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[资源] Advanced.Engineering.Mathematics,.Peter.V.O'Neil,.7ed,.Cengage,.2012

Contents
Preface xi
PART 1 Ordinary Differential Equations 1
CHAPTER 1 First-Order Differential Equations 3
1.1 Terminology and Separable Equations 3
1.2 Linear Equations 16
1.3 Exact Equations 21
1.4 Homogeneous, Bernoulli, and Riccati Equations 26
1.4.1 The Homogeneous Differential Equation 26
1.4.2 The Bernoulli Equation 27
1.4.3 The Riccati Equation 28
1.5 Additional Applications 30
1.6 Existence and Uniqueness Questions 40
CHAPTER 2 Linear Second-Order Equations 43
2.1 The Linear Second-Order Equation 43
2.2 The Constant Coefficient Case 50
2.3 The Nonhomogeneous Equation 55
2.3.1 Variation of Parameters 55
2.3.2 Undetermined Coefficients 57
2.3.3 The Principle of Superposition 60
2.4 Spring Motion 61
2.4.1 Unforced Motion 62
2.4.2 Forced Motion 66
2.4.3 Resonance 67
2.4.4 Beats 69
2.4.5 Analogy with an Electrical Circuit 70
2.5 Euler’s Differential Equation 72
CHAPTER 3 The Laplace Transform 77
3.1 Definition and Notation 77
3.2 Solution of Initial Value Problems 81
3.3 Shifting and the Heaviside Function 84
v
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
vi Contents
3.3.1 The First Shifting Theorem 84
3.3.2 The Heaviside Function and Pulses 86
3.3.3 Heaviside’s Formula 93
3.4 Convolution 96
3.5 Impulses and the Delta Function 102
3.6 Solution of Systems 106
3.7 Polynomial Coefficients 112
3.7.1 Differential Equations with Polynomial Coefficients 112
3.7.2 Bessel Functions 114
CHAPTER 4 Series Solutions 121
4.1 Power Series Solutions 121
4.2 Frobenius Solutions 126
CHAPTER 5 Approximation of Solutions 137
5.1 Direction Fields 137
5.2 Euler’s Method 139
5.3 Taylor and Modified Euler Methods 142
PART 2 Vectors, Linear Algebra, and Systems
of Linear Differential Equations 145
CHAPTER 6 Vectors and Vector Spaces 147
6.1 Vectors in the Plane and 3-Space 147
6.2 The Dot Product 154
6.3 The Cross Product 159
6.4 The Vector Space Rn 162
6.5 Orthogonalization 175
6.6 Orthogonal Complements and Projections 177
6.7 The Function Space C[a, b] 181
CHAPTER 7 Matrices and Linear Systems 187
7.1 Matrices 187
7.1.1 Matrix Multiplication from Another Perspective 191
7.1.2 Terminology and Special Matrices 192
7.1.3 Random Walks in Crystals 194
7.2 Elementary Row Operations 198
7.3 Reduced Row Echelon Form 203
7.4 Row and Column Spaces 208
7.5 Homogeneous Systems 213
7.6 Nonhomogeneous Systems 220
7.7 Matrix Inverses 226
7.8 Least Squares Vectors and Data Fitting 232
7.9 LU Factorization 237
7.10 Linear Transformations 240
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Contents vii
CHAPTER 8 Determinants 247
8.1 Definition of the Determinant 247
8.2 Evaluation of Determinants I 252
8.3 Evaluation of Determinants II 255
8.4 A Determinant Formula for A−1 259
8.5 Cramer’s Rule 260
8.6 The Matrix Tree Theorem 262
CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices 267
9.1 Eigenvalues and Eigenvectors 267
9.2 Diagonalization 277
9.3 Some Special Types of Matrices 284
9.3.1 Orthogonal Matrices 284
9.3.2 Unitary Matrices 286
9.3.3 Hermitian and Skew-Hermitian Matrices 288
9.3.4 Quadratic Forms 290
CHAPTER 10 Systems of Linear Differential Equations 295
10.1 Linear Systems 295
10.1.1 The Homogeneous System X =AX. 296
10.1.2 The Nonhomogeneous System 301
10.2 Solution of X =AX for Constant A 302
10.2.1 Solution When A Has a Complex Eigenvalue 306
10.2.2 Solution When A Does Not Have n Linearly Independent Eigenvectors 308
10.3 Solution of X =AX+G 312
10.3.1 Variation of Parameters 312
10.3.2 Solution by Diagonalizing A 314
10.4 Exponential Matrix Solutions 316
10.5 Applications and Illustrations of Techniques 319
10.6 Phase Portraits 329
10.6.1 Classification by Eigenvalues 329
10.6.2 Predator/Prey and Competing Species Models 338
PART 3 Vector Analysis 343
CHAPTER 11 Vector Differential Calculus 345
11.1 Vector Functions of One Variable 345
11.2 Velocity and Curvature 349
11.3 Vector Fields and Streamlines 354
11.4 The Gradient Field 356
11.4.1 Level Surfaces, Tangent Planes, and Normal Lines 359
11.5 Divergence and Curl 362
11.5.1 A Physical Interpretation of Divergence 364
11.5.2 A Physical Interpretation of Curl 365
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
viii Contents
CHAPTER 12 Vector Integral Calculus 367
12.1 Line Integrals 367
12.1.1 Line Integral With Respect to Arc Length 372
12.2 Green’s Theorem 374
12.3 An Extension of Green’s Theorem 376
12.4 Independence of Path and Potential Theory 380
12.5 Surface Integrals 388
12.5.1 Normal Vector to a Surface 389
12.5.2 Tangent Plane to a Surface 392
12.5.3 Piecewise Smooth Surfaces 392
12.5.4 Surface Integrals 393
12.6 Applications of Surface Integrals 395
12.6.1 Surface Area 395
12.6.2 Mass and Center of Mass of a Shell 395
12.6.3 Flux of a Fluid Across a Surface 397
12.7 Lifting Green’s Theorem to R3 399
12.8 The Divergence Theorem of Gauss 402
12.8.1 Archimedes’s Principle 404
12.8.2 The Heat Equation 405
12.9 Stokes’s Theorem 408
12.9.1 Potential Theory in 3-Space 410
12.9.2 Maxwell’s Equations 411
12.10 Curvilinear Coordinates 414
PART 4 Fourier Analysis, Special Functions,
and Eigenfunction Expansions 425
CHAPTER 13 Fourier Series 427
13.1 Why Fourier Series? 427
13.2 The Fourier Series of a Function 429
13.2.1 Even and Odd Functions 436
13.2.2 The Gibbs Phenomenon 438
13.3 Sine and Cosine Series 441
13.3.1 Cosine Series 441
13.3.2 Sine Series 443
13.4 Integration and Differentiation of Fourier Series 445
13.5 Phase Angle Form 452
13.6 Complex Fourier Series 457
13.7 Filtering of Signals 461
CHAPTER 14 The Fourier Integral and Transforms 465
14.1 The Fourier Integral 465
14.2 Fourier Cosine and Sine Integrals 468
14.3 The Fourier Transform 470
14.3.1 Filtering and the Dirac Delta Function 481
14.3.2 The Windowed Fourier Transform 483
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Contents ix
14.3.3 The Shannon Sampling Theorem 485
14.3.4 Low-Pass and Bandpass Filters 487
14.4 Fourier Cosine and Sine Transforms 490
14.5 The Discrete Fourier Transform 492
14.5.1 Linearity and Periodicity of the DFT 494
14.5.2 The Inverse N-Point DFT 494
14.5.3 DFT Approximation of Fourier Coefficients 495
14.6 Sampled Fourier Series 498
14.7 DFT Approximation of the Fourier Transform 501
CHAPTER 15 Special Functions and Eigenfunction Expansions 505
15.1 Eigenfunction Expansions 505
15.1.1 Bessel’s Inequality and Parseval’s Theorem 515
15.2 Legendre Polynomials 518
15.2.1 A Generating Function for Legendre Polynomials 521
15.2.2 A Recurrence Relation for Legendre Polynomials 523
15.2.3 Fourier-Legendre Expansions 525
15.2.4 Zeros of Legendre Polynomials 528
15.2.5 Distribution of Charged Particles 530
15.2.6 Some Additional Results 532
15.3 Bessel Functions 533
15.3.1 The Gamma Function 533
15.3.2 Bessel Functions of the First Kind 534
15.3.3 Bessel Functions of the Second Kind 538
15.3.4 Displacement of a Hanging Chain 540
15.3.5 Critical Length of a Rod 542
15.3.6 Modified Bessel Functions 543
15.3.7 Alternating Current and the Skin Effect 546
15.3.8 A Generating Function for Jν(x) 548
15.3.9 Recurrence Relations 549
15.3.10 Zeros of Bessel Functions 550
15.3.11 Fourier-Bessel Expansions 552
15.3.12 Bessel’s Integrals and the Kepler Problem 556
PART 5 Partial Differential Equations 563
CHAPTER 16 The Wave Equation 565
16.1 Derivation of the Wave Equation 565
16.2 Wave Motion on an Interval 567
16.2.1 Zero Initial Velocity 568
16.2.2 Zero Initial Displacement 570
16.2.3 Nonzero Initial Displacement and Velocity 572
16.2.4 Influence of Constants and Initial Conditions 573
16.2.5 Wave Motion with a Forcing Term 575
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
x Contents
16.3 Wave Motion in an Infinite Medium 579
16.4 Wave Motion in a Semi-Infinite Medium 585
16.4.1 Solution by Fourier Sine or Cosine Transform 586
16.5 Laplace Transform Techniques 587
16.6 Characteristics and d’Alembert’s Solution 594
16.6.1 Forward and Backward Waves 596
16.6.2 Forced Wave Motion 599
16.7 Vibrations in a Circular Membrane I 602
16.7.1 Normal Modes of Vibration 604
16.8 Vibrations in a Circular Membrane II 605
16.9 Vibrations in a Rectangular Membrane 608
CHAPTER 17 The Heat Equation 611
17.1 Initial and Boundary Conditions 611
17.2 The Heat Equation on [0, L] 612
17.2.1 Ends Kept at Temperature Zero 612
17.2.2 Insulated Ends 614
17.2.3 Radiating End 615
17.2.4 Transformation of Problems 618
17.2.5 The Heat Equation with a Source Term 619
17.2.6 Effects of Boundary Conditions and Constants 622
17.3 Solutions in an Infinite Medium 626
17.3.1 Problems on the Real Line 626
17.3.2 Solution by Fourier Transform 627
17.3.3 Problems on the Half-Line 629
17.3.4 Solution by Fourier Sine Transform 630
17.4 Laplace Transform Techniques 631
17.5 Heat Conduction in an Infinite Cylinder 636
17.6 Heat Conduction in a Rectangular Plate 638
CHAPTER 18 The Potential Equation 641
18.1 Laplace’s Equation 641
18.2 Dirichlet Problem for a Rectangle 642
18.3 Dirichlet Problem for a Disk 645
18.4 Poisson’s Integral Formula 648
18.5 Dirichlet Problem for Unbounded Regions 649
18.5.1 The Upper Half-Plane 650
18.5.2 The Right Quarter-Plane 652
18.6 A Dirichlet Problem for a Cube 654
18.7 Steady-State Equation for a Sphere 655
18.8 The Neumann Problem 659
18.8.1 A Neumann Problem for a Rectangle 660
18.8.2 A Neumann Problem for a Disk 662
18.8.3 A Neumann Problem for the Upper Half-Plane 664
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Contents xi
PART 6 Complex Functions 667
CHAPTER 19 Complex Numbers and Functions 669
19.1 Geometry and Arithmetic of Complex Numbers 669
19.2 Complex Functions 676
19.2.1 Limits, Continuity, and Differentiability 677
19.2.2 The Cauchy-Riemann Equations 680
19.3 The Exponential and Trigonometric Functions 684
19.4 The Complex Logarithm 689
19.5 Powers 690
CHAPTER 20 Complex Integration 695
20.1 The Integral of a Complex Function 695
20.2 Cauchy’s Theorem 700
20.3 Consequences of Cauchy’s Theorem 703
20.3.1 Independence of Path 703
20.3.2 The Deformation Theorem 704
20.3.3 Cauchy’s Integral Formula 706
20.3.4 Properties of Harmonic Functions 709
20.3.5 Bounds on Derivatives 710
20.3.6 An Extended Deformation Theorem 711
20.3.7 A Variation on Cauchy’s Integral Formula 713
CHAPTER 21 Series Representations of Functions 715
21.1 Power Series 715
21.2 The Laurent Expansion 725
CHAPTER 22 Singularities and the Residue Theorem 729
22.1 Singularities 729
22.2 The Residue Theorem 733
22.3 Evaluation of Real Integrals 740
22.3.1 Rational Functions 740
22.3.2 Rational Functions Times Cosine or Sine 742
22.3.3 Rational Functions of Cosine and Sine 743
22.4 Residues and the Inverse Laplace Transform 746
22.4.1 Diffusion in a Cylinder 748
CHAPTER 23 Conformal Mappings and Applications 751
23.1 Conformal Mappings 751
23.2 Construction of Conformal Mappings 765
23.2.1 The Schwarz-Christoffel Transformation 773
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
xii Contents
23.3 Conformal Mapping Solutions of Dirichlet Problems 776
23.4 Models of Plane Fluid Flow 779
APPENDIX A MAPLE Primer 789
Answers to Selected Problems 801
Index 867
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