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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 (TM) and MATLAB (R) 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 Moebius 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

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