Advanced Engineering Mathematics 10th Edition (1)

Advanced Engineering Mathematics 10th Edition (1)

(Parte 1 de 5)

Systems of Units. Some Important Conversion Factors

The most important systems of units are shown in the table below. The mks system is also known as the International System of Units(abbreviated SI), and the abbreviations sec (instead of s), gm (instead of g), and nt (instead of N) are also used.

System of unitsLengthMassTimeForce cgs systemcentimeter (cm)gram (g)second (s)dyne mks systemmeter (m)kilogram (kg)second (s)newton (nt) Engineering systemfoot (ft)slugsecond (s)pound (lb)

For further details see, for example, D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics.9th ed., Hoboken,

N. J: Wiley, 2011. See also AN American National Standard, ASTM/IEEE Standard Metric Practice, Institute of Electrical and Electronics Engineers, Inc. (IEEE), 445 Hoes Lane, Piscataway, N. J. 08854, website at

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eax sin bx dx (asinbx bcosbx) c

eax cos bx dx x a dx x a dx x a dx


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Professor of Mathematics Ohio State University Columbus, Ohio

In collaboration with


EDWARD J. NORMINTON Associate Professor of Mathematics Carleton University Ottawa, Ontario ffirs.qxd 1/8/10 3:50 PM Page v

PUBLISHER Laurie Rosatone PROJECT EDITORShannon Corliss MARKETING MANAGERJonathan Cottrell CONTENT MANAGER Lucille Buonocore PRODUCTION EDITORBarbara Russiello MEDIA EDITORMelissa Edwards MEDIA PRODUCTION SPECIALISTLisa Sabatini TEXT AND COVER DESIGNMadelyn Lesure PHOTO RESEARCHERSheena Goldstein COVER PHOTO©Denis Jr. Tangney/iStockphoto

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ISBN 978-0-470-45836-5

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PREF A CE See also

Purpose and Structure of the Book

This book provides a comprehensive, thorough, and up-to-date treatment of engineering mathematics.It is intended to introduce students of engineering, physics, mathematics, computer science, and related fields to those areas of applied mathematics that are most relevant for solving practical problems. A course in elementary calculus is the sole prerequisite.(However, a concise refresher of basic calculus for the student is included on the inside cover and in Appendix 3.)

The subject matter is arranged into seven parts as follows:

A.Ordinary Differential Equations (ODEs)in Chapters 1–6 B.Linear Algebra. Vector Calculus.See Chapters 7–10 C.Fourier Analysis. Partial Differential Equations (PDEs).See Chapters 1 and 12 D.Complex Analysisin Chapters 13–18 E.Numeric Analysisin Chapters 19–21 F.Optimization, Graphsin Chapters 2 and 23 G.Probability, Statisticsin Chapters 24 and 25.

These are followed by five appendices: 1.References, 2.Answers to Odd-Numbered Problems, 3.Auxiliary Materials (see also inside covers of book), 4.Additional Proofs, 5.Table of Functions. This is shown in a block diagram on the next page.

The parts of the book are kept independent. In addition, individual chapters are kept as independent as possible. (If so needed, any prerequisites—to the level of individual sections of prior chapters—are clearly stated at the opening of each chapter.) We give the instructor maximum flexibility in selecting the materialand tailoring it to his or her need. The book has helped to pave the way for the present development of engineering mathematics. This new edition will prepare the student for the current tasks and the future by a modern approach to the areas listed above. We provide the material and learning tools for the students to get a good foundation of engineering mathematics that will help them in their careers and in further studies.

General Features of the Book Include:

•Simplicity of examplesto make the book teachable—why choose complicated examples when simple ones are as instructive or even better?

•Independence of parts and blocks of chaptersto provide flexibility in tailoring courses to specific needs.

•Self-contained presentation, except for a few clearly marked places where a proof would exceed the level of the book and a reference is given instead.

•Gradual increase in difficulty of material with no jumps or gapsto ensure an enjoyable teaching and learning experience.

•Modern standard notationto help students with other courses, modern books, and journals in mathematics, engineering, statistics, physics, computer science, and others.

Furthermore, we designed the book to be a single, self-contained, authoritative, and convenient sourcefor studying and teaching applied mathematics, eliminating the need for time-consuming searches on the Internet or time-consuming trips to the library to get a particular reference book.

vii fpref.qxd 1/8/10 3:16 PM Page vii viii Preface

Maple Computer Guide Mathematica Computer Guide

Student Solutions Manual and Study Guide

Instructor’s Manual

Chaps. 1–6 Ordinary Differential Equations (ODEs)

Chaps. 1–4 Basic Material

Chap. 5 Chap. 6 Series SolutionsLaplace Transforms

Chaps. 7–10 Linear Algebra. Vector Calculus

Chap. 7Chap. 9 Matrices, Vector Differential Linear Systems Calculus

Chap. 8Chap. 10 Eigenvalue ProblemsVector Integral Calculus

Chaps. 1–12

Fourier Analysis. Partial Differential Equations (PDEs)

Chap. 1 Fourier Analysis

Chap. 12 Partial Differential Equations

Chaps. 13–18

Complex Analysis, Potential Theory

Chaps. 13–17 Basic Material

Chap. 18 Potential Theory

Chaps. 19–21 Numeric Analysis

Chap. 19Chap. 20Chap. 21

Numerics inNumericNumerics for GeneralLinear AlgebraODEs and PDEs

Chaps. 2–23 Optimization, Graphs

Chap. 22Chap. 23 Linear ProgrammingGraphs, Optimization

Chaps. 24–25 Probability, Statistics

Chap. 24 Data Analysis. Probability Theory

Chap. 25 Mathematical Statistics fpref.qxd 1/8/10 3:16 PM Page viii

Four Underlying Themes of the Book

The driving force in engineering mathematics is the rapid growth of technology and the sciences. New areas—often drawing from several disciplines—come into existence. Electric cars, solar energy, wind energy, green manufacturing, nanotechnology, risk management, biotechnology, biomedical engineering, computer vision, robotics, space travel, communication systems, green logistics, transportation systems, financial engineering, economics, and many other areas are advancing rapidly. What does this mean for engineering mathematics? The engineer has to take a problem from any diverse area and be able to model it. This leads to the first of four underlying themes of the book.

1. Modelingis the process in engineering, physics, computer science, biology, chemistry, environmental science, economics, and other fields whereby a physical situation or some other observation is translated into a mathematical model. This mathematical model could be a system of differential equations, such as in population control (Sec. 4.5), a probabilistic model (Chap. 24), such as in risk management, a linear programming problem (Secs. 2.2–2.4) in minimizing environmental damage due to pollutants, a financial problem of valuing a bond leading to an algebraic equation that has to be solved by Newton’s method (Sec. 19.2), and many others.

The next step is solving the mathematical problemobtained by one of the many techniques covered in Advanced Engineering Mathematics.

The third step is interpreting the mathematical resultin physical or other terms to see what it means in practice and any implications.

Finally, we may have to make a decisionthat may be of an industrial nature or recommend a public policy. For example, the population control model may imply thepolicy to stop fishing for 3 years. Or the valuation of the bond may lead to a recommendation to buy. The variety is endless, but the underlying mathematics is surprisingly powerful and able to provide advice leading to the achievement of goals toward the betterment of society, for example, by recommending wise policies concerning global warming, better allocation of resources in a manufacturing process, or making statistical decisions (such as in Sec. 25.4 whether a drug is effective in treating a disease).

While we cannot predict what the future holds, we do know that the student has to practice modeling by being given problems from many different applications as is done in this book. We teach modeling from scratch, right in Sec. 1.1, and give many examples in Sec. 1.3, and continue to reinforce the modeling process throughout the book.

2. Judicious use of powerful software for numerics(listed in the beginning of PartE) and statistics (Part G) is of growing importance. Projects in engineering and industrial companies may involve large problems of modeling very complex systems with hundreds of thousands of equations or even more. They require the use of such software. However, our policy has always been to leave it up to the instructor to determine the degree of use of computers, from none or little use to extensive use. More on this below.

3. The beauty of engineering mathematics.Engineering mathematics relies on relatively few basic concepts and involves powerful unifying principles. We point them out whenever they are clearly visible, such as in Sec. 4.1 where we “grow” a mixing problem from one tank to two tanks and a circuit problem from one circuit to two circuits, thereby also increasing the number of ODEs from one ODE to two ODEs. This is an example of an attractive mathematical model because the “growth” in the problem is reflected by an “increase” in ODEs.

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4. To clearly identify the conceptual structure of subject matters.For example, complex analysis (in Part D) is a field that is not monolithic in structure but was formed by three distinct schools of mathematics. Each gave a different approach, which we clearly mark. The first approach is solving complex integrals by Cauchy’s integral formula (Chaps. 13 and 14), the second approach is to use the Laurent series and solve complex integrals by residue integration (Chaps. 15 and 16), and finally we use a geometric approach of conformal mapping to solve boundary value problems (Chaps. 17 and 18). Learning the conceptual structure and terminology of the different areas of engineering mathematics is very important for three reasons: a.It allows the student to identify a new problem and put it into the right group of problems. The areas of engineering mathematics are growing but most often retain their conceptual structure. b.The student can absorb new information more rapidlyby being able to fit it into the conceptual structure. c.Knowledge of the conceptual structure and terminology is also important when using the Internet to search for mathematical information. Since the search proceeds by putting in key words (i.e., terms) into the search engine, the student has to remember the important concepts (or be able to look them up in the book) that identify the application and area of engineering mathematics.

Big Changes in This Edition

Problem Sets Changed

The problem sets have been revised and rebalanced with some problem sets having more problems and some less, reflecting changes in engineering mathematics. There is a greater emphasis on modeling. Now there are also problems on the discrete Fourier transform (inSec. 1.9).

Series Solutions of ODEs, Special Functions and Fourier Analysis Reorganized

Chap. 5, on series solutions of ODEs and special functions, has been shortened. Chap. 1 onFourier Analysis now contains Sturm–Liouville problems, orthogonal functions, and orthogonal eigenfunction expansions (Secs. 1.5, 1.6), where they fit better conceptually (rather than in Chap. 5), being extensions of Fourier’s idea of using orthogonal functions.

Openings of Parts and Chapters Rewritten As Well As Parts of Sections

In order to give the student a better idea of the structure of the material (see Underlying Theme 4 above), we have entirely rewritten the openings of parts and chapters. Furthermore, large parts or individual paragraphs of sections have been rewritten or new sentences inserted into the text. This should give the students a better intuitive understanding of the material (see Theme 3 above), let them draw conclusions on their own, and be able to tackle more advanced material. Overall, we feel that the book has become more detailed and leisurely written.

Student Solutions Manual and Study Guide Enlarged

(Parte 1 de 5)