-Cartesian coordinates 835
Chapter 16 Review
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 859xii Contents
17.3.4 Complex exponential form for Fourier series 864
Half- and Quarter-Range Expansions 869
Manipulation of Fourier Series (Optional) 873
17.6 Vector Space Approach 881
17.7 The Sturm-Liouvilie Theory 887
Sturm-Liouville problem 887
Lagrange identity and proofs (optional)
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
18 DIFFUSION EQUATION 943
18.1 Introduction 943
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
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
Fourier and Laplace Transforms (Optional ) 981
The Method of Images (Optional ) 992
18.5. 1 Illustration of the method 992
18.5.2 Mathematical basis for the method 994
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
19 WAVE EQUATION 1017 {
19.1 Introduction 1017
19.2 Separation of Variables; Vibrating String 1023
19.2.1 Solution by separation of variables
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
Contents xiii
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
Complex Numbers and the Complex Plane 1109
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
Polar Form, Additional Elementary Functions, and Multi-valuedness
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 of z 1129
21.4.5 General powers of z 1130
21.4.6 Obtaining single-valued functions by branch cuts 1131
21.4.7 More about branch cuts (optional ) 1132
The Differential Calculus and Analyticity 1136
Chapter 21 Review
22 CONFORMAL MAPPING 1150
22.1 Introduction 1150
22.2 The Idea Behind Conformal Mapping
22.3 The Bilinear Transformation 1158
1 150xiv Contents
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
Cauchy’s Theorem 1189
23.4 Fundamental Theorem of the Complex Integral Calculus 1195
23.5 Cauchy Integral Formula 1199
Chapter 23 Review 1207
Definition and properties 1182
Bounds
24 TAYLOR SERIES, LAURENT SERIES, AND THE RESIDUE THEOREM 1209
Introduction 1209
Complex Series and Taylor Series 1209
24.2.1 Complex series 1209
24.2.2 Taylor series 1214
Laurent Series 1225
Classification of Singularities 1234
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
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