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Advanced Engineering Mathematics (Seventh Edition) by Peter V. O'Neil



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Advanced Engineering Mathematics (Seventh Edition) written by Peter V. O'Neil , University of Alabama, at Birmingham. This seventh edition of Advanced Engineering Mathematics differs from the sixth in four ways.
First, based on reviews and user comments, new material has been added, including the following. Orthogonal projections and least squares approximations of vectors and functions. This provides a unifying theme in recognizing partial sums of eigenfunction expansions as projections onto subspaces, as well as understanding lines of best fit to data points. Orthogonalization and the production of orthogonal bases. LU factorization of matrices. Linear transformations and matrix representations. Application of the Laplace transform to the solution of Bessel’s equation and to problems involving wave motion and diffusion. Expanded treatment of properties and applications of Legendre polynomials and Bessel functions, including a solution of Kepler’s problem and a model of alternating current flow. Heaviside’s formula for the computation of inverse Laplace transforms. A complex integral formula for the inverse Laplace transform, including an application to heat diffusion in a slab. Vector operations in orthogonal curvilinear coordinates. Application of vector integral theorems to the development of Maxwell’s equations. An application of the Laplace transform convolution to a replacement scheduling problem.
The second new feature of this edition is the interaction of the text with MapleTM. An appendix (called A Maple Primer) is included on the use of MapleTM and references to the use of MapleTM are made throughout the text. Third, there is an added emphasis on constructing and analyzing models, using ordinary and partial differential equations, integral transforms, special functions, eigenfunction expansions, and matrix and complex function methods. Finally, the answer section in the back of the book has been expanded to provide more information to the student. This edition is also shorter and more convenient to use than preceding editions. The chapters comprising Part 8 of the Sixth Edition, Counting and Probability, and Statistics, are now available on the 7e book website for instructors and students.

Advanced Engineering Mathematics (Seventh Edition) written by Peter V. O'Neil cover the following topics.

  • Part 1 Ordinary Differential Equations

  • 1. First-Order Differential Equations
    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

  • 2. Linear Second-Order Equations
    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

  • 3. The Laplace Transform
    3.1 Definition and Notation 77
    3.2 Solution of Initial Value Problems 81
    3.3 Shifting and the Heaviside Function 84
    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

  • 4. Series Solutions
    4.1 Power Series Solutions 121
    4.2 Frobenius Solutions 126

  • 5. Approximation of Solutions
    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

  • 6. Vectors and Vector Spaces
    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

  • 7. Matrices and Linear Systems
    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

  • 8. Determinants
    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

  • 9. Eigenvalues, Diagonalization, and Special Matrices
    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

  • 10. Systems of Linear Differential Equations
    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

  • 11. Vector Differential Calculus
    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

  • 12. Vector Integral Calculus
    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

  • 13. Fourier Series
    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

  • 14. The Fourier Integral and Transforms
    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
    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

  • 15. Special Functions and Eigenfunction Expansions
    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

  • 16. The Wave Equation
    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
    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

  • 17. The Heat Equation
    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

  • 18. The Potential Equation
    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

  • PART 6 Complex Functions

  • 19. Complex Numbers and Functions
    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

  • 20. Complex Integration
    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

  • 21. Series Representations of Functions 21.1 Power Series 715 21.2 The Laurent Expansion 725

  • 22. Singularities and the Residue Theorem
    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

  • 23. Conformal Mappings and Applications
    23.1 Conformal Mappings 751
    23.2 Construction of Conformal Mappings 765
    23.2.1 The Schwarz-Christoffel Transformation 773
    23.3 Conformal Mapping Solutions of Dirichlet Problems 776
    23.4 Models of Plane Fluid Flow 779

  • Answers to Selected Problems

  • Index

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