# Advanced engineering mathematics 10th edition, Study notes for Engineering Mathematics. Vinoba Bhave University

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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 units Length Mass Time Force

cgs system centimeter (cm) gram (g) second (s) dyne

mks system meter (m) kilogram (kg) second (s) newton (nt)

Engineering system foot (ft) slug second (s) pound (lb)

1 inch (in.)  2.540000 cm 1 foot (ft)  12 in.  30.480000 cm

1 yard (yd)  3 ft  91.440000 cm 1 statute mile (mi)  5280 ft  1.609344 km

1 nautical mile  6080 ft  1.853184 km

1 acre  4840 yd2  4046.8564 m2 1 mi2  640 acres  2.5899881 km2

1 fluid ounce  1/128 U.S. gallon  231/128 in.3  29.573730 cm3

1 U.S. gallon  4 quarts (liq)  8 pints (liq)  128 fl oz  3785.4118 cm3

1 British Imperial and Canadian gallon  1.200949 U.S. gallons  4546.087 cm3

1 slug  14.59390 kg

1 pound (lb)  4.448444 nt 1 newton (nt)  105 dynes

1 British thermal unit (Btu)  1054.35 joules 1 joule  107 ergs

1 calorie (cal)  4.1840 joules

1 kilowatt-hour (kWh)  3414.4 Btu  3.6 • 106 joules

1 horsepower (hp)  2542.48 Btu/h  178.298 cal/sec  0.74570 kW

1 kilowatt (kW)  1000 watts  3414.43 Btu/h  238.662 cal/s

°F  °C • 1.8  32 1°  60  3600  0.017453293 radian

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 www.ieee.org.

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Integration

uv dx  uv   uv dx (by parts) xn dx   c (n  1)  dx  ln x  c eax dx  eax  c sin x dx  cos x  c cos x dx  sin x  c  tan x dx  ln cos x  c cot x dx  ln sin x  c sec x dx  ln sec x  tan x  c csc x dx  ln csc x  cot x  c   arctan  c

  arcsin  c

  arcsinh  c

  arccosh  c

sin2 x dx  1_2 x  1_4 sin 2x  c cos2 x dx  1_2 x  1_4 sin 2x  c  tan2 x dx  tan x  x  c cot2 x dx  cot x  x  c  ln x dx  x ln x  x  c eax sin bx dx

 (a sin bx  b cos bx)  c

eax cos bx dx  (a cos bx  b sin bx)  c

eax

a2  b2

eax

a2  b2

x  a

dx  x2  a2

x  a

dx  x2  a2

x  a

dx  a2 x2

x  a

1  a

dx  x2  a2

1 a

1 x

xn1

n  1

Differentiation

(cu)  cu (c constant)

(u  v)  u  v

(uv)  uv  uv

( ) 

 • (Chain rule)

(xn)  nxn1

(ex)  ex

(eax)  aeax

(ax)  ax ln a

(sin x)  cos x

(cos x)  sin x

(tan x)  sec2 x

(cot x)  csc2 x

(sinh x)  cosh x

(cosh x)  sinh x

(ln x) 

(loga x) 

(arcsin x) 

(arccos x)  

(arctan x) 

(arccot x)   1

 1  x2

1  1  x2

1  1  x2

1  1  x2

loga e 

x

1  x

dy  dx

du  dy

du  dx

uv  uv 

v2 u  v

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10T H E D I T I O N

ADVANCED ENGINEERING MATHEMATICS ERWIN KREYSZIG Professor of Mathematics Ohio State University Columbus, Ohio

In collaboration with

HERBERT KREYSZIG New York, New York

EDWARD J. NORMINTON Associate Professor of Mathematics Carleton University Ottawa, Ontario

JOHN WILEY & SONS, INC.

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PUBLISHER Laurie Rosatone PROJECT EDITOR Shannon Corliss MARKETING MANAGER Jonathan Cottrell CONTENT MANAGER Lucille Buonocore PRODUCTION EDITOR Barbara Russiello MEDIA EDITOR Melissa Edwards MEDIA PRODUCTION SPECIALIST Lisa Sabatini TEXT AND COVER DESIGN Madelyn Lesure PHOTO RESEARCHER Sheena Goldstein COVER PHOTO © Denis Jr. Tangney/iStockphoto

Cover photo shows the Zakim Bunker Hill Memorial Bridge in Boston, MA.

This book was set in Times Roman. The book was composed by PreMedia Global, and printed and bound by RR Donnelley & Sons Company, Jefferson City, MO. The cover was printed by RR Donnelley & Sons Company, Jefferson City, MO.

This book is printed on acid free paper.

Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper specifications and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website: www.wiley.com/go/citizenship.

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Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at: www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative.

ISBN 978-0-470-45836-5

Printed in the United States of America

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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 11 and 12 D. Complex Analysis in Chapters 13–18 E. Numeric Analysis in Chapters 19–21 F. Optimization, Graphs in Chapters 22 and 23 G. Probability, Statistics in 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 material and 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 examples to make the book teachable—why choose complicated

examples when simple ones are as instructive or even better?

Independence of parts and blocks of chapters to 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 gaps to ensure an enjoyable teaching and learning experience.

Modern standard notation to 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 source for 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.

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viii Preface

GUIDES AND MANUALS Maple Computer Guide

Mathematica Computer Guide

Student Solutions Manual and Study Guide

Instructor’s Manual

PART A Chaps. 1–6

Ordinary Differential Equations (ODEs)

Chaps. 1–4 Basic Material

Chap. 5 Chap. 6 Series Solutions Laplace Transforms

PART B Chaps. 7–10

Linear Algebra. Vector Calculus

Chap. 7 Chap. 9 Matrices, Vector Differential

Linear Systems Calculus

Chap. 8 Chap. 10 Eigenvalue Problems Vector Integral Calculus

PARTS AND CHAPTERS OF THE BOOK

PART C Chaps. 11–12

Fourier Analysis. Partial Differential Equations (PDEs)

Chap. 11 Fourier Analysis

Chap. 12 Partial Differential Equations

PART D Chaps. 13–18

Complex Analysis, Potential Theory

Chaps. 13–17 Basic Material

Chap. 18 Potential Theory

PART E Chaps. 19–21

Numeric Analysis

Chap. 19 Chap. 20 Chap. 21 Numerics in Numeric Numerics for

General Linear Algebra ODEs and PDEs

PART F Chaps. 22–23

Optimization, Graphs

Chap. 22 Chap. 23 Linear Programming Graphs, Optimization

PART G Chaps. 24–25

Probability, Statistics

Chap. 24 Data Analysis. Probability Theory

Chap. 25 Mathematical Statistics

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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. Modeling is 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. 22.2–22.4) 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 problem obtained by one of the many techniques covered in Advanced Engineering Mathematics.

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

Finally, we may have to make a decision that may be of an industrial nature or recommend a public policy. For example, the population control model may imply the policy 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 Part E) 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 rapidly by 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 (in Sec. 11.9).

Series Solutions of ODEs, Special Functions and Fourier Analysis Reorganized Chap. 5, on series solutions of ODEs and special functions, has been shortened. Chap. 11 on Fourier Analysis now contains Sturm–Liouville problems, orthogonal functions, and orthogonal eigenfunction expansions (Secs. 11.5, 11.6), 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 Upon the explicit request of the users, the answers provided are more detailed and complete. More explanations are given on how to learn the material effectively by pointing out what is most important.

More Historical Footnotes, Some Enlarged Historical footnotes are there to show the student that many people from different countries working in different professions, such as surveyors, researchers in industry, etc., contributed

5

4

3

2

1

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to the field of engineering mathematics. It should encourage the students to be creative in their own interests and careers and perhaps also to make contributions to engineering mathematics.

Further Changes and New Features • Parts of Chap. 1 on first-order ODEs are rewritten. More emphasis on modeling, also

new block diagram explaining this concept in Sec. 1.1. Early introduction of Euler’s method in Sec. 1.2 to familiarize student with basic numerics. More examples of separable ODEs in Sec. 1.3.

• For Chap. 2, on second-order ODEs, note the following changes: For ease of reading, the first part of Sec. 2.4, which deals with setting up the mass-spring system, has been rewritten; also some rewriting in Sec. 2.5 on the Euler–Cauchy equation.

• Substantially shortened Chap. 5, Series Solutions of ODEs. Special Functions: combined Secs. 5.1 and 5.2 into one section called “Power Series Method,” shortened material in Sec. 5.4 Bessel’s Equation (of the first kind), removed Sec. 5.7 (Sturm–Liouville Problems) and Sec. 5.8 (Orthogonal Eigenfunction Expansions) and moved material into Chap. 11 (see “Major Changes” above).

• New equivalent definition of basis (Sec. 7.4).

• In Sec. 7.9, completely new part on composition of linear transformations with two new examples. Also, more detailed explanation of the role of axioms, in connection with the definition of vector space.

• New table of orientation (opening of Chap. 8 “Linear Algebra: Matrix Eigenvalue Problems”) where eigenvalue problems occur in the book. More intuitive explanation of what an eigenvalue is at the begining of Sec. 8.1.

• Better definition of cross product (in vector differential calculus) by properly identifying the degenerate case (in Sec. 9.3).

Chap. 11 on Fourier Analysis extensively rearranged: Secs. 11.2 and 11.3 combined into one section (Sec. 11.2), old Sec. 11.4 on complex Fourier Series removed and new Secs. 11.5 (Sturm–Liouville Problems) and 11.6 (Orthogonal Series) put in (see “Major Changes” above). New problems (new!) in problem set 11.9 on discrete Fourier transform.

New section 12.5 on modeling heat flow from a body in space by setting up the heat equation. Modeling PDEs is more difficult so we separated the modeling process from the solving process (in Sec. 12.6).

Introduction to Numerics rewritten for greater clarity and better presentation; new Example 1 on how to round a number. Sec. 19.3 on interpolation shortened by removing the less important central difference formula and giving a reference instead.

• Large new footnote with historical details in Sec. 22.3, honoring George Dantzig, the inventor of the simplex method.

Traveling salesman problem now described better as a “difficult” problem, typical of combinatorial optimization (in Sec. 23.2). More careful explanation on how to compute the capacity of a cut set in Sec. 23.6 (Flows on Networks).

• In Chap. 24, material on data representation and characterization restructured in terms of five examples and enlarged to include empirical rule on distribution of

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xii Preface

data, outliers, and the z-score (Sec. 24.1). Furthermore, new example on encription (Sec. 24.4).

• Lists of software for numerics (Part E) and statistics (Part G) updated.

• References in Appendix 1 updated to include new editions and some references to websites.

Use of Computers The presentation in this book is adaptable to various degrees of use of software, Computer Algebra Systems (CAS’s), or programmable graphic calculators, ranging from no use, very little use, medium use, to intensive use of such technology. The choice of how much computer content the course should have is left up to the instructor, thereby exhibiting our philosophy of maximum flexibility and adaptability. And, no matter what the instructor decides, there will be no gaps or jumps in the text or problem set. Some problems are clearly designed as routine and drill exercises and should be solved by hand (paper and pencil, or typing on your computer). Other problems require more thinking and can also be solved without computers. Then there are problems where the computer can give the student a hand. And finally, the book has CAS projects, CAS problems and CAS experiments, which do require a computer, and show its power in solving problems that are difficult or impossible to access otherwise. Here our goal is to combine intelligent computer use with high-quality mathematics. The computer invites visualization, experimentation, and independent discovery work. In summary, the high degree of flexibility of computer use for the book is possible since there are plenty of problems to choose from and the CAS problems can be omitted if desired.

Note that information on software (what is available and where to order it) is at the beginning of Part E on Numeric Analysis and Part G on Probability and Statistics. Since Maple and Mathematica are popular Computer Algebra Systems, there are two computer guides available that are specifically tailored to Advanced Engineering Mathematics: E. Kreyszig and E.J. Norminton, Maple Computer Guide, 10th Edition and Mathematica Computer Guide, 10th Edition. Their use is completely optional as the text in the book is written without the guides in mind.

Suggestions for Courses: A Four-Semester Sequence The material, when taken in sequence, is suitable for four consecutive semester courses, meeting 3 to 4 hours a week:

1st Semester ODEs (Chaps. 1–5 or 1–6) 2nd Semester Linear Algebra. Vector Analysis (Chaps. 7–10) 3rd Semester Complex Analysis (Chaps. 13–18) 4th Semester Numeric Methods (Chaps. 19–21)

Suggestions for Independent One-Semester Courses The book is also suitable for various independent one-semester courses meeting 3 hours a week. For instance,

Introduction to ODEs (Chaps. 1–2, 21.1) Laplace Transforms (Chap. 6) Matrices and Linear Systems (Chaps. 7–8)

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Vector Algebra and Calculus (Chaps. 9–10) Fourier Series and PDEs (Chaps. 11–12, Secs. 21.4–21.7) Introduction to Complex Analysis (Chaps. 13–17) Numeric Analysis (Chaps. 19, 21) Numeric Linear Algebra (Chap. 20) Optimization (Chaps. 22–23) Graphs and Combinatorial Optimization (Chap. 23) Probability and Statistics (Chaps. 24–25)

Acknowledgments We are indebted to former teachers, colleagues, and students who helped us directly or indirectly in preparing this book, in particular this new edition. We profited greatly from discussions with engineers, physicists, mathematicians, computer scientists, and others, and from their written comments. We would like to mention in particular Professors Y. A. Antipov, R. Belinski, S. L. Campbell, R. Carr, P. L. Chambré, Isabel F. Cruz, Z. Davis, D. Dicker, L. D. Drager, D. Ellis, W. Fox, A. Goriely, R. B. Guenther, J. B. Handley, N. Harbertson, A. Hassen, V. W. Howe, H. Kuhn, K. Millet, J. D. Moore, W. D. Munroe, A. Nadim, B. S. Ng, J. N. Ong, P. J. Pritchard, W. O. Ray, L. F. Shampine, H. L. Smith, Roberto Tamassia, A. L. Villone, H. J. Weiss, A. Wilansky, Neil M. Wigley, and L. Ying; Maria E. and Jorge A. Miranda, JD, all from the United States; Professors Wayne H. Enright, Francis. L. Lemire, James J. Little, David G. Lowe, Gerry McPhail, Theodore S. Norvell, and R. Vaillancourt; Jeff Seiler and David Stanley, all from Canada; and Professor Eugen Eichhorn, Gisela Heckler, Dr. Gunnar Schroeder, and Wiltrud Stiefenhofer from Europe. Furthermore, we would like to thank Professors John B. Donaldson, Bruce C. N. Greenwald, Jonathan L. Gross, Morris B. Holbrook, John R. Kender, and Bernd Schmitt; and Nicholaiv Villalobos, all from Columbia University, New York; as well as Dr. Pearl Chang, Chris Gee, Mike Hale, Joshua Jayasingh, MD, David Kahr, Mike Lee, R. Richard Royce, Elaine Schattner, MD, Raheel Siddiqui, Robert Sullivan, MD, Nancy Veit, and Ana M. Kreyszig, JD, all from New York City. We would also like to gratefully acknowledge the use of facilities at Carleton University, Ottawa, and Columbia University, New York.

Furthermore we wish to thank John Wiley and Sons, in particular Publisher Laurie Rosatone, Editor Shannon Corliss, Production Editor Barbara Russiello, Media Editor Melissa Edwards, Text and Cover Designer Madelyn Lesure, and Photo Editor Sheena Goldstein for their great care and dedication in preparing this edition. In the same vein, we would also like to thank Beatrice Ruberto, copy editor and proofreader, WordCo, for the Index, and Joyce Franzen of PreMedia and those of PreMedia Global who typeset this edition.

Suggestions of many readers worldwide were evaluated in preparing this edition. Further comments and suggestions for improving the book will be gratefully received.

KREYSZIG

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xv

C O N T E N T S

P A R T A Ordinary Differential Equations (ODEs) 1

CHAPTER 1 First-Order ODEs 2 1.1 Basic Concepts. Modeling 2 1.2 Geometric Meaning of y  ƒ(x, y). Direction Fields, Euler’s Method 9 1.3 Separable ODEs. Modeling 12 1.4 Exact ODEs. Integrating Factors 20 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 27 1.6 Orthogonal Trajectories. Optional 36 1.7 Existence and Uniqueness of Solutions for Initial Value Problems 38 Chapter 1 Review Questions and Problems 43 Summary of Chapter 1 44

CHAPTER 2 Second-Order Linear ODEs 46 2.1 Homogeneous Linear ODEs of Second Order 46 2.2 Homogeneous Linear ODEs with Constant Coefficients 53 2.3 Differential Operators. Optional 60 2.4 Modeling of Free Oscillations of a Mass–Spring System 62 2.5 Euler–Cauchy Equations 71 2.6 Existence and Uniqueness of Solutions. Wronskian 74 2.7 Nonhomogeneous ODEs 79 2.8 Modeling: Forced Oscillations. Resonance 85 2.9 Modeling: Electric Circuits 93 2.10 Solution by Variation of Parameters 99 Chapter 2 Review Questions and Problems 102 Summary of Chapter 2 103

CHAPTER 3 Higher Order Linear ODEs 105 3.1 Homogeneous Linear ODEs 105 3.2 Homogeneous Linear ODEs with Constant Coefficients 111 3.3 Nonhomogeneous Linear ODEs 116 Chapter 3 Review Questions and Problems 122 Summary of Chapter 3 123

CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods 124 4.0 For Reference: Basics of Matrices and Vectors 124 4.1 Systems of ODEs as Models in Engineering Applications 130 4.2 Basic Theory of Systems of ODEs. Wronskian 137 4.3 Constant-Coefficient Systems. Phase Plane Method 140 4.4 Criteria for Critical Points. Stability 148 4.5 Qualitative Methods for Nonlinear Systems 152 4.6 Nonhomogeneous Linear Systems of ODEs 160 Chapter 4 Review Questions and Problems 164 Summary of Chapter 4 165

CHAPTER 5 Series Solutions of ODEs. Special Functions 167 5.1 Power Series Method 167 5.2 Legendre’s Equation. Legendre Polynomials Pn(x) 175

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5.3 Extended Power Series Method: Frobenius Method 180 5.4 Bessel’s Equation. Bessel Functions J(x) 187 5.5 Bessel Functions of the Y(x). General Solution 196 Chapter 5 Review Questions and Problems 200 Summary of Chapter 5 201

CHAPTER 6 Laplace Transforms 203 6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting) 204 6.2 Transforms of Derivatives and Integrals. ODEs 211 6.3 Unit Step Function (Heaviside Function).

Second Shifting Theorem (t-Shifting) 217 6.4 Short Impulses. Dirac’s Delta Function. Partial Fractions 225 6.5 Convolution. Integral Equations 232 6.6 Differentiation and Integration of Transforms.

ODEs with Variable Coefficients 238 6.7 Systems of ODEs 242 6.8 Laplace Transform: General Formulas 248 6.9 Table of Laplace Transforms 249 Chapter 6 Review Questions and Problems 251 Summary of Chapter 6 253

P A R T B Linear Algebra. Vector Calculus 255

CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems 256

7.1 Matrices, Vectors: Addition and Scalar Multiplication 257 7.2 Matrix Multiplication 263 7.3 Linear Systems of Equations. Gauss Elimination 272 7.4 Linear Independence. Rank of a Matrix. Vector Space 282 7.5 Solutions of Linear Systems: Existence, Uniqueness 288 7.6 For Reference: Second- and Third-Order Determinants 291 7.7 Determinants. Cramer’s Rule 293 7.8 Inverse of a Matrix. Gauss–Jordan Elimination 301 7.9 Vector Spaces, Inner Product Spaces. Linear Transformations. Optional 309 Chapter 7 Review Questions and Problems 318 Summary of Chapter 7 320

CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems 322 8.1 The Matrix Eigenvalue Problem.

Determining Eigenvalues and Eigenvectors 323 8.2 Some Applications of Eigenvalue Problems 329 8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices 334 8.4 Eigenbases. Diagonalization. Quadratic Forms 339 8.5 Complex Matrices and Forms. Optional 346 Chapter 8 Review Questions and Problems 352 Summary of Chapter 8 353

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CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl 354 9.1 Vectors in 2-Space and 3-Space 354 9.2 Inner Product (Dot Product) 361 9.3 Vector Product (Cross Product) 368 9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives 375 9.5 Curves. Arc Length. Curvature. Torsion 381 9.6 Calculus Review: Functions of Several Variables. Optional 392 9.7 Gradient of a Scalar Field. Directional Derivative 395 9.8 Divergence of a Vector Field 402 9.9 Curl of a Vector Field 406 Chapter 9 Review Questions and Problems 409 Summary of Chapter 9 410

CHAPTER 10 Vector Integral Calculus. Integral Theorems 413 10.1 Line Integrals 413 10.2 Path Independence of Line Integrals 419 10.3 Calculus Review: Double Integrals. Optional 426 10.4 Green’s Theorem in the Plane 433 10.5 Surfaces for Surface Integrals 439 10.6 Surface Integrals 443 10.7 Triple Integrals. Divergence Theorem of Gauss 452 10.8 Further Applications of the Divergence Theorem 458 10.9 Stokes’s Theorem 463 Chapter 10 Review Questions and Problems 469 Summary of Chapter 10 470

P A R T C Fourier Analysis. Partial Differential Equations (PDEs) 473

CHAPTER 11 Fourier Analysis 474 11.1 Fourier Series 474 11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions 483 11.3 Forced Oscillations 492 11.4 Approximation by Trigonometric Polynomials 495 11.5 Sturm–Liouville Problems. Orthogonal Functions 498 11.6 Orthogonal Series. Generalized Fourier Series 504 11.7 Fourier Integral 510 11.8 Fourier Cosine and Sine Transforms 518 11.9 Fourier Transform. Discrete and Fast Fourier Transforms 522 11.10 Tables of Transforms 534 Chapter 11 Review Questions and Problems 537 Summary of Chapter 11 538

CHAPTER 12 Partial Differential Equations (PDEs) 540 12.1 Basic Concepts of PDEs 540 12.2 Modeling: Vibrating String, Wave Equation 543 12.3 Solution by Separating Variables. Use of Fourier Series 545 12.4 D’Alembert’s Solution of the Wave Equation. Characteristics 553 12.5 Modeling: Heat Flow from a Body in Space. Heat Equation 557

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12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem 558

12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms 568

12.8 Modeling: Membrane, Two-Dimensional Wave Equation 575 12.9 Rectangular Membrane. Double Fourier Series 577 12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series 585 12.11 Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential 593 12.12 Solution of PDEs by Laplace Transforms 600 Chapter 12 Review Questions and Problems 603 Summary of Chapter 12 604

P A R T D Complex Analysis 607

CHAPTER 13 Complex Numbers and Functions. Complex Differentiation 608

13.1 Complex Numbers and Their Geometric Representation 608 13.2 Polar Form of Complex Numbers. Powers and Roots 613 13.3 Derivative. Analytic Function 619 13.4 Cauchy–Riemann Equations. Laplace’s Equation 625 13.5 Exponential Function 630 13.6 Trigonometric and Hyperbolic Functions. Euler’s Formula 633 13.7 Logarithm. General Power. Principal Value 636 Chapter 13 Review Questions and Problems 641 Summary of Chapter 13 641

CHAPTER 14 Complex Integration 643 14.1 Line Integral in the Complex Plane 643 14.2 Cauchy’s Integral Theorem 652 14.3 Cauchy’s Integral Formula 660 14.4 Derivatives of Analytic Functions 664 Chapter 14 Review Questions and Problems 668 Summary of Chapter 14 669

CHAPTER 15 Power Series, Taylor Series 671 15.1 Sequences, Series, Convergence Tests 671 15.2 Power Series 680 15.3 Functions Given by Power Series 685 15.4 Taylor and Maclaurin Series 690 15.5 Uniform Convergence. Optional 698 Chapter 15 Review Questions and Problems 706 Summary of Chapter 15 706

CHAPTER 16 Laurent Series. Residue Integration 708 16.1 Laurent Series 708 16.2 Singularities and Zeros. Infinity 715 16.3 Residue Integration Method 719 16.4 Residue Integration of Real Integrals 725 Chapter 16 Review Questions and Problems 733 Summary of Chapter 16 734

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CHAPTER 17 Conformal Mapping 736 17.1 Geometry of Analytic Functions: Conformal Mapping 737 17.2 Linear Fractional Transformations (Möbius Transformations) 742 17.3 Special Linear Fractional Transformations 746 17.4 Conformal Mapping by Other Functions 750 17.5 Riemann Surfaces. Optional 754 Chapter 17 Review Questions and Problems 756 Summary of Chapter 17 757

CHAPTER 18 Complex Analysis and Potential Theory 758 18.1 Electrostatic Fields 759 18.2 Use of Conformal Mapping. Modeling 763 18.3 Heat Problems 767 18.4 Fluid Flow 771 18.5 Poisson’s Integral Formula for Potentials 777 18.6 General Properties of Harmonic Functions.

Uniqueness Theorem for the Dirichlet Problem 781 Chapter 18 Review Questions and Problems 785 Summary of Chapter 18 786

P A R T E Numeric Analysis 787 Software 788

CHAPTER 19 Numerics in General 790 19.1 Introduction 790 19.2 Solution of Equations by Iteration 798 19.3 Interpolation 808 19.4 Spline Interpolation 820 19.5 Numeric Integration and Differentiation 827 Chapter 19 Review Questions and Problems 841 Summary of Chapter 19 842

CHAPTER 20 Numeric Linear Algebra 844 20.1 Linear Systems: Gauss Elimination 844 20.2 Linear Systems: LU-Factorization, Matrix Inversion 852 20.3 Linear Systems: Solution by Iteration 858 20.4 Linear Systems: Ill-Conditioning, Norms 864 20.5 Least Squares Method 872 20.6 Matrix Eigenvalue Problems: Introduction 876 20.7 Inclusion of Matrix Eigenvalues 879 20.8 Power Method for Eigenvalues 885 20.9 Tridiagonalization and QR-Factorization 888 Chapter 20 Review Questions and Problems 896 Summary of Chapter 20 898

CHAPTER 21 Numerics for ODEs and PDEs 900 21.1 Methods for First-Order ODEs 901 21.2 Multistep Methods 911 21.3 Methods for Systems and Higher Order ODEs 915

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21.4 Methods for Elliptic PDEs 922 21.5 Neumann and Mixed Problems. Irregular Boundary 931 21.6 Methods for Parabolic PDEs 936 21.7 Method for Hyperbolic PDEs 942 Chapter 21 Review Questions and Problems 945 Summary of Chapter 21 946

P A R T F Optimization, Graphs 949

CHAPTER 22 Unconstrained Optimization. Linear Programming 950 22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent 951 22.2 Linear Programming 954 22.3 Simplex Method 958 22.4 Simplex Method: Difficulties 962 Chapter 22 Review Questions and Problems 968 Summary of Chapter 22 969

CHAPTER 23 Graphs. Combinatorial Optimization 970 23.1 Graphs and Digraphs 970 23.2 Shortest Path Problems. Complexity 975 23.3 Bellman’s Principle. Dijkstra’s Algorithm 980 23.4 Shortest Spanning Trees: Greedy Algorithm 984 23.5 Shortest Spanning Trees: Prim’s Algorithm 988 23.6 Flows in Networks 991 23.7 Maximum Flow: Ford–Fulkerson Algorithm 998 23.8 Bipartite Graphs. Assignment Problems 1001 Chapter 23 Review Questions and Problems 1006 Summary of Chapter 23 1007

P A R T G Probability, Statistics 1009 Software 1009

CHAPTER 24 Data Analysis. Probability Theory 1011 24.1 Data Representation. Average. Spread 1011 24.2 Experiments, Outcomes, Events 1015 24.3 Probability 1018 24.4 Permutations and Combinations 1024 24.5 Random Variables. Probability Distributions 1029 24.6 Mean and Variance of a Distribution 1035 24.7 Binomial, Poisson, and Hypergeometric Distributions 1039 24.8 Normal Distribution 1045 24.9 Distributions of Several Random Variables 1051 Chapter 24 Review Questions and Problems 1060 Summary of Chapter 24 1060

CHAPTER 25 Mathematical Statistics 1063 25.1 Introduction. Random Sampling 1063 25.2 Point Estimation of Parameters 1065 25.3 Confidence Intervals 1068

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25.4 Testing Hypotheses. Decisions 1077 25.5 Quality Control 1087 25.6 Acceptance Sampling 1092 25.7 Goodness of Fit. 2-Test 1096 25.8 Nonparametric Tests 1100 25.9 Regression. Fitting Straight Lines. Correlation 1103 Chapter 25 Review Questions and Problems 1111 Summary of Chapter 25 1112

APPENDIX 1 References A1

APPENDIX 2 Answers to Odd-Numbered Problems A4

APPENDIX 3 Auxiliary Material A63 A3.1 Formulas for Special Functions A63 A3.2 Partial Derivatives A69 A3.3 Sequences and Series A72 A3.4 Grad, Div, Curl, 2 in Curvilinear Coordinates A74