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## Overview

Now you can make rigorous mathematical topics accessible to your students by emphasizing visuals, numerous examples, and interesting mathematical models with O'Neil’s ADVANCED ENGINEERING MATHEMATICS, 8E. New "Math in Context" broadens the engineering connections for your students by clearly demonstrating how mathematical concepts are applied to current engineering problems. You have the flexibility to select additional topics that are best for your individual course, including many new web modules while minimizing the cost of the printed book.

- SPECIALIZED AND ADVANCED TOPICS ADDED AS WEB MODULES. In order to broaden coverage while keeping the length and cost of the book down, certain topics have been added as convenient web modules rather than appearing in the printed text. These modules include applications of complex analysis to the Dirichlet problem and to inverses of Laplace transforms, Lyapunov functions, the discrete Fourier transform, Maxwell’s equations, numerical methods for solving differential equations, LU factoring of matrices, limit cycles for systems of differential equations, and models of plane fluid flow.
- NEW “MATH IN CONTEXT” FEATURES CONNECT THE MATHEMATICAL APPLICATIONS TO CURRENT ENGINEERING PROBLEMS. These special features specifically relate mathematical concepts and methods to real world problems that students will find in the workplace.
- THIS EDITION PRESENTS THE SOLUTION OF INITIAL VALUE PROBLEMS FOR WAVE MOTION AND DIFFUSION PHENOMENA UNDER A VARIETY OF CONDITIONS. This edition considers nonhomogeneous conditions, forcing terms, convection and insulation effects, and other conditions encountered in applications.
- COVERAGE INCLUDES EXPANDED TREATMENT OF SPECIAL FUNCTIONS. This edition includes even more material about both special functions and their applications to help you better prepare your students.
- THE BOOK OFFERS A GENERAL APPROACH TO THE SOLUTION OF PROBLEMS USING EIGENFUNCTION EXPANSIONS.
- NEW ORGANIZATION CLARIFIES AND CLEARLY DIFFERENTIATES TOPICS. This edition’s refined organization allows your students to focus on a particular idea and its applications before progressing to new topics.
- NEW EXAMPLES AND APPLICATIONS CLARIFY HOW TO SOLVE SPECIFIC TYPES OF PROBLEMS. These new examples apply the mathematical techniques that students are learning to real-world settings, such as the analysis of wave motion and diffusion processes.

- AN INSTRUCTOR’S SOLUTIONS MANUAL OFFERS COMPLETE SOLUTIONS. You will find detailed solutions to virtually all problems in the book to save you time and effort.
- TABLES OF TRANSFORMS PROVIDE QUICK REFERENCE. Your students will find tables for Fourier, Laplace, Fourier cosine and sine for use in solving problems within the book.
- A GUIDE TO POST-CALCULUS NOTATION ASSISTS STUDENTS. This helpful guide provides support for your students as they reference new terms and symbols
- DETAILED EXAMPLES HIGHLIGHT BOTH MATHEMATICAL IDEAS AND THEIR APPLICATIONS. Valuable examples throughout develop and emphasize mathematical concepts as well as practical engineering uses.
- TWO ADDITIONAL WEB CHAPTERS DETAIL STATISTICS AND PROBABILITY. These chapters are conveniently posted on the book’s website and include both problems and solutions.
- STUDENT SOLUTIONS MANUAL CONTAINS WORKED-OUT SOLUTIONS TO HALF OF THE TEXT PROBLEMS. These solutions are specifically developed to assist your students in truly understanding the ideas involved, rather than simply listing the mechanical steps.
- THE BOOK’S SEVEN-PART ORGANIZATION BEST FITS MOST COURSE ORGANIZATIONS. Seven distinctive parts clearly focus and differentiate the mathematical ideas and methods while giving you the flexibility to select the sections best suited for your course and student needs.
- DETAILED EXAMPLES ILLUSTRATE THE USE OF NOTATION AND THE THEORY. The numerous examples clarify notation, theory and the underlying computations, followed by the numerical calculations themselves.

1. First-Order Differential Equations.

Terminology and Separable Equations. Singular Solutions, Linear Equations. Exact Equations. Homogeneous, Bernoulli and Riccati Equations.

2. Second-Order Differential Equations.

The Linear Second-Order Equation. The Constant Coefficient Homogeneous Equation. Particular Solutions of the Nonhomogeneous Equation. The Euler Differential Equation, Series Solutions. Frobenius Series Solutions.

3. The Laplace Transform.

Definition and Notation. Solution of Initial Value Problems. The Heaviside Function and Shifting Theorems. Convolution. Impulses and the Dirac Delta Function. Systems of Linear Differential Equations.

4. Eigenfunction Expansions.

Eigenvalues, Eigenfunctions, and Sturm-Liouville Problems. Eigenfunction Expansions, Fourier Series.

Part II: PARTIAL DIFFERENTIAL EQUATIONS.

5. The Heat Equation.

Diffusion Problems in a Bounded Medium. The Heat Equation with a Forcing Term F(x,t). The Heat Equation on the Real Line. A Reformulation of the Solution on the Real Line. The Heat Equation on a Half-Line, The Two-Dimensional Heat Equation.

6. The Wave Equation.

Wave Motion on a Bounded Interval. The Effect of c on the Motion. Wave Motion with a Forcing Term F(x). Wave Motion in an Unbounded Medium. The Wave Equation on the Real Line. d’Alembert’s Solution and Characteristics. The Wave Equation with a Forcing Term K(x,t). The Wave Equation in Higher Dimensions.

7. Laplace’s Equation.

The Dirichlet Problem for a Rectangle. Dirichlet Problem for a Disk. The Poisson Integral Formula. The Dirichlet Problem for Unbounded Regions. A Dirichlet Problem in 3 Dimensions. The Neumann Problem. Poisson’s Equation.

8. Special Functions and Applications.

Legendre Polynomials. Bessel Functions. Some Applications of Bessel Functions.

9. Transform Methods of Solution.

Laplace Transform Methods. Fourier Transform Methods. Fourier Sine and Cosine Transforms.

Part III: MATRICES AND LINEAR ALGEBRA.

10. Vectors and the Vector Space Rn.

Vectors in the Plane and 3 – Space. The Dot Product. The Cross Product. n-Vectors and the Algebraic Structure of Rn. Orthogonal Sets and Orthogonalization. Orthogonal Complements and Projections.

11. Matrices, Determinants and Linear Systems.

Matrices and Matrix Algebra. Row Operations and Reduced Matrices. Solution of Homogeneous Linear Systems. Solution of Nonhomogeneous Linear Systems. Matrix Inverses. Determinants, Cramer’s Rule. The Matrix Tree Theorem.

12. Eigenvalues, Diagonalization and Special Matrices.

Eigenvalues and Eigenvectors. Diagonalization. Special Matrices and Their Eigenvalues and Eigenvectors. Quadratic Forms.

PART IV: SYSTEMS OF DIFFERENTIAL EQUATIONS.

13. Systems of Linear Differential Equations.

Linear Systems. Solution of X’ = AX When A Is Constant. Exponential Matrix Solutions. Solution of X’ = AX + G for Constant A.

14. Nonlinear Systems and Qualitative Analysis.

Nonlinear Systems and Phase Portraits. Critical Points and Stability. Almost Linear Systems, Linearization.

Part V: VECTOR ANALYSIS.

15. Vector Differential Calculus.

Vector Functions of One Variable. Velocity, Acceleration, and Curvature. The Gradient Field. Divergence and Curl. Streamlines of a Vector Field.

16. Vector Integral Calculus.

Line Integrals. Green’s Theorem. Independence of Path and Potential Theory. Surface Integrals. Applications of Surface Integrals. Gauss’s Divergence Theorem. Stokes’s Theorem.

PART VI: FOURIER ANALYSIS.

17. Fourier Series.

Fourier Series On [-L, L]. Fourier Sine and Cosine Series. Integration and Differentiation of Fourier Series. Properties of Fourier Coefficients. Phase Angle Form. Complex Fourier Series, Filtering of Signals.

18. Fourier Transforms.

The Fourier Transform. Fourier Sine and Cosine Transforms.

PART VII: COMPLEX FUNCTIONS.

19. Complex Numbers and Functions.

Geometry and Arithmetic of Complex Numbers. Complex Functions, Limits. The Exponential and Trigonometric Functions. The Complex Logarithm. Powers.

20. Integration.

The Integral of a Complex Function. Cauchy’s Theorem. Consequences of Cauchy’s Theorem.

21. Series Representations of Functions.

Power Series. The Laurent Expansion.

22. Singularities and the Residue Theorem.

Classification of Singularities. The Residue Theorem. Evaluation of Real Integrals.

23. Conformal Mappings.

The Idea of a Conformal Mapping. Construction of Conformal Mappings.

Notation.

ANSWERS TO SELECTED PROBLEMS.

“The material is written in a coherent fashion…. I like the clarity of explanation.”

Cengage provides a range of supplements that are updated in coordination with the main title selection. For more information about these supplements, contact your Learning Consultant.

### FOR INSTRUCTORS

#### Instructor’s Solution Manual

ISBN: 9781337115094

This helpful Instructor’s Solution Manual saves you time as it provides detailed answers to nearly all of the book’s problems.

#### Instructor's Companion Website

ISBN: 9781337288453

Find everything you need for your course in one place. This password-protected Instructor’s website contains the Instructor’s Solutions Manual, Lecture Note PowerPoint® slides, two additional web chapters, and more than 20 supplementary web modules.