Math refresher for scientists and engineers

MATH REFRESHER

FOR SCIENTISTS

AND ENGINEERS

Third Edition

JOHN R. FANCHI

A JOHN WILEY & SONS, INC., PUBLICATION

MATH REFRESHER

FOR SCIENTISTS

AND ENGINEERS

MATH REFRESHER

FOR SCIENTISTS

AND ENGINEERS

Third Edition

JOHN R. FANCHI

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright # 2006 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data.

Fanchi, John R.

Math refresher for scientists and engineers/John R. Fanchi.—3rd ed.

p. cm.

Includes bibliographical references and index.

ISBN-13: 978-0-471-75715-3

ISBN-10: 0-471-75715-2

1. Mathematics. I. Title

QA37.2.F35 2006

5120

.1—dc22 2005056262

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

permission.

CONTENTS

PREFACE xi

1 ALGEBRA 1

1.1 Algebraic Axioms / 1

1.2 Algebraic Operations / 6

1.3 Exponents and Roots / 7

1.4 Quadratic Equations / 9

1.5 Logarithms / 10

1.6 Factorials / 12

1.7 Complex Numbers / 13

1.8 Polynomials and Partial Fractions / 17

2 GEOMETRY, TRIGONOMETRY,

AND HYPERBOLIC FUNCTIONS 21

2.1 Geometry / 21

2.2 Trigonometry / 26

2.3 Common Coordinate Systems / 32

2.4 Euler’s Equation and Hyperbolic Functions / 34

2.5 Series Representations / 37

3 ANALYTIC GEOMETRY 41

3.1 Line / 41

vii

3.2 Conic Sections / 44

3.3 Polar Form of Complex Numbers / 48

4 LINEAR ALGEBRA I 51

4.1 Rotation of Axes / 51

4.2 Matrices / 53

4.3 Determinants / 61

5 LINEAR ALGEBRA II 65

5.1 Vectors / 65

5.2 Vector Spaces / 69

5.3 Eigenvalues and Eigenvectors / 71

5.4 Matrix Diagonalization / 74

6 DIFFERENTIAL CALCULUS 79

6.1 Limits / 79

6.2 Derivatives / 82

6.3 Finite Difference Concept / 87

7 PARTIAL DERIVATIVES 93

7.1 Partial Differentiation / 94

7.2 Vector Analysis / 96

7.3 Analyticity and the Cauchy –Riemann Equations / 103

8 INTEGRAL CALCULUS 107

8.1 Indefinite Integrals / 107

8.2 Definite Integrals / 109

8.3 Solving Integrals / 112

8.4 Numerical Integration / 114

9 SPECIAL INTEGRALS 117

9.1 Line Integral / 117

9.2 Double Integral / 119

9.3 Fourier Analysis / 121

9.4 Fourier Integral and Fourier Transform / 124

9.5 Time Series and Z Transform / 127

9.6 Laplace Transform / 130

10 ORDINARY DIFFERENTIAL EQUATIONS 133

10.1 First-Order ODE / 133

viii CONTENTS

10.2 Higher-Order ODE / 141

10.3 Stability Analysis / 142

10.4 Introduction to Nonlinear Dynamics and Chaos / 145

11 ODE SOLUTION TECHNIQUES 151

11.1 Higher-Order ODE with Constant Coefficients / 151

11.2 Variation of Parameters / 155

11.3 Cauchy Equation / 157

11.4 Series Methods / 158

11.5 Laplace Transform Method / 164

12 PARTIAL DIFFERENTIAL EQUATIONS 167

12.1 Boundary Conditions / 168

12.2 PDE Classification Scheme / 168

12.3 Analytical Solution Techniques / 169

12.4 Numerical Solution Methods / 175

13 INTEGRAL EQUATIONS 181

13.1 Classification / 181

13.2 Integral Equation Representation

of a Second-Order ODE / 182

13.3 Solving Integral Equations: Neumann Series Method / 185

13.4 Solving Integral Equations with Separable Kernels / 187

13.5 Solving Integral Equations with Laplace Transforms / 188

14 CALCULUS OF VARIATIONS 191

14.1 Calculus of Variations with One Dependent Variable / 191

14.2 The Beltrami Identity and the Brachistochrone Problem / 195

14.3 Calculus of Variations with Several Dependent Variables / 198

14.4 Calculus of Variations with Constraints / 200

15 TENSOR ANALYSIS 203

15.1 Contravariant and Covariant Vectors / 203

15.2 Tensors / 207

15.3 The Metric Tensor / 210

15.4 Tensor Properties / 213

16 PROBABILITY 219

16.1 Set Theory / 219

16.2 Probability Defined / 222

CONTENTS ix

16.3 Properties of Probability / 222

16.4 Probability Distribution Defined / 227

17 PROBABILITY DISTRIBUTIONS 229

17.1 Joint Probability Distribution / 229

17.2 Expectation Values and Moments / 233

17.3 Multivariate Distributions / 235

17.4 Example Probability Distributions / 240

18 STATISTICS 245

18.1 Probability and Frequency / 245

18.2 Ungrouped Data / 247

18.3 Grouped Data / 249

18.4 Statistical Coefficients / 250

18.5 Curve Fitting, Regression, and Correlation / 252

19 SOLUTIONS TO EXERCISES 257

S1 Algebra / 257

S2 Geometry, Trigonometry, and Hyperbolic Functions / 262

S3 Analytic Geometry / 267

S4 Linear Algebra I / 269

S5 Linear Algebra II / 276

S6 Differential Calculus / 280

S7 Partial Derivatives / 286

S8 Integral Calculus / 290

S9 Special Integrals / 293

S10 Ordinary Differential Equations / 299

S11 ODE Solution Techniques / 306

S12 Partial Differential Equations / 312

S13 Integral Equations / 316

S14 Calculus of Variations / 323

S15 Tensor Analysis / 327

S16 Probability / 331

S17 Probability Distributions / 333

S18 Statistics / 337

REFERENCES 339

INDEX 343

x CONTENTS

PREFACE

Math Refresher for Scientists and Engineers, Third Edition is intended for

people with technical backgrounds who would like to refresh their math

skills. This book is unique because it contains in one source an overview of the

essential elements of a wide range of mathematical topics that are normally found

in separate texts. The first edition began with relatively simple concepts in

college algebra and trigonometry and then proceeded to more advanced concepts

ranging from calculus to linear algebra (including matrices) and differential

equations. Numerical methods were interspersed throughout the presentation.

The second edition added chapters that discussed probability and statistics. In this

third edition, three new chapters with exercises and solutions have been added.

The new material includes chapters on integral equations, the calculus of variations,

and tensor analysis. Furthermore, the discussion of integral transforms has been

expanded, a section on partial fractions has been added, and several new exercises

have been included.

Math Refresher for Scientists and Engineers, Third Edition is designed for the

adult learner and is suitable for reference, self-review, adult education, and

college review. It is especially useful for the professional who wants to understand

the latest technology, the engineer who is preparing to take a professional engineer￾ing exam, or students who wish to refresh their math background. The focus of the

book is on modern, practical applications and exercises rather than theory.

Chapters are organized to include a review of important principles and methods.

Interwoven with the review are examples, exercises, and applications. Examples

are intended to clarify concepts. Exercises are designed to make you an active

participant in the review process. Keeping in mind that your time is valuable, I

have restricted the number of exercises to a number that should complement and

xi

supplement the text while minimizing nonessential repetition. Solutions to exercises

are separated from the exercises to allow for a self-paced review.

Applications provide an integration of concepts and methods from one or more

sections. In many cases, they introduce you to modern topics. Applications may

include material that has not yet been covered in the book, but should be familiar

if you are using the book as a “refresher course.” You may wish to return to the

applications after a first reading of the rest of the text.

I developed much of the material in this book as course notes for continuing edu￾cation courses in Denver and Houston. I would like to thank Kathy Fanchi, Cindee

Calton, Chris Fanchi, and Stephanie Potter for their assistance in the development of

this material. Any written comments or suggestions for improving the material are

welcome.

JOHN R. FANCHI

XII PREFACE