Advanced Engineering Mathematics [3 ed.] 0471507296, 9780471507291

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Advanced Engineering Mattiematies THIRD EDITION

By Ervtrin Kreyszig

WILEY INTERNATIONAL EDITION

Advanced Engineering Mathematics

Third Edition

Advanced Engineering Mathematics Erivin Kreyszig Professor of Mathematics

Ohio State University Columbus, Ohio

John Wiley and Sons, Inc. New York London Sydney •

•

•

Toronto

Copyright

©

1962, 1967, 1972, by John Wiley

All rights reserved.

No

part of this

&

Sons, Inc.

Published simultaneously in Canada.

book may be reproduced by any means, nor

transmitted, nor translated into a machine language with-

out the written permission of the publisher. Library of Congress Catalogue Card

Number: 71-172951

Printed in the United States of America. 10 9 8 7

Preface

This book

Purpose of the book.

is

intended to introduce students of engineering

and physics to those areas of mathematics which, from a

modern point of view,

seem to be the most important in connection with practical problems.

Topics

New

ideas of

are chosen according to the frequency of occurrence

modern mathematical

in

apphcations.

training, as expressed in recent

education, were taken into account.

The book should

symposia on engineering suit

those institutions that

have offered extended mathematical training for a long time as well as those that intend to follow the general trend of broadening the program of instruction in mathematics.

A

course in elementary calculus

The material included

in the

is

the sole prerequisite.

book has formed

the basis of various lecture courses

given to undergraduate and graduate students of engineering, physics, and mathe-

matics in this country, in Canada, and in Europe.

Changes in the third edition. The present edition differs essentially from the and second editions as follows. The problems have been changed. A new chapter on numerical methods for engineers (Chap. 18) has been added. It is suitable for a course meeting three hours a week. The material on linear algebra and analysis (vectors and matrices in Chaps. 5 to 8) has been revised and amplified. The new presentation emphasizes algebraic as well as geometric aspects. Chapter 5 also contains an elementary introduction to vector spaces, inner product spaces, and some other, concepts that are important in functional analysis and its applications. These major changes are suggested and motivated by the development of

first

engineering mathematics during the past decade, especially the widespread application of automatic computers to the solution of engineering problems and the increasing importance of methods of linear algebra.

Other improvements include a more compact consideration of Bessel functions and Legendre polynomials. In particular, the orthogonality of these functions is now discussed much earlier (in Chap. 3) in connection with the Sturm-Liouville

VI

Preface

theory.

Appendix

extended.

A

1,

containing references to the literature, has been updated and

collection of frequently used formulas for elementary

and higher

functions has been added as Appendix 3; this replaces and extends Chap.

The numerical

in

the

have been placed together in Appendix 4 in order to facilitate the use of the book. Content and arrangement. The arrangement of the subject matter in major parts can be seen from the diagram. previous editions.

tables

Ordinary

Linear

Linear

differential

algebra:

algebra:

equations Chaps. 1,2

Vectors

Matrices

Chap. 5

Chap. 6

Power series

Laplace trans-

method

formation

Chap. 3

Chap. 4

Vector calculus

Chap. 7

Integral

theorems Chap. 8

Fourier

Preface considerable complexity.

Answers

to

odd-numbered problems

vii

are included at the

end o( the book, starting on page 794. Tables of functions are included in Appendix Suggestions for a sequence of courses.

on page 833.

4.

The material may be taken

sequence

equations (Chaps. 1^).

Firs! semester.

Ordinary

Second semester.

Linear algebra and vector analysis (Chaps. 5-8).

difTerential

Fourier series and partial differential equations (Chaps.

Third semester.

Numerical methods (Chap. Fourth semester.

A

in

for four consecutive semester courses, meeting 3-5 hours a

and is then suitable week as follows:

Complex

analysis (Chaps.

9,

10),

18). 1

1-17).

course on engineering statistics (3-5 hours a week;

Chap.

19)

may

be taught

during any of those four semesters or afterwards. Independent one-semester courses.

pendent one-semester courses meeting

The book 3

is

also suitable for various inde-

hours a week; for example:

Introduction to ordinary differential equations (Chaps.

Laplace transformation (Chap.

Vector algebra and calculus (Chaps.

5, 7).

Matrices and systems of linear equations (Chap.

8).

Fourier series and partial differential equations (Chaps.

Complex

analysis (Chaps.

Numerical analysis (Chap. Shorter courses. at the beginning of

1

1, 2).

4).

9, 10).

1-16). 18).

Sections that

may

be omitted

in a shorter

course are indicated

each chapter.

Which topics should be contained in a book how and should these topics be arranged and presented? of the present type, these basic questions we may take a look at the historical To find some answer to engineering mathematics. This development shows the following development of two interesting features: Principles for selection of topics.

1 it

.

is

Mathematics has become more and more important easy to conjecture that

this

to engineering science,

trend will also continue

in

the future.

and

In fact,

modern engineering are so complex that most of them cannot be solved solely by using physical intuition and past experience. The empirical approach has been successful in the solution of many problems in the past, but fails as soon as high speeds, large forces, high temperatures or other abnormal conditions are involved, and the situation becomes still more critical by the fact that various modern materials Experimental work has (plastics, alloys, etc.) have unusual physical properties. become complicated, time-consuming, and expensive. Here mathematics offers aid in planning constructions and experiments, in evaluating experimental data, and in reducing the work and cost offinding solutions.

prohlenis

in

Preface

via 2.

Mathematical

niethocls which

were developed for purely theoretical reasons

suddenly became of great importance in engineering mathematics. Examples are the theory of matrices, conformal mapping, and the theory of differential equations having periodic solutions.

What

are the reflections of this situation on the teaching of engineering mathe-

will need more and more mathematics, should we try more and more topics in our courses, devoting less and less time to each topic? Or should we concentrate on a few carefully selected basic ideas of general

matics? Since the engineer to include

which are especially suitable for teaching the student mathe-

practical importance

matical thinking and developing his Sixty or eighty years ago

own

creative ability?

no one was able to predict that conformal mapping

or matrices would ever be of importance in the mathematical part of engineering

work.

Similarly,

it

is

difficult to

new mathematical

conjecture which

have applications to engineering twenty or

thirty years

from now.

theories will

But no matter

what happens in that respect, if a student has a good training in the fundamentals of mathematics he will meet the future needs because he will be able to get acquainted with new methods by his own further study. It follows that the most important objective and purpose in engineering mathematics seems to be that the student becomes familiar with mathematical thinking. He should learn to recognize the guiding principles and ideas "behind the scenes," which are more important than formal manipulations. He should get the impression that mathematics is not a collection of tricks and recipes but a systematic science of practical importance, resting on a relatively small number of basic concepts and involving powerful unifying methods. He should soon convince himself of the necessity for applying mathematical procedures to engineering problems, and he will find that the theory and its applications are related to each other like a tree

and

its fruits.

The student

1.

see that the application of

will

problem consists

mathematics to an engineering

essentially of three phases:

Translation of the given physical information into a mathematical form.

way we obtain a mathematical model of the physical differential equation,

situation.

This model

In this

may

be a

a system of linear equations, or some other mathematical

expression. 2. Treatment of the model by mathematical methods. This will lead of the given problem in mathematical form. 3. Interpretation of the mathematical result in physical terms.

All three steps is

such that

it

to the solution

seem to be of equal importance, and the presentation

will

help the student to develop

skill in

carrying out

In this connection, preference has been given to applications

in this

all

book

these steps.

which are of a general

nature. In some considerations it will be unavoidable to rely on results whose proofs are beyond the level of a book of the present type. In any case these points are marked distinctly, because hiding difficulties and oversimplifying matters is no real help to the student in preparing him for his professional work.

ix

Preface

These are some of the guiding principles material in this book.

1

made

I

used

in selecting

and presenting the and present

the choice with greatest care, using past

teaching and research experience and resisting the temptation to include "everything which

is

Particular

important"

eft'orts

increases

level

ditlicull theoretical

engineering mathematics. in

presenting the topics as simply, clearly, and

choice of the notations.

this refers also to the

accurately as possible;

chapter the

in

have been made

gradually,

In each

avoiding jumps and accumulations of

considerations.

am indebted to many of my former teachers, colleagues, Acknowledgement. and students for advice and help in preparing this book. Various parts of the manuscript were distributed to my classes in mimeographed form and returned to me, together with suggestions for improvement. Discussions with various engineers and mathematicians were of great help to me; I want to mention in particular Professors S. Bergman, P. L. Chambre, A. Cronheim, J. W. Dettman, R. G. Helsel, 1

Mann,!. Marx,W. D. Munroe,H.-W. Pu,T. Rado(t), Todd, H. J. Weiss, A. Wilansky, all in this country, Professor H. S. M. Coxeter of Toronto, and Professors B. Baule, H. Behnke, H. Florian, H. Graf, F. Hohenberg, K. Klotter, M. Pinl, F. Reutter, C. Schmieden,

E.C. Klipple.H. Kuhn.H. H. A. Smith,

J.

B.

P. Spencer, J.

H. linger, A. Walther(t), H. Wielandt, inadequate acknowledgment of Finally,

1

want

to thank

my

all in

Europe.

I

can offer here only an

appreciation.

John Wiley and Sons

their great care in preparing this edition

for their effective cooperation

and

of the book.

Suggestions of many readers were evaluated in preparing the present edition. Any further comment and suggestion for improvement of the book will be gratefully received.

Erwin Kreyszig

Coil ten ts

Chapter

1

Ordinary Differential Equations of the First Order 1.1

Basic Concepts and Ideas,

1.2

Geometrical Considerations.

1.3

Separable Equations, 10

1.4

Equations Reducible to Separable Form, 18 Exact Differential Equations, 19

1.5

1

1

Isoclines, 8

1.6

Integrating Factors, 22

1.7

Linear First-Order Differential Equations, 24

1.8

Variation of Parameters, 28

1.9

Electric Circuits, 30

Orthogonal Trajectories, 35 Method, 39 1.12 Existence and Uniqueness of Solutions, 42 1.10 Families of Curves. 1.11

Picard's Iteration

Chapter 2

47

Ordinary Linear Differential Equations 2.1

2.2

Homogeneous Linear Equations of the Second Order, 48 Homogeneous Second-Order Equations with Constant Coefficients, 51

2.3

2.4

General Solution. Fundamental System, 53 the Characteristic Equation.

Complex Roots of

Initia'l

Value Problem, 56 2.5

Double Root of the Characteristic Equation, 60

2.6

Free Oscillations, 62

Cauchy Equation, 69 Existence and Uniqueness of Solutions, 71 Homogeneous Linear Equations of Arbitrary Order, 77 2.9 2.10 Homogeneous Linear Equations of Arbitrary Order 2.7

2.8

with Constant Coefficients, 80

Contents

xii

2.11

2.12

Nonhomogeneous Linear Equations, 82 A Method for Solving Nonhomogeneous Linear Equations, 84

2.13 Forced Oscillations. Resonance, 87

2.14 Electric Circuits, 93 2.15

Complex Method for Obtaining Particular Solutions, 97 Method for Solving Nonhomogeneous Equations, 99

2.16 General

Chapter 3

Power

Series Solutions of Differential Equations

3.1

The Power

3.2

Theoretical Basis of the

3.3

Legendre's Equation. Legendre Polynomials, 111

3.4

Extended Power Series Method. Indicial Equation, 114 Bessel's Equation. Bessel Functions of the First Kind, 125

3.5

3.6 3.7

3.8

3.9

Series

Method, 102 Power

Series

102

Method, 106

Bessel Functions of the Second Kind, 130 Orthogonal Sets of Functions, 134 Sturm-Liouville Problem, 138 Orthogonality of Legendre Polynomials and Bessel Functions, 142

Chapter 4

147

Laplace Transformation 4.1

Laplace Transform. Inverse Transform.

4.2

Laplace Transforms of Derivatives and Integrals, 152

4.3

Transformation of Ordinary Differential Equations, 156

4.4

Partial Fractions, 158

4.5

Examples and Applications, 162 Differentiation and Integration of Transforms, 167

4.6 4.7

Unit Step Function, 169

4.8

Shifting

4.9

Periodic Functions, 177

4.10 Table of

on the

r-axis,

Linearity, 147

173

Some Laplace Transforms,

185

Chapter 5 Linear Algebra Part

I:

Vectors

5.1

Scalars and Vectors, 189

5.2

Components of

5.3

5.4

Addition of Vectors, Multiplication by Scalars, 193 Vector Spaces. Linear Dependence and Independence, 196

5.5

Inner Product (Dot Product), 200

a Vector, 191

5.6

Inner Product Spaces, 205

5.7

Vector Product (Cross Product), 206

188

1

Contents

Terms of Components, 208

5.8

Vector Products

5.9

Scalar Triple Product.

in

xiii

Other Repeated Products, 213

Chapter 6 Linear Algebra Part

Matrices and Determinants

II:

6.1

Basic Concepts, 218

6.2

Addition of Matrices, Multiplication by Numbers, 221

6.3

6.4

Transpose of a Matrix. Special Matrices, 222 Matrix Multiplication, 224

6.5

Systems of Linear Equations.

6.6

Determinants of Second and Third Order, 238

Gauss Elimination, 233

6.7

Determinants of Arbitrary Order, 243

6.8

Rank of

6.9

Systems of Linear Equations: Existence and General Properties

a Matrix.

Row

218

Equivalence, 253

of Solutions, 256 6.10 Linear Dependence and Rank.

Singular Matrices, 258

Systems of Linear Equations: Solution by Determinants, 261 6.12 The Inverse of a Matrix, 265 6.13 Bilinear, Quadratic, Hermitian, and Skew-Hermitian Forms, 269 6.11

6.14 Eigenvalues, Eigenvectors, 273 6.15 Eigenvalues of Hermitian, Skew-Hermitian, and Unitary Matrices, 279

Chapter 7 Vector Fields

Vector Differential Calculus. 7.1

Scalar Fields and Vector Fields, 285

7.2

Vector Calculus, 288

7.3

Curves, 291

7.4

Arc Length, 294

7.5

Tangent.

7.6

Velocity and Acceleration, 299

7.7

Chain Rule and Mean Value Theorem for Functions

285

Curvature and Torsion, 296

of Several Variables, 303 Directional Derivative. Gradient of a Scalar Field, 306 Transformation of Coordinate Systems and Vector Components, 313 7.10 Divergence of a Vector Field, 316

7.8

7.9

7.1

Curl of a Vector Field, 320

Chapter 8 Line and Surface Integrals.

Integral Theorems

8.1

Line Integral, 322

8.2

Evaluation of Line Integrals, 325

8.3

Double

8.4

Transformation of Double Integrals into Line Integrals, 336

Integrals,

329

322

xiv

Contents

8.5

Surfaces, 341

8.6

Tangent Plane.

8.7

Surface Integrals, 349

Fundamental Form. Area, 343

First

Divergence Theorem of Gauss, 354 Consequences and Applications of the Divergence Theorem, 358 8.10 Stokes's Theorem, 364 8.11 Consequences and Applications of Stolces's Theorem, 367 8.12 Line Integrals Independent of Path, 369 8.8

Triple Integrals.

8.9

Chapter 9 Fourier Series and Integrals

317

Trigonometric

9.1

Periodic Functions.

9.2

Fourier Series.

9.3

Functions Having Arbitrary Period, 386 Even and Odd Functions, 389

9.4

Series,

377

Euler Formulas, 380

9.8

Half-Range Expansions, 393 Determination of Fourier Coefficients Without Integration, 396 Forced Oscillations, 402 Approximation by Trigonometric Polynomials. Square Error, 405

9.9

The Fourier

9.5

9.6 9.7

Integral,

407

Chapter 10

415

Partial Differential Equations 10.1

Basic Concepts, 415

10.2

Vibrating String.

10.3

Separation of Variables (Product Method), 419

10.4

D'AIembert's Solution of the

One-Dimensional Wave Equation, 417

Wave

Equation, 426

One-Dimensional Heat Flow, 430 10.6 Heat Flow in an Infinite Bar, 434 10.7 Vibrating Membrane. Two-Dimensional Wave Equation, 438 10.8 Rectangular Membrane, 440 10.9 Laplacian in Polar Coordinates, 447 10.10 Circular Membrane. Bessel's Equation, 449 10.11 Laplace's Equation. Potential, 454 10.12 Laplace's Equation in Spherical Coordinates. Legendre's Equation, 457

10.5

Chapter 11

461

Complex Analytic Functions 11.1

Complex Numbers. Triangle

11.2

Limit.

11.3

Cauchy-Riemann Equations. Laplace's Equation, 474

Derivative.

Inequality, 462

Analytic Function, 470

Contents

xv

Root, 478

11.4

Rational Functions.

11.5

Exponential Function, 481

11.6

Trigonometric and Hyperbolic Functions, 483

11.7

Logarithm. General Power, 487

Chapter 12

490

Conformal Mapping

12.2

Mapping, 490 Conformal Mapping, 494

12.3

Linear Fractional Transformations, 498

12.4

Special Linear Fractional Transformations, 502

12.5

Mapping by Other Elementary Functions, 506 Riemann Surfaces, 513

12.1

12.6

Chapter 13

Complex

517

Integrals

13.1

Line Integral in the Complex Plane, 517

13.2

Basic Properties of the

13.3

Cauchy's Integral

13.4 13.5 13.6

Complex Line

Integral,

522

Theorem, 524 Line Integrals by Indefinite Integration, 531 Evaluation of Cauchy's Integral Formula, 534 The Derivatives of an Analytic Function, 536

Chapter 14 Sequences and Series 14.1

540

Sequences, 540

548

14.2

Series,

14.3

Tests for Convergence and Divergence of Series, 552

14.4

Operations on Series, 558

14.5

Power

14.6

Functions Represented by Power Series, 569

Series,

562

Chapter 15

574

Taylor and Laurent Series 15.1

Taylor

15.2

Taylor Series of Elementary Functions, 579

15.3

Practical

15.4

Series,

574

Methods for Obtaining Power Uniform Convergence, 584

15.5

Laurent

15.6

Behavior of Functions

Series, 591 at Infinity,

596

Series, 581

xvi

Contents

Chapter 16 Integration by the 16.1

Method of Residues

599

Zeros and Singularities, 599

16.2

Residues, 603

16.3

The Residue Theorem, 606

16.4

Evaluation of Real Integrals, 608

Chapter 17

Complex Analytic Functions and Potential Theory 17.1

Electrostatic Fields, 616

17.2

Two-Dimensional Fluid Flow, 620 General Properties of Harmonic Functions, 626 Poisson's Integral Formula, 629

17.3

17.4

616

Chapter 18

Numerical Analysis

633

Automatic Computers, 633

18.1

Errors and Mistakes.

18.2

Solution of Equations by Iteration, 638

18.3

Finite Differences, 644

18.4

Interpolation, 648

18.5

18.7

Numerical Integration and Differentiation. 653 Numerical Methods for First-Order Differential Equations, 661 Numerical Methods for Second-Order Differential Equations, 668

18.8

Systems of Linear Equations.

18.9

Systems of Linear Equations.

18.6

18.10 Systems of Linear Equations. 18.11

Method of Least Squares,

Gauss Elimination, 672 Solution by Iteration, 675 Ill-conditioning, 679

681

18.12 Inclusion of Matrix Eigenvalues, 684

18.13 Determination of Eigenvalues by Iteration, 688

18.14 Asymptotic Expansions, 691

Chapter 19 Probability 19.1

19.2 19.3

19.4

and

699

Statistics

Nature and Purpose of Mathematical Statistics, 699 Tabular and Graphical Representation of Samples, 701

Sample Mean and Sample Variance, 707 Random Experiments, Outcomes, Events, 710

19.5

Probability, 714

19.6

Permutations and Combinations, 718

19.7

Random Variables. Mean and Variance

19.8

Discrete and Continuous Distributions, 721

of a Distribution, 726

Contents 19.9

Binomial. Poisson, and Hypergeometric Distributions, 730

19.10

Normal

19.11

Distributions of Several

19.12

Random

xvii

Distribution, 734

Sampling.

Random Variables, 740 Random Numbers, 746

19.13 Estimation of Parameters, 747

19.14 Confidence Intervals, 751 19.15 Testing of Hypotheses, Decisions, 760

19.16 Quality Control, 769

19.17 Acceptance Sampling, 774 19.18

Goodness of

Fit.

;(;2-Test,

779

19.19 Nonparametric Tests, 781

19.20 Pairs of Measurements.

Appendix

Fitting Straight Lines, 783

1

References

789

Appendix 2 Answers to Odd-Numbered Problems

794

Appendix 3

Some Formulas for

Special Functions

826

Appendix 4 Tables

833

Index

849

Advanced Engineering Mathematics

1

CHAPTER

Ordinary Differential Equations of the First Order

Differential equations are of

matics because

many

fundamental importance

and

physical laws

in engineering

mathe-

relations appear mathematically in the

form of such equations. We shall consider various physical and geometrical problems which lead to differential equations and the most important standard methods for solving such equations. These methods will in general involve integration.

We

shall

pay particular attention to the derivation of

differential equations

This transition from the physical problem to a

from given physical situations.

corresponding "mathematical model"

is

of great practical importance and

will

be illustrated by typical examples.

The

first

and

tions,

four chapters of the book are devoted to ordinary differential equa-

in the

present chapter

we

shall start with the simplest of these equations,

Corresponding numerical methods for obtaining approximate solutions are included in Sec. 18.6, which is independent of

the so-called equations of the

first

order.

the other sections in Chap. 18. Prerequisite for the present chapter: integral calculus.

Sections which

may

be omitted

References: Appendix

Answers

/,/

to

1,

in

a shorter course: 1.8-1.12.

Part B.

Problems: Appendix

2.

Basic Concepts and Ideas we

and explain the basic concepts that matter in illustrate these concepts by examples. Then we shall consider two simple practical problems taken from physics and geometry. This will give us a first idea of the nature and purpose of differential equations and their application. By an ordinary differential equation we mean a relation which involves one or In this section

shall define

connection with differential equations and

several derivatives of

may

also involve y

an unspecified function y of x with respect to x; the relation given functions of x, and constants.

itself,

1

Ordinal V Differential Equations of the First Order

2

[Section 1-1

For example,

=

y'

(1)

+

y"

(2)

+

xY'y'

(3)

cos

X,

= 0, = [x^ +

42/

2e-y"

l)y^

are ordinary differential equations.

The term

ordinary distinguishes

from a partial

it

differential equation,

involves partial derivatives of an unspecified function of two or variables.

For example,

dx^ is

which

more independent

dy-

In the present chapter

a partial differential equation.

we

shall consider

only

ordinary differential equations.

An

ordinary differential equation

said to be of order n,

is

if

the «th derivative

of y with respect to x is the highest derivative of ?/ in that equation. The notion of the order of a differential equation leads to a useful classification of the equations into equations of

first

order, second order, etc.

Thus

( 1

) is

a

first-

order equation, (2) is of the second order, and (3) is of the third order. In the present chapter we shall consider first-order equations. Equations of

second and higher order

A

will

be discussed

in

Chaps. 2-4.

function

=

y

(4)

g{x)

is

called a solution of a given first-order differential equation

a




30.

I

follows that the value

that

=

7"(0)

+

In

70f-3''

T(t)

It

ce-^'

k can be determined from the given information T(i)

Step,

Using

k

is

T(t)= 4tlt

in order that

obtained by separating variables;

is

=

T = 31°C

is

70e-