Advanced engineering mathematics 10th edition, Study notes of Engineering Mathematics

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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 yd^2  4046.8564 m^2 1 mi^2  640 acres  2.5899881 km^2

1 fluid ounce  1/128 U.S. gallon  231/128 in.^3  29.573730 cm^3

1 U.S. gallon  4 quarts (liq)  8 pints (liq)  128 fl oz  3785.4118 cm^3

1 British Imperial and Canadian gallon  1.200949 U.S. gallons  4546.087 cm^3

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.

ADVANCED

ENGINEERING

MATHEMATICS

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.

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ISBN 978-0-470-45836-

Printed in the United States of America

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P R E F A C E

See also http://www.wiley.com/college/kreyszig

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– B. Linear Algebra. Vector Calculus. See Chapters 7– C. Fourier Analysis. Partial Differential Equations (PDEs). See Chapters 11 and 12 D. Complex Analysis in Chapters 13– E. Numeric Analysis in Chapters 19– 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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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

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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 , 10 th Edition and Mathematica Computer Guide , 10 th 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)

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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• P A R T A Ordinary Differential Equations (ODEs) C O N T E N T S - CHAPTER 1 First-Order ODEs - 1.1 Basic Concepts. Modeling - 1.2 Geometric Meaning of y   ƒ( x , y ). Direction Fields, Euler’s Method - 1.3 Separable ODEs. Modeling - 1.4 Exact ODEs. Integrating Factors - 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics - 1.6 Orthogonal Trajectories. Optional - 1.7 Existence and Uniqueness of Solutions for Initial Value Problems - Chapter 1 Review Questions and Problems - Summary of Chapter - CHAPTER 2 Second-Order Linear ODEs - 2.1 Homogeneous Linear ODEs of Second Order - 2.2 Homogeneous Linear ODEs with Constant Coefficients - 2.3 Differential Operators. Optional - 2.4 Modeling of Free Oscillations of a Mass–Spring System - 2.5 Euler–Cauchy Equations - 2.6 Existence and Uniqueness of Solutions. Wronskian - 2.7 Nonhomogeneous ODEs - 2.8 Modeling: Forced Oscillations. Resonance - 2.9 Modeling: Electric Circuits - 2.10 Solution by Variation of Parameters - Chapter 2 Review Questions and Problems - Summary of Chapter - CHAPTER 3 Higher Order Linear ODEs - 3.1 Homogeneous Linear ODEs - 3.2 Homogeneous Linear ODEs with Constant Coefficients - 3.3 Nonhomogeneous Linear ODEs - Chapter 3 Review Questions and Problems - Summary of Chapter - CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods - 4.0 For Reference: Basics of Matrices and Vectors - 4.1 Systems of ODEs as Models in Engineering Applications
  • 4.2 Basic Theory of Systems of ODEs. Wronskian
  • 4.3 Constant-Coefficient Systems. Phase Plane Method
  • 4.4 Criteria for Critical Points. Stability
  • 4.5 Qualitative Methods for Nonlinear Systems
  • 4.6 Nonhomogeneous Linear Systems of ODEs
    • Chapter 4 Review Questions and Problems
    • Summary of Chapter
    • CHAPTER 5 Series Solutions of ODEs. Special Functions
    • 5.1 Power Series Method
    • 5.2 Legendre’s Equation. Legendre Polynomials Pn ( x )
    • 5.3 Extended Power Series Method: Frobenius Method
    • 5.4 Bessel’s Equation. Bessel Functions J ( x )
  • 5.5 Bessel Functions of the Y ( x ). General Solution
    • Chapter 5 Review Questions and Problems
    • Summary of Chapter
    • CHAPTER 6 Laplace Transforms
    • 6.1 Laplace Transform. Linearity. First Shifting Theorem ( s -Shifting)
  • 6.2 Transforms of Derivatives and Integrals. ODEs - Second Shifting Theorem ( t -Shifting) 6.3 Unit Step Function (Heaviside Function).
  • 6.4 Short Impulses. Dirac’s Delta Function. Partial Fractions
  • 6.5 Convolution. Integral Equations - ODEs with Variable Coefficients 6.6 Differentiation and Integration of Transforms.
  • 6.7 Systems of ODEs
  • 6.8 Laplace Transform: General Formulas
  • 6.9 Table of Laplace Transforms
  • Chapter 6 Review Questions and Problems
  • Summary of Chapter
• P A R T B Linear Algebra. Vector Calculus - Linear Systems CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. - 7.1 Matrices, Vectors: Addition and Scalar Multiplication
  • 7.2 Matrix Multiplication
  • 7.3 Linear Systems of Equations. Gauss Elimination
  • 7.4 Linear Independence. Rank of a Matrix. Vector Space
  • 7.5 Solutions of Linear Systems: Existence, Uniqueness
  • 7.6 For Reference: Second- and Third-Order Determinants
  • 7.7 Determinants. Cramer’s Rule
  • 7.8 Inverse of a Matrix. Gauss–Jordan Elimination
  • 7.9 Vector Spaces, Inner Product Spaces. Linear Transformations. Optional
  • Chapter 7 Review Questions and Problems
  • Summary of Chapter
  • CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems - Determining Eigenvalues and Eigenvectors 8.1 The Matrix Eigenvalue Problem.
  • 8.2 Some Applications of Eigenvalue Problems
    • 8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices
    • 8.4 Eigenbases. Diagonalization. Quadratic Forms
    • 8.5 Complex Matrices and Forms. Optional
    • Chapter 8 Review Questions and Problems
    • Summary of Chapter