Advanced Engineering Mathematics

**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. A Student Solutions Manual is also available. \* 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 Contents 3 Part One: Review Material 15 Chapter 1. Review of Prerequisites 16 1.1 Real Numbers, Mathematical Induction, and Mathematical Conventions 17 1.2 Complex Numbers 23 1.3 The Complex Plane 28 1.4 Modulus and Argument Representation of Complex Numbers 31 1.5 Roots of Complex Numbers 35 1.6 Partial Fractions 40 1.7 Fundamentals of Determinants 44 1.8 Continuity in One or More Variables 48 1.9 Differentiability of Functions of One or More Variables 51 1.10 Tangent Line and Tangent Plane Approximations to Functions 53 1.11 Integrals 54 1.12 Taylor and Maclaurin Theorems 56 1.13 Cylindrical and Spherical Polar Coordinates and Change of Variables in Partial Differentiation 59 1.14 Inverse Functions and the Inverse Function Theorem 62 Part Two: Vectors and Matrices 65 Chapter 2. Vectors and Vector Spaces 66 2.1 Vectors, Geometry, and Algebra 67 2.2 The Dot Product (Scalar Product) 81 2.3 The Cross Product (Vector Product) 88 2.4 Linear Dependence and Independence of Vectors and Triple Products 93 2.5 n-Vectors and the Vector Space Rn 99 2.6 Linear Independence, Basis, and Dimension 106 2.7 Gram–Schmidt Orthogonalization Process 112 Chapter 3. Matrices and Systems of Linear Equations 115 3.1 Matrices 116 3.2 Some Problems That Give Rise to Matrices 130 3.3 Determinants 143 3.4 Elementary Row Operations, Elementary Matrices, and Their Connection with Matrix Multiplication 153 3.5 The Echelon and Row-Reduced Echelon Forms of a Matrix 157 3.6 Row and Column Spaces and Rank 162 3.7 The Solution of Homogeneous Systems of Linear Equations 165 3.8 The Solution of Nonhomogeneous Systems of Linear Equations 168 3.9 The Inverse Matrix 173 3.10 Derivative of a Matrix 181 Chapter 4. Eigenvalues, Eigenvectors, and Diagonalization 186 4.1 Characteristic Polynomial, Eigenvalues, and Eigenvectors 187 4.2 Diagonalization of Matrices 205 4.3 Special Matrices with Complex Elements 214 4.4 Quadratic Forms 219 4.5 The Matrix Exponential 224 Part Three: Ordinary Differential Equations 234 Chapter 5. First Order Differential Equations 235 5.1 Background to Ordinary Differential Equations 236 5.2 Some Problems Leading to Ordinary Differential Equations 241 5.3 Direction Fields 248 5.4 Separable Equations 250 5.5 Homogeneous Equations 255 5.6 Exact Equations 258 5.7 Linear First Order Equations 261 5.8 The Bernoulli Equation 267 5.9 The Riccati Equation 270 5.10 Existence and Uniqueness of Solutions 272 Chapter 6. Second and Higher Order Linear Differential Equations and Systems 276 6.1 Homogeneous Linear Constant Coefficient Second Order Equations 277 6.2 Oscillatory Solutions 287 6.3 Homogeneous Linear Higher Order Constant Coefficient Equations 298 6.4 Undetermined Coefficients Particular Integrals 309 6.5 Cauchy–Euler Equation 316 6.6 Variation of Parameters and the Green’s Function 318 6.7 Finding a Second Linearly Independent Solution from a Known Solution The Reduction of Order Method 328 6.8 Reduction to the Standard Form u'' + f (x)u = 0 331 6.9 Systems of Ordinary Differential Equations An Introduction 333 6.10 A Matrix Approach to Linear Systems of Differential Equations 340 6.11 Nonhomogeneous Systems 345 6.12 Autonomous Systems of Equations 358 Chapter 7. The Laplace Transform 386 7.1 Laplace Transform Fundamental Ideas 386 7.2 Operational Properties of the Laplace Transform 397 7.3 Systems of Equations and Applications of the Laplace Transform 422 7.4 The Transfer Function, Control Systems, and Time Lags 444 Chapter 8. Series Solutions of Differential Equations, Special Functions, and Sturm–Liouville Equations 450 8.1 A First Approach to Power Series Solutions of Differential Equations 450 8.2 A General Approach to Power Series Solutions of Homogeneous Equations 454 8.3 Singular Points of Linear Differential Equations 468 8.4 The Frobenius Method 470 8.5 The Gamma Function Revisited 487 8.6 Bessel Function of the First Kind Jn(x) 492 8.7 Bessel Functions of the Second Kind Yv(x) 502 8.8 Modified Bessel Functions Iv(x) and Kv(x) 508 8.9 A Critical Bending Problem Is There a Tallest Flagpole? 511 8.10 Sturm–Liouville Problems, Eigenfunctions, and Orthogonality 516 8.11 Eigenfunction Expansions and Completeness 533 Part Four: Fourier series, Integrals, and The Fourier Transform 549 Chapter 9. Fourier Series 550 9.1 Introduction to Fourier Series 550 9.2 Convergence of Fourier Series and Their Integration and Differentiation 564 9.3 Fourier Sine and Cosine Series on 0