# Advanced Engineering Mathematics

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Advanced Engineering Mathematics provides comprehensive and contemporary coverage of key mathematical ideas, techniques, and their widespread applications, for students majoring in engineering, computer science, mathematics and physics. Using a wide range of examples throughout the book, Jeffrey illustrates how to construct simple mathematical models, how to apply mathematical reasoning to select a particular solution from a range of possible alternatives, and how to determine which solution has physical significance. Jeffrey includes material that is not found in works of a similar nature, such as the use of the matrix exponential when solving systems of ordinary differential equations. The text provides many detailed, worked examples following the introduction of each new idea, and large problem sets provide both routine practice, and, in many cases, greater challenge and insight for students. Most chapters end with a set of computer projects that require the use of any CAS (such as Maple or Mathematica) that reinforce ideas and provide insight into more advanced problems.

## Key Features

- Comprehensive coverage of frequently used integrals, functions and fundamental mathematical results
- Contents selected and organized to suit the needs of students, scientists, and engineers
- Contains tables of Laplace and Fourier transform pairs
- New section on numerical approximation
- New section on the z-transform
- Easy reference system

## Readership

Engineers, engineering academics, and mathematicians

## Table of Contents

- Preface xv

CHAPTER 1 Review of Prerequisites 4

1.1 Real Numbers, Mathematical Induction, and Mathematical Conventions 4

1.2 Complex Numbers 10

1.3 The Complex Plane 15

1.4 Modulus and Argument Representation of Complex Numbers 18

1.5 Roots of Complex Numbers 22

1.6 Partial Fractions 27

1.7 Fundamentals of Determinants 31

1.8 Continuity in One or More Variables 35

1.9 Differentiability of Functions of One or More Variables 38

1.10 Tangent Line and Tangent Plane Approximations to Functions 40

1.11 Integrals 41

1.12 Taylor and Maclaurin Theorems 43

1.13 Cylindrical and Spherical Polar Coordinates and Change of Variables in Partial Differentiation 46

1.14 Inverse Functions and the Inverse Function Theorem 49

CHAPTER 2 Vectors and Vector Spaces 55

2.1 Vectors, Geometry, and Algebra 56

2.2 The Dot Product (Scalar Product) 70

2.3 The Cross Product 77

2.4 Linear Dependence and Independence of Vectors and Triple Products 82

2.5 n-Vectors and the Vector Space R 88

2.6 Linear Independence, Basis, and Dimension 95

2.7 Gram-Schmidt Orthogonalization Process 101

CHAPTER 3 Matrices and Systems of Linear Equations 105

3.1 Matrices 106

3.2 Some Problems That Give Rise to Matrices 120

3.3 Determinants 133

3.4 Elementary Row Operations, Elementary Matrices, and Their Connection with Matrix Multiplication 143

3.5 The Echelon and Row-Reduced Echelon Forms of a Matrix 147

3.6 Row and Column Spaces and Rank 152

3.7 The Solution of Homogeneous Systems of Linear Equations 155

3.8 The Solution of Nonhomogeneous Systems of Linear Equations 158

3.9 The Inverse Matrix 163

3.10 Derivative of a Matrix 171

CHAPTER 4 Eigenvalues, Eigenvectors, and Diagonalization 177

4.1 Characteristic Polynomial, Eigenvalues, and Eigenvectors 178

4.2 Diagonalization of Matrices 196

4.3 Special Matrices with Complex Elements 205

4.4 Quadratic Forms 210

4.5 The Matrix Exponential 215

CHAPTER 5 First Order Differential Equations 227

5.1 Background to Ordinary Differential Equations 228

5.2 Some Problems Leading to Ordinary Differential Equations 233

5.3 Direction Fields 240

5.4 Separable Equations 242

.5 Homogeneous Equations 247

5.6 Exact Equations 250

5.7 Linear First Order Equations 253

5.8 The Bernoulli Equation 259

5.9 The Riccati Equation 262

5.10 Existence and Uniqueness of Solutions 264

CHAPTER 6 Second and Higher Order Linear Differential Equations and Systems 269

6.1 Homogeneous Linear Constant Coefficient Second Order Equations 270

6.2 Oscillatory Solutions 280

6.3 Homogeneous Linear Higher Order Constant Coefficient Equations 291

6.4 Undetermined Coefficients: Particular Integrals 302

.5 Cauchy-Euler Equation 309

6.6 Variation of Parameters and the Green's Function 311

6.7 Finding a Second Linearly Independent Solution from a Known Solution: The Reduction of Order Method 321

6.8 Reduction to the Standard Form u+f (x)u =0 324

6.9 Systems of Ordinary Differential Equations: An Introduction 326

6.10 A Matrix Approach to Linear Systems of Differential Equations 333

6.11 Nonhomogeneous Systems 338

6.12 Autonomous Systems of Equations 351

CHAPTER 7 The Laplace Transform 379

7.1 Laplace Transform: Fundamental Ideas 379

7.2 Operational Properties of the Laplace Transform 390

7.3 Systems of Equations and Applications of the Laplace Transform 415

7.4 The Transfer Function, Control Systems, and Time Lags 437

CHAPTER 8 Series Solutions of Differential Equations, Special Functions, and Sturm-Liouville Equations 443

8.1 A First Approach to Power Series Solutions of Differential Equations 443

8.2 A General Approach to Power Series Solutions of Homogeneous Equations 447

8.3 Singular Points of Linear Differential Equations 461

8.4 The Frobenius Method 463

8.5 The Gamma Function Revisited 480

8.6 Bessel Function of the First Kind Jn(x) 485

8.7 Bessel Functions of the Second Kind Yv (x) 495

8.8 Modified Bessel Functions Iv(x) and Kv(x) 501

8.9 A Critical Bending Problem: Is There a Tallest Flagpole? 504

8.10 Sturm-Liouville Problems, Eigenfunctions, and Orthogonality 509

8.11 Eigenfunction Expansions and Completeness 526

CHAPTER 9 Fourier Series 545

9.1 Introduction to Fourier Series 545

9.2 Convergence of Fourier Series and Their Integration and Differentiation 559

9.3 Fourier Sine and Cosine Series on 0 568

9.4 Other Forms of Fourier Series 572

9.5 Frequency and Amplitude Spectra of a Function 577

9.6 Double Fourier Series 581

CHAPTER 10 Fourier Integrals and the Fourier Transform 589

10.1 The Fourier Integral 589

10.2 The Fourier Transform 595

10.3 Fourier Cosine and Sine Transforms 611

CHAPTER 11 Vector Differential Calculus 625

11.1 Scalar and Vector Fields, Limits, Continuity, and Differentiability 626

11.2 Integration of Scalar and Vector Functions of a Single Real Variable 636

11.3 Directional Derivatives and the Gradient Operator 644

11.4 Conservative Fields and Potential Functions 650

11.5 Divergence and Curl of a Vector 659

11.6 Orthogonal Curvilinear Coordinates 665

CHAPTER 12 Vector Integral Calculus 677

12.1 Background to Vector Integral Theorems 678

12.2 Integral Theorems 680

12.3 Transport Theorems 697

12.4 Fluid Mechanics Applications of Transport Theorems 704

CHAPTER 13 Analytic Functions 711

13.1 Complex Functions and Mappings 711

13.2 Limits, Derivatives, and Analytic Functions 717

13.3 Harmonic Functions and Laplace's Equation 730

13.4 Elementary Functions, Inverse Functions, and Branches 735

CHAPTER 14 Complex Integration 745

14.1 Complex Integrals 745

14.2 Contours, the Cauchy-Goursat Theorem, and Contour Integrals 755

14.3 The Cauchy Integral Formulas 769

14.4 Some Properties of Analytic Functions 775

CHAPTER 15 Laurent Series, Residues, and Contour Integration 791

15.1 Complex Power Series and Taylor Series 791

15.2 Uniform Convergence 811

15.3 Laurent Series and the Classification of Singularities 816Laurent Series and the Classification of Singularities 816

15.4 Residues and the Residue Theorem 830

15.5 Evaluation of Real Integrals by Means of Residues 839

CHAPTER 16 The Laplace Inversion Integral 863

16.1 The Inversion Integral for the Laplace Transform 863

CHAPTER 17 Conformal Mapping and Applications to Boundary Value Problems 877

17.1 Conformal Mapping 877

17.2 Conformal Mapping and Boundary Value Problems 904

CHAPTER 18 Partial Differential Equations 927

18.1 What Is a Partial Differential Equation? 927

18.2 The Method of Characteristics 934

18.3 Wave Propagation and First Order PDEs 942

18.4 Generalizing Solutions: Conservation Laws and Shocks 951

18.5 The Three Fundamental Types of Linear Second Order PDE 956

18.6 Classification and Reduction to Standard Form of a Second Order Constant Coefficient Partial Differential Equation for u(x, y) 964

18.7 Boundary Conditions and Initial Conditions 975

18.8 Waves and the One-Dimensional Wave Equation 978

18.9 The DÕAlembert Solution of the Wave Equation and Applications 981

18.10 Separation of Variables 988

18.11 Some General Results for the Heat and Laplace Equation 1025

18.12 An Introduction to Laplace and Fourier Transform Methods for PDEs 1030

CHAPTER 19 Numerical Mathematics 1045

19.1 Decimal Places and Significant Figures 1046

19.2 Roots of Nonlinear Functions 1047

19.3 Interpolation and Extrapolation 1058

19.4 Numerical Integration 1065

19.5 Numerical Solution of Linear Systems of Equations 1077

19.6 Eigenvalues and Eigenvectors 1090

19.7 Numerical Solution of Differential Equations 1095

Answers 1109

References 1143

Index 1147

## Product details

- No. of pages: 1184
- Language: English
- Copyright: © Academic Press 2001
- Published: June 13, 2001
- Imprint: Academic Press
- eBook ISBN: 9780080522968

## About the Author

### Alan Jeffrey

#### Affiliations and Expertise

University at Newcastle Upon Tyne, UK

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