Elsevier S & T

Advanced Engineering Mathematics

Jeffrey, Alan

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Table of contents
  • Title Pageiii
  • Copyright Pageiv
  • Contentsvii
  • Prefacexv
  • Part One: Review Material1
  • Chapter 1. Review of Prerequisites3
  • 1.1 Real Numbers, Mathematical Induction, and Mathematical Conventions4
  • 1.2 Complex Numbers10
  • 1.3 The Complex Plane15
  • 1.4 Modulus and Argument Representation of Complex Numbers18
  • 1.5 Roots of Complex Numbers22
  • 1.6 Partial Fractions27
  • 1.7 Fundamentals of Determinants31
  • 1.8 Continuity in One or More Variables35
  • 1.9 Differentiability of Functions of One or More Variables38
  • 1.10 Tangent Line and Tangent Plane Approximations to Functions40
  • 1.11 Integrals41
  • 1.12 Taylor and Maclaurin Theorems43
  • 1.13 Cylindrical and Spherical Polar Coordinates and Change of Variables in Partial Differentiation46
  • 1.14 Inverse Functions and the Inverse Function Theorem49
  • Part Two: Vectors and Matrices53
  • Chapter 2. Vectors and Vector Spaces55
  • 2.1 Vectors, Geometry, and Algebra56
  • 2.2 The Dot Product (Scalar Product)70
  • 2.3 The Cross Product (Vector Product)77
  • 2.4 Linear Dependence and Independence of Vectors and Triple Products82
  • 2.5 n-Vectors and the Vector Space Rn88
  • 2.6 Linear Independence, Basis, and Dimension95
  • 2.7 Gram–Schmidt Orthogonalization Process101
  • Chapter 3. Matrices and Systems of Linear Equations105
  • 3.1 Matrices106
  • 3.2 Some Problems That Give Rise to Matrices120
  • 3.3 Determinants133
  • 3.4 Elementary Row Operations, Elementary Matrices, and Their Connection with Matrix Multiplication143
  • 3.5 The Echelon and Row-Reduced Echelon Forms of a Matrix147
  • 3.6 Row and Column Spaces and Rank152
  • 3.7 The Solution of Homogeneous Systems of Linear Equations155
  • 3.8 The Solution of Nonhomogeneous Systems of Linear Equations158
  • 3.9 The Inverse Matrix163
  • 3.10 Derivative of a Matrix171
  • Chapter 4. Eigenvalues, Eigenvectors, and Diagonalization177
  • 4.1 Characteristic Polynomial, Eigenvalues, and Eigenvectors178
  • 4.2 Diagonalization of Matrices196
  • 4.3 Special Matrices with Complex Elements205
  • 4.4 Quadratic Forms210
  • 4.5 The Matrix Exponential215
  • Part Three: Ordinary Differential Equations225
  • Chapter 5. First Order Differential Equations227
  • 5.1 Background to Ordinary Differential Equations228
  • 5.2 Some Problems Leading to Ordinary Differential Equations233
  • 5.3 Direction Fields240
  • 5.4 Separable Equations242
  • 5.5 Homogeneous Equations247
  • 5.6 Exact Equations250
  • 5.7 Linear First Order Equations253
  • 5.8 The Bernoulli Equation259
  • 5.9 The Riccati Equation262
  • 5.10 Existence and Uniqueness of Solutions264
  • Chapter 6. Second and Higher Order Linear Differential Equations and Systems269
  • 6.1 Homogeneous Linear Constant Coefficient Second Order Equations270
  • 6.2 Oscillatory Solutions280
  • 6.3 Homogeneous Linear Higher Order Constant Coefficient Equations291
  • 6.4 Undetermined Coefficients Particular Integrals302
  • 6.5 Cauchy–Euler Equation309
  • 6.6 Variation of Parameters and the Green’s Function311
  • 6.7 Finding a Second Linearly Independent Solution from a Known Solution The Reduction of Order Meth321
  • 6.8 Reduction to the Standard Form u'' + f (x)u = 0324
  • 6.9 Systems of Ordinary Differential Equations An Introduction326
  • 6.10 A Matrix Approach to Linear Systems of Differential Equations333
  • 6.11 Nonhomogeneous Systems338
  • 6.12 Autonomous Systems of Equations351
  • Chapter 7. The Laplace Transform379
  • 7.1 Laplace Transform Fundamental Ideas379
  • 7.2 Operational Properties of the Laplace Transform390
  • 7.3 Systems of Equations and Applications of the Laplace Transform415
  • 7.4 The Transfer Function, Control Systems, and Time Lags437
  • Chapter 8. Series Solutions of Differential Equations, Special Functions, and Sturm–Liouville Equa443
  • 8.1 A First Approach to Power Series Solutions of Differential Equations443
  • 8.2 A General Approach to Power Series Solutions of Homogeneous Equations447
  • 8.3 Singular Points of Linear Differential Equations461
  • 8.4 The Frobenius Method463
  • 8.5 The Gamma Function Revisited480
  • 8.6 Bessel Function of the First Kind Jn(x)485
  • 8.7 Bessel Functions of the Second Kind Yν(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 Orthogonality509
  • 8.11 Eigenfunction Expansions and Completeness526
  • Part Four: Fourier series, Integrals, and The Fourier Transform543
  • Chapter 9. Fourier Series545
  • 9.1 Introduction to Fourier Series545
  • 9.2 Convergence of Fourier Series and Their Integration and Differentiation559
  • 9.3 Fourier Sine and Cosine Series on 0<_x<_ L568
  • 9.4 Other Forms of Fourier Series572
  • 9.5 Frequency and Amplitude Spectra of a Function577
  • 9.6 Double Fourier Series581
  • Chapter 10. Fourier Integrals and the Fourier Transform589
  • 10.1 The Fourier Integral589
  • 10.2 The Fourier Transform595
  • 10.3 Fourier Cosine and Sine Transforms611
  • Part Five: Vector Calculus623
  • Chapter 11. Vector Differential Calculus625
  • 11.1 Scalar and Vector Fields, Limits, Continuity, and Differentiability626
  • 11.2 Integration of Scalar and Vector Functions of a Single Real Variable636
  • 11.3 Directional Derivatives and the Gradient Operator644
  • 11.4 Conservative Fields and Potential Functions650
  • 11.5 Divergence and Curl of a Vector659
  • 11.6 Orthogonal Curvilinear Coordinates665
  • Chapter 12. Vector Integral Calculus677
  • 12.1 Background to Vector Integral Theorems678
  • 12.2 Integral Theorems680
  • 12.3 Transport Theorems697
  • 12.4 Fluid Mechanics Applications of Transport Theorems704
  • Part Six: Complex Analysis709
  • Chapter 13.Analytic Functions711
  • 13.1 Complex Functions and Mappings711
  • 13.2 Limits, Derivatives, and Analytic Functions717
  • 13.3 Harmonic Functions and Laplace’s Equation730
  • 13.4 Elementary Functions, Inverse Functions, and Branches735
  • Chapter 14.Complex Integration745
  • 14.1 Complex Integrals745
  • 14.2 Contours, the Cauchy–Goursat Theorem, and Contour Integrals755
  • 14.3 The Cauchy Integral Formulas769
  • 14.4 Some Properties of Analytic Functions775
  • Chapter 15. Laurent Series, Residues, and Contour Integration791
  • 15.1 Complex Power Series and Taylor Series791
  • 15.2 Uniform Convergence811
  • 15.3 Laurent Series and the Classification of Singularities816
  • 15.4 Residues and the Residue Theorem830
  • 15.5 Evaluation of Real Integrals by Means of Residues839
  • Chapter 16. The Laplace Inversion Integral863
  • 16.1 The Inversion Integral for the Laplace Transform863
  • Chapter 17. Conformal Mapping and Applications to Boundary Value Problems877
  • 17.1 Conformal Mapping877
  • 17.2 Conformal Mapping and Boundary Value Problems904
  • Part Seven: Partial Differential Equations925
  • Chapter 18. Partial Differential Equations927
  • 18.1 What Is a Partial Differential Equation?927
  • 18.2 The Method of Characteristics934
  • 18.3 Wave Propagation and First Order Pdes942
  • 18.4 Generalizing Solutions Conservation Laws and Shocks951
  • 18.5 The Three Fundamental Types of Linear Second Order Pde956
  • 18.6 Classification and Reduction to Standard Form of a Second Order Constant Coefficient Partial Di964
  • 18.7 Boundary Conditions and Initial Conditions975
  • 18.8 Waves and the One-Dimensional Wave Equation978
  • 18.9 The D’Alembert Solution of the Wave Equation and Applications981
  • 18.10 Separation of Variables988
  • 18.11 Some General Results for the Heat and Laplace Equation1025
  • 18.12 An Introduction to Laplace and Fourier Transform Methods for PDEs1030
  • Part Eight: Numerical Mathematics1043
  • Chapter 19. Numerical Mathematics1045
  • 19.1 Decimal Places and Significant Figures1046
  • 19.2 Roots of Nonlinear Functions1047
  • 19.3 Interpolation and Extrapolation1058
  • 19.4 Numerical Integration1065
  • 19.5 Numerical Solution of Linear Systems of Equations1077
  • 19.6 Eigenvalues and Eigenvectors1090
  • 19.7 Numerical Solution of Differential Equations1095
  • Answers1109
  • References1143
  • Index1147
Book details
  • Vendor Elsevier S & T
  • SKU 9780123825926
  • ISBN-13 9780080522968
  • Author Jeffrey, Alan
  • Category Mathematics
  • Subject Applied

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Description eBook delivery & access Reviews
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