# Advanced Engineering Mathematics

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## Book Description

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.

## Table of Contents

**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

## Author(s)

### 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.

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## Reviews

—Vladimir A. Dobrushkin, University of Rhode Island"… 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.""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