1st Edition

Advanced Engineering Mathematics

By Lawrence Turyn Copyright 2014
    1456 Pages 353 B/W Illustrations
    by CRC Press

    Beginning with linear algebra and later expanding into calculus of variations, Advanced Engineering Mathematics provides accessible and comprehensive mathematical preparation for advanced undergraduate and beginning graduate students taking engineering courses. This book offers a review of standard mathematics coursework while effectively integrating science and engineering throughout the text. It explores the use of engineering applications, carefully explains links to engineering practice, and introduces the mathematical tools required for understanding and utilizing software packages.

    • Provides comprehensive coverage of mathematics used by engineering students
    • Combines stimulating examples with formal exposition and provides context for the mathematics presented
    • Contains a wide variety of applications and homework problems
    • Includes over 300 figures, more than 40 tables, and over 1500 equations
    • Introduces useful Mathematica™ and MATLAB® procedures
    • Presents faculty and student ancillaries, including an online student solutions manual, full solutions manual for instructors, and full-color figure sides for classroom presentations

    Advanced Engineering Mathematics covers ordinary and partial differential equations, matrix/linear algebra, Fourier series and transforms, and numerical methods. Examples include the singular value decomposition for matrices, least squares solutions, difference equations, the z-transform, Rayleigh methods for matrices and boundary value problems, the Galerkin method, numerical stability, splines, numerical linear algebra, curvilinear coordinates, calculus of variations, Liapunov functions, controllability, and conformal mapping.

    This text also serves as a good reference book for students seeking additional information. It incorporates Short Takes sections, describing more advanced topics to readers, and Learn More about It sections with direct references for readers wanting more in-depth information.

    Linear Algebraic Equations, Matrices, and Eigenvalues

    Solving Systems and Row Echelon Forms

    Matrix Addition, Multiplication, and Transpose

    Homogeneous Systems, Spanning Set, and Basic Solutions

    Solutions of Nonhomogeneous Systems

    Inverse Matrix

    Determinant, Adjugate Matrix, and Cramer’s Rule

    Linear Independence, Basis and Dimension

    Key Terms


    Matrix Theory

    Eigenvalues and Eigenvectors

    Basis of Eigenvectors and Diagonalization

    Inner Product and Orthogonal Sets of Vectors

    Orthonormal Bases and Orthogonal Matrices

    Least Squares Solutions

    Symmetric Matrices, Definite Matrices, and Applications

    Factorizations: QR and SVD

    Factorizations: LU and Cholesky

    Rayleigh Quotient

    Short Take: Inner Product and Hilbert Spaces

    Key Terms


    Scalar ODEs I: Homogeneous Problems

    Linear First-Order ODEs

    Separable and Exact ODEs

    Second-Order Linear Homogeneous ODEs

    Higher-Order Linear ODEs

    Cauchy–Euler ODEs

    Key Terms


    Scalar ODEs II: Nonhomogeneous Problems

    Nonhomogeneous ODEs

    Forced Oscillations

    Variation of Parameters

    Laplace Transforms: Basic Techniques

    Laplace Transforms: Unit Step and Other Techniques

    Scalar Difference Equations

    Short Take: z-Transforms

    Key Terms


    Linear Systems of ODEs

    Systems of ODEs

    Solving Linear Homogenous Systems of ODEs

    Complex or Deficient Eigenvalues

    Nonhomogeneous Linear Systems

    Nonresonant Nonhomogeneous Systems

    Linear Control Theory: Complete Controllability

    Linear Systems of Difference Equations

    Short Take: Periodic Linear Differential Equations

    Key Terms


    Geometry, Calculus, and Other Tools

    Dot Product, Cross Product, Lines, and Planes

    Trigonometry, Polar, Cylindrical, and Spherical Coordinates

    Curves and Surfaces

    Partial Derivatives

    Tangent Plane and Normal Vector

    Area, Volume, and Linear Transformations

    Differential Operators and Curvilinear Coordinates

    Rotating Coordinate Frames

    Key Terms


    Integral Theorems, Multiple Integrals, and Applications

    Integrals for a Function of a Single Variable

    Line Integrals

    Double Integrals, Green’s Theorem, and Applications

    Triple Integrals and Applications

    Surface Integrals and Applications

    Integral Theorems: Divergence, Stokes, and Applications

    Probability Distributions

    Key Terms


    Numerical Methods I

    Solving a Scalar Equation

    Solving a System of Equations

    Approximation of Integrals

    Numerical Solution of Ax = b

    Linear Algebraic Eigenvalue Problems

    Approximations of Derivatives

    Approximate Solutions of ODE-IVPs

    Approximate Solutions of Two Point BVPs


    Key Terms


    Fourier Series

    Orthogonality and Fourier Coefficients

    Fourier Cosine and Sine Series

    Generalized Fourier Series

    Complex Fourier Series and Fourier Transform

    Discrete Fourier and Fast Fourier Transforms

    Sturm–Liouville Problems

    Rayleigh Quotient

    Parseval’s Theorems and Applications

    Key Terms


    Partial Differential Equations Models

    Integral and Partial Differential Equations

    Heat Equations

    Potential Equations

    Wave Equations

    D’AlembertWave Solutions

    Short Take: Conservation of Energy in a Finite String

    Key Terms


    Separation of Variables for PDEs

    Heat Equation in One Space Dimension

    Wave Equation in One Space Dimension

    Laplace Equation in a Rectangle

    Eigenvalues of the Laplacian and Applications

    PDEs in Polar Coordinates

    PDEs in Cylindrical and Spherical Coordinates

    Key Terms


    Numerical Methods II

    Finite Difference Methods for Heat Equations

    Numerical Stability

    Finite Difference Methods for Potential Equations

    Finite Difference Methods for the Wave Equation

    Short Take: Galerkin Method

    Key Terms



    Functions of a Single Variable

    Functions of Several Variables

    Linear Programming Problems

    Simplex Procedure

    Nonlinear Programming

    Rayleigh–Ritz Method

    Key Terms


    Calculus of Variations

    Minimization Problems

    Necessary Conditions

    Problems with Constraints

    Eigenvalue Problems

    Short Take: Finite Element Methods

    Key Terms


    Functions of a Complex Variable

    Complex Numbers, Roots, and Functions

    Analyticity, Harmonic Function, and Harmonic Conjugate

    Elementary Functions

    Trigonometric Functions

    Taylor and Laurent Series

    Zeros and Poles

    Complex Integration and Cauchy’s Integral Theorem

    Cauchy’s Integral Formulas and Residues

    Real Integrals by Complex Integration Methods

    Key Terms

    Conformal Mapping

    Conformal Mappings and the Laplace Equation

    Möbius Transformations

    Solving Laplace’s Equation Using Conformal Maps

    Key Terms


    Integral Transform Methods

    Applications to Partial Differential Equations

    Inverse Laplace Transform

    Hankel Transforms

    Key Terms


    Nonlinear Ordinary Differential Equations

    Phase Line and Phase Plane

    Stability of an Equilibrium Point

    Variation of Parameters Using Linearization

    Liapunov Functions

    Short Take: LaSalle Invariance Principle

    Limit Cycles

    Existence, Uniqueness, and Continuous Dependence

    Short Take: Horseshoe Map and Chaos

    Short Take: Delay Equations

    Key Terms





    Dr. Larry Turyn is a professor of mathematics and statistics at Wright State University in Dayton, Ohio, where he has taught for 31 years. He earned degrees from Brown University and the Columbia University Fu Foundation School of Engineering and Applied Science. He has also been a Fellow and sessional instructor at the University of Calgary. At Wright State University he has developed several courses in engineering mathematics, differential equations, and applied analysis. Dr. Turyn has authored 26 articles in the fields of differential equations, eigenvalue problems, and applied mathematics.

    "… great expositions of many topics that are usually omitted in similar books but are important in applications. For instance, least square solutions are presented at great detail. Another strength of Turyn's book is a collection of exercises. … the selection of topics which makes the book very attractive."
    —Vladimir A. Dobrushkin, University of Rhode Island

    "The author has considerable experience teaching mathematical methods to engineers and he has produced an effective textbook based on that experience. The topics are broad, standard and appropriate. The exposition is aimed at the engineering student who has limited background in rigorous mathematics but who has experience in both application and computation."
    —Paul Eloe, University of Dayton

    "… well organized and its stuff is concisely presented. It covers almost every topic that should appear in an engineering textbook. It contains many examples to help students to understand. The material is presented in a conductive way and easy to follow. This book will be an ideal option for both first-time and advanced learners, thanks to its clarity in presentation and comprehensiveness in contents."
    —Xiaojun Yuan, Institute of Network Coding, The Chinese University of Hong Kong

    "The materials are well-written and self-contained. Examples are appropriate for better understanding of the theorems and definitions that are presented. Exercise problems are of varied difficulties, and they are suitable for the related topics presented in the book."
    —Muhammad N. Islam, University of Dayton