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
References
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
References
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
Reference
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
References
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
References
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
Reference
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
Reference
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
Splines
Key Terms
References
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
References
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
Reference
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
References
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
Reference
Optimization
Functions of a Single Variable
Functions of Several Variables
Linear Programming Problems
Simplex Procedure
Nonlinear Programming
Rayleigh–Ritz Method
Key Terms
References
Calculus of Variations
Minimization Problems
Necessary Conditions
Problems with Constraints
Eigenvalue Problems
Short Take: Finite Element Methods
Key Terms
References
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
References
Integral Transform Methods
Applications to Partial Differential Equations
Inverse Laplace Transform
Hankel Transforms
Key Terms
References
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
Reference
Appendices
Index
Biography
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