Advanced Engineering Mathematics (eBook)

Cover 1
Title Page 4
Copyright Page 5
Contents 8
Preface 16
Part One: Review Material 22
Chapter 1. Review of Prerequisites 24
1.1 Real Numbers, Mathematical Induction, and Mathematical Conventions 25
1.2 Complex Numbers 31
1.3 The Complex Plane 36
1.4 Modulus and Argument Representation of Complex Numbers 39
1.5 Roots of Complex Numbers 43
1.6 Partial Fractions 48
1.7 Fundamentals of Determinants 52
1.8 Continuity in One or More Variables 56
1.9 Differentiability of Functions of One or More Variables 59
1.10 Tangent Line and Tangent Plane Approximations to Functions 61
1.11 Integrals 62
1.12 Taylor and Maclaurin Theorems 64
1.13 Cylindrical and Spherical Polar Coordinates and Change of Variables in Partial Differentiation 67
1.14 Inverse Functions and the Inverse Function Theorem 70
Part Two: Vectors and Matrices 74
Chapter 2. Vectors and Vector Spaces 76
2.1 Vectors, Geometry, and Algebra 77
2.2 The Dot Product (Scalar Product) 91
2.3 The Cross Product (Vector Product) 98
2.4 Linear Dependence and Independence of Vectors and Triple Products 103
2.5 n-Vectors and the Vector Space Rn 109
2.6 Linear Independence, Basis, and Dimension 116
2.7 Gram–Schmidt Orthogonalization Process 122
Chapter 3. Matrices and Systems of Linear Equations 126
3.1 Matrices 127
3.2 Some Problems That Give Rise to Matrices 141
3.3 Determinants 154
3.4 Elementary Row Operations, Elementary Matrices, and Their Connection with Matrix Multiplication 164
3.5 The Echelon and Row-Reduced Echelon Forms of a Matrix 168
3.6 Row and Column Spaces and Rank 173
3.7 The Solution of Homogeneous Systems of Linear Equations 176
3.8 The Solution of Nonhomogeneous Systems of Linear Equations 179
3.9 The Inverse Matrix 184
3.10 Derivative of a Matrix 192
Chapter 4. Eigenvalues, Eigenvectors, and Diagonalization 198
4.1 Characteristic Polynomial, Eigenvalues, and Eigenvectors 199
4.2 Diagonalization of Matrices 217
4.3 Special Matrices with Complex Elements 226
4.4 Quadratic Forms 231
4.5 The Matrix Exponential 236
Part Three: Ordinary Differential Equations 246
Chapter 5. First Order Differential Equations 248
5.1 Background to Ordinary Differential Equations 249
5.2 Some Problems Leading to Ordinary Differential Equations 254
5.3 Direction Fields 261
5.4 Separable Equations 263
5.5 Homogeneous Equations 268
5.6 Exact Equations 271
5.7 Linear First Order Equations 274
5.8 The Bernoulli Equation 280
5.9 The Riccati Equation 283
5.10 Existence and Uniqueness of Solutions 285
Chapter 6. Second and Higher Order Linear Differential Equations and Systems 290
6.1 Homogeneous Linear Constant Coefficient Second Order Equations 291
6.2 Oscillatory Solutions 301
6.3 Homogeneous Linear Higher Order Constant Coefficient Equations 312
6.4 Undetermined Coefficients Particular Integrals 323
6.5 Cauchy–Euler Equation 330
6.6 Variation of Parameters and the Green’s Function 332
6.7 Finding a Second Linearly Independent Solution from a Known Solution The Reduction of Order Method 342
6.8 Reduction to the Standard Form u'' + f (x)u = 0 345
6.9 Systems of Ordinary Differential Equations An Introduction 347
6.10 A Matrix Approach to Linear Systems of Differential Equations 354
6.11 Nonhomogeneous Systems 359
6.12 Autonomous Systems of Equations 372
Chapter 7. The Laplace Transform 400
7.1 Laplace Transform Fundamental Ideas 400
7.2 Operational Properties of the Laplace Transform 411
7.3 Systems of Equations and Applications of the Laplace Transform 436
7.4 The Transfer Function, Control Systems, and Time Lags 458
Chapter 8. Series Solutions of Differential Equations, Special Functions, and Sturm–Liouville Equations 464
8.1 A First Approach to Power Series Solutions of Differential Equations 464
8.2 A General Approach to Power Series Solutions of Homogeneous Equations 468
8.3 Singular Points of Linear Differential Equations 482
8.4 The Frobenius Method 484
8.5 The Gamma Function Revisited 501
8.6 Bessel Function of the First Kind Jn(x) 506
8.7 Bessel Functions of the Second Kind Y.(x) 516
8.8 Modified Bessel Functions Iv(x) and Kv(x) 522
8.9 A Critical Bending Problem Is There a Tallest Flagpole? 525
8.10 Sturm–Liouville Problems, Eigenfunctions, and Orthogonality 530
8.11 Eigenfunction Expansions and Completeness 547
Part Four: Fourier series, Integrals, and The Fourier Transform 564
Chapter 9. Fourier Series 566
9.1 Introduction to Fourier Series 566
9.2 Convergence of Fourier Series and Their Integration and Differentiation 580
9.3 Fourier Sine and Cosine Series on 0< _x<
9.4 Other Forms of Fourier Series 593
9.5 Frequency and Amplitude Spectra of a Function 598
9.6 Double Fourier Series 602
Chapter 10. Fourier Integrals and the Fourier Transform 610
10.1 The Fourier Integral 610
10.2 The Fourier Transform 616
10.3 Fourier Cosine and Sine Transforms 632
Part Five: Vector Calculus 644
Chapter 11. Vector Differential Calculus 646
11.1 Scalar and Vector Fields, Limits, Continuity, and Differentiability 647
11.2 Integration of Scalar and Vector Functions of a Single Real Variable 657
11.3 Directional Derivatives and the Gradient Operator 665
11.4 Conservative Fields and Potential Functions 671
11.5 Divergence and Curl of a Vector 680
11.6 Orthogonal Curvilinear Coordinates 686
Chapter 12. Vector Integral Calculus 698
12.1 Background to Vector Integral Theorems 699
12.2 Integral Theorems 701
12.3 Transport Theorems 718
12.4 Fluid Mechanics Applications of Transport Theorems 725
Part Six: Complex Analysis 730
Chapter 13.Analytic Functions 732
13.1 Complex Functions and Mappings 732
13.2 Limits, Derivatives, and Analytic Functions 738
13.3 Harmonic Functions and Laplace’s Equation 751
13.4 Elementary Functions, Inverse Functions, and Branches 756
Chapter 14.Complex Integration 766
14.1 Complex Integrals 766
14.2 Contours, the Cauchy–Goursat Theorem, and Contour Integrals 776
14.3 The Cauchy Integral Formulas 790
14.4 Some Properties of Analytic Functions 796
Chapter 15. Laurent Series, Residues, and Contour Integration 812
15.1 Complex Power Series and Taylor Series 812
15.2 Uniform Convergence 832
15.3 Laurent Series and the Classification of Singularities 837
15.4 Residues and the Residue Theorem 851
15.5 Evaluation of Real Integrals by Means of Residues 860
Chapter 16. The Laplace Inversion Integral 884
16.1 The Inversion Integral for the Laplace Transform 884
Chapter 17. Conformal Mapping and Applications to Boundary Value Problems 898
17.1 Conformal Mapping 898
17.2 Conformal Mapping and Boundary Value Problems 925
Part Seven: Partial Differential Equations 946
Chapter 18. Partial Differential Equations 948
18.1 What Is a Partial Differential Equation? 948
18.2 The Method of Characteristics 955
18.3 Wave Propagation and First Order Pdes 963
18.4 Generalizing Solutions Conservation Laws and Shocks 972
18.5 The Three Fundamental Types of Linear Second Order Pde 977
18.6 Classification and Reduction to Standard Form of a Second Order Constant Coefficient Partial Differential Equation for u(x, y) 985
18.7 Boundary Conditions and Initial Conditions 996
18.8 Waves and the One-Dimensional Wave Equation 999
18.9 The D’Alembert Solution of the Wave Equation and Applications 1002
18.10 Separation of Variables 1009
18.11 Some General Results for the Heat and Laplace Equation 1046
18.12 An Introduction to Laplace and Fourier Transform Methods for PDEs 1051
Part Eight: Numerical Mathematics 1064
Chapter 19. Numerical Mathematics 1066
19.1 Decimal Places and Significant Figures 1067
19.2 Roots of Nonlinear Functions 1068
19.3 Interpolation and Extrapolation 1079
19.4 Numerical Integration 1086
19.5 Numerical Solution of Linear Systems of Equations 1098
19.6 Eigenvalues and Eigenvectors 1111
19.7 Numerical Solution of Differential Equations 1116
Answers 1130
References 1164
Index 1168