modern engineering mathematics

ModernEngineeringMathematics

Fourth Edition

Glyn Jamesand ; 'David BurleyDick ClementsPhil DykeJohn SearlJerry Wright

Coventry University

University of SheffieldUniversity of BristolUniversity of PlymouthUniversity of EdinburghAT&T Shannon Laboratory

PEARSON

PrenticeHall

Harlow, England • London • New York • Boston • San Francisco • Toronto • Sydney • Singapore • Hong KongTokyo • Seoul • Taipei • New Delhi • Cape Town • Madrid • Mexico City • Amsterdam • Munich • Paris • Milan

tr. •

Contents

Preface :About the authors

XXI

xxiv

Chapter 1 Numbers, Algebra and Geometry

1.1 Introduction

1*2 Number and arithmetic1.2.1 Number line1.2.2 Rules of arithmetic1.2.3 Exercises (1-5)1.2.4* Inequalities1.2.5 Modulus and intervals1.2.6 Exercises (6-10)

223778

11

1.3 Algebra '1.3.1 Algebraic manipulation1.3.2 Exercises (11-16)1.3.3 Equations, inequalities and identities1.3.4 Exercises (17-28) ; '1.3.5 Suffix, sigma and pi notation1.3.6 Factorial notation and the binomial expansion1.3.7 Exercises (29-31)

1.4 Geometry1.4.1 Coordinates1.4.2 Straight lines ;1.4.3 Circles

•,. 1.4.4 Exercises (32-38)1.4.5 Conies1.4.6 Exercises (39-41)

1213192027273033

Numbers and accuracyt5.1 Representation of numbers1.5,2 Rounding, decimal places and significant figures

33.333335383844

444547

Vi CONTENTS

1.5.3 Estimating the effect of rounding errors1.5.4 Exercises (42-55)1.5.5 Computer arithmetic1.5.6 Exercises (56-58)

49545556

1.6 Engineering applications 56

1.7 Review exercises (1-25) 59

Chapter 2 Functions

2.1 Introduction 64

2.2 Basic definitions : :

2.2.1 Concept of a function2.2.2 Exercises (1-6)2.2.3 Inverse functions2.2.4 Composite functions .2.2.5 Exercises (7-13)2.2.6 Odd, even and periodic functions2.2.7 Exercises (14-15)

64

64737478818287

2.3 Linear and quadratic functions 87

2.3.1 Linear functions 872.3.2 Least squares fit of a linear function to experimental data 892.3.3 Exercises (16-22) 932.3.4 The quadratic function 942.3.5 Exercises (23-28) 97

2.4 Polynomial functions

2.4.1 Basic properties2.4.2 Factorization2.4.3 Nested multiplication and synthetic division2.4.4 Roots of polynomial equations2.4.5 Exercises (29-37)

2.5 Rational functions

2.5.1 Partial fractions2.5.2 Exercises (38-41)2.5.3 Asymptotes2.5.4 Parametric representation2.5.5 Exercises (42-46)

9899100102105112

114

116122123126128

2.6 Circular functions

2.6.1 Trigonometric ratios2.6.2 Exercises (47-53)

128129131

CONTENTS Vii _--s"

2.6.3 Circular functions2.6.4 Trigonometric identities2.6.5 Amplitude and phase '2.6.6 Exercises (54-65)2.6.7 Inverse circular (trigonometric) functions2.6.8 Polar coordinates2.6.9 Exercises (66-70)

132138142145146148151

2.7 Exponential, logarithmic and hyperbolic functions

2.7.1 Exponential functions2.7.2 Logarithmic functions,2.7.3 Exercises (71-79) ; '.2.7A Hyperbolic functions2.7.5 Inverse hyperbolic functions2.7.6 Exercises (80-87)

152

152155157157162164

2.8 Irrational functions

2.8.1 Algebraic functions2.8.2 Implicit functions2.8.3 Piecewise defined functions2.8.4 Exercises (88-97)

164

165166170172

2.9 Numerical evaluation of functions

2.9.1 Tabulated functions and interpolation2.9.2 Exercises (98-103) :

173

174178

2.10 Engineering application: a design problem 179


Chapter 3 Complex Numbers

3.1 Introduction 185

3.2 Properties

3.2.1 The Argand diagram3.2.2 The arithmetic of complex numbers3.2.3 Complex conjugate3.2.4 Modulus and argument3.2.5 Exercises (1-14)3.2.6 Polar form of a complex number3.2.7 Euler's formula

186

186187190191195196200

VIII CONTENTS

3.2.8 Exercises (15-18) 2023.2.9 Relationship between circular and hyperbolic functions 2023.2.10 Logarithm of a complex number 2063.2.11 Exercises (19-24) 207

3.3 Powers of complex numbers 208

3.3.1 De Moivre's theorem 2083.3.2 Powers of trigonometric functions and multiple angles 2123.3.3 Exercises (25-32) 215

3.4 Loci in the complex plane 216

3.4.1 Straight lines • 2163.4.2 Circles • 2173.4.3 More general loci 2193.4.4 Exercises (33-41) 220_

3.5 Functions of a complex variable 221

3.5.1 Exercises (42-45) , 223

3.6 Engineering application: alternating currents in electrical networks 223

.3.6.1 Exercises (46-47) : _225

3.7 Review'exercises (1-34) : 225

Chapter 4 Vector Algebra

4.1 Introduction

4.2 Basic definitions and results

4.2.1 Cartesian coordinates4.2.2 Scalars and vectors4.2.3 Addition of vectors4.2.4 Cartesian components and basic properties4.2.5 Complex numbers as vectors4.2.6 Exercises (1-16)4.2.7, The scalar product4.2.8 Exercises (17-30)4.2.9 The vector product4.2.10 Exercises (31-42)4.2.11 Triple products4.2.12 Exercises (43-51)

230

231

231233235241247249251257258268269275

CONTENTS ix

Iff 4.3 The vector treatment of the geometry of lines and planes

4.3.1 Vector equation of a line4.3.2 Vector equation of a plane4.3.3 Exercises (52-67)

4.4 Engineering application: spin-dryer suspension

4.4.1 Point-particle model

4.5 Engineering application: cable stayed bridge

4.5.1 A simple stayed bridge

276

276283286

287

287

290

290


Chapter 5 Matrix Algebra


5,2 Definitions and properties

5.2.1 Definitions5.2.2 Basic operations of matrices5.2.3 Exercises (1-10) •5.2.4 Matrix multiplication5.2.5 Exercises (11-16)5.2.6 Properties of matrix multiplication5.2.7 Exercises (17-33)

5.3 Determinants

5.3.1 Exercises (34-50)

5.4 The inverse matrix

5.4.1 Exercises (51-59)

5.5 Linear equations

5.5.1 Exercises (60-71)5.5.2 The solution of linear equations: elimination methods5.5.3 Exercises (72-80)5.5.4 The solution of linear equations: iterative methods5.5.5 Exercises (81-86)

5.6 Rank

5.6.1 Exercises (87-95)

299

301304308310314315325

328

340

341

345

347

354356369371377

377

385

X CONTENTS

5.7 The eigenvalue problem

5.7.1 The characteristic equation5.7.2 Eigenvalues and eigenvectors5.7.3 Exercises (96-97)5.7.4 Repeated eigenvalues5.7.5 Exercises (98-102)5.7.6 Some useful properties of eigenvalues5.7.7 Symmetric matrices5.7.8 Exercises (103-107)

387

387389395396400400402403

5.8 Engineering application: spring systems

5.8.1 A two-particle system5.8.2 An n-particle system

5.9

403

404404

Engineering application: steady heat transfer throughcomposite materials

5.9.1 Introduction5.9.2 Heat conduction5.9.3 The three-layer situation5.9.4 Many-layer situation

407

407408408410


Chapter 6 An Introduction to Discrete Mathematics


6.2 Set theory

6.2.1 Definitions and notation6.2.2 Union and intersection6.2.3 Exercises (1-8)6.2.4 Algebra of sets6.2.5 Exercises (9-17)

418419420422422427

6.3 Switching and logic circuits

6.3.1 Switching circuits6.3.2 Algebra of switching circuits6.3.3 Exercises (18-29)6.3.4 Logic circuits6.3.5 Exercises (30-31)

429429430436437441

6.4 Propositional logic and methods of proof

6.4.1 Propositions6.4.2 Compound propositions6.4.3 Algebra of statements

442

442444447

CONTENTS XI

6.4.4 Exercises (32-37)6.4.5 Implications and proofs6.4.6 Exercises (38-4:7)

450450456

6.5 Engineering application: expert systems

6.6 Engineering application: control

6.7 Review exercises (1-23)

457

459

462

Chapter 7 Sequences, Series and Limits


7.2 Sequences and series

7.2.1 Notation7.2.2 Graphical representation of sequences7.2.3 Exercises (1-13)

467

467469472

7.3 Finite sequences and series ,:.

7.3.1 Arithmetical sequences and series7.3.2^ Geometric sequences and series7.3.3 Other finite series7.3.4 Exercises (14-25)

474

474475477480

7.4 Recurrence relations

7.4.1 First-order linear recurrence relations withconstant coefficients

7.4.2 Exercises (26-28) .7.4.3 Second-order linear recurrence relations with

constant coefficients7.4.4 Exercises (29-35)

481

482485

486494

7.5 Limit of a sequence

7.5.1 Convergent sequences7.5.2 Properties of convergent sequences7.5.3 Computation of limits7.5.4 Exercises (36-40)

494

495497499501

7.6 Infinite series

7.6.1 Convergence of infinite series7.6.2 Testa for convergence of positive series7.6.3 The absolute convergence of general series7.6.4 Exercises (41-49)

502

502504507508

Xii CONTENTS

7.7 Power series

7.7.1 Convergence of power series7.7.2 Special power series7.7.3 Exercises (50-56)

7.8 Functions of a real variable

7.8.1 Limit of a function of a real variable7.8.2 One-sided limits7.8.3 Exercises (57-61)

7.9 Continuity of functions of a real variable

7.9.1 Properties of continuous functions7.9.2 Continuous and discontinuous functions7.9.3 Numerical location of zeros7.9.4 Exercises (62-69)

7.10 Engineering application: insulator chain

509

509511517

518

518522524

525

525527529532

532

7.11 Engineering application: approximating functions andPade approximants 533


Chapter 8 Differentiation and Integration

8.1 Introduction

8.2 Differentiation

8.2.1 Rates of change8.2.2 Definition of a derivative8.2.3 Interpretation as the slope of a tangent8.2.4 Differentiate functions8.2.5 Speed, velocity and acceleration8.2.6 Exercises (1-7)8.2.7 Mathematical modelling using derivatives8.2.8 Exercises (8-18)

8.3 Techniques of differentiation

8.3.1 Basic rules of differentiation8.3.2 Derivative of xr

8.3.3 Exercises (19-23)8.3.4 Differentiation of polynomial functions

540

541

541542544546547548549556

557

558560564564

CONTENTS xiii

8.3.5 Differention of rational functions8.3.6 Differentiation of composite functions8.3.7 Differentiation of inverse functions8.3.8 Exercises (24-31)8.3.9 Differentiation of circular functions8.3.10 Extended form of the chain rule8.3.11 Exercises (32-34)8.3.12 Differentiation of exponential and related functions8.3.13 Exercises (35-43)8.3.14 Parametric and implicit differentiation8.3.15 Exercises (44-54)

8.4 Higher derivatives

8.4.1 The second derivative8.4.2 Exercises (55-67)8.4.3 Curvature of plane curves8.4.4 Exercises (68-71)

567568573574575579581581586586591

592

592596597600

8.5 Applications to optimization problems

8.5.1 Optimal values8.5.2 Exercises (72-81)

600

600609

8.6 Numerical differentiation

8.6.1 The chord approximation8.6.2 Exercises (82-86)

611

611613

8.7 Integration

8.7.1 Basic ideas and definitions8.7.2 Mathematical modelling using integration8.7.3 Exercises (87-95)8.7.4 Definite and indefinite integrals8.7.5 The Fundamental Theorem of Calculus8.7.6 Exercise (96)

8.8 Techniques of integration

8.8.1 Integration as antiderivative8.8.2 Exercises (97-104)8.8.3 Integration by parts8.8.4 Exercises (105-107)8.8.5 Integration by substitution8.8.6 Exercises (108-116)

8.9 Applications of integration

8.9.1 Volume of a solid of revolution8.9.2 Centroid of a Diane area

613

613616620620623625

625

625636637640640645

646

646647

Xiv CONTENTS

8.9.3 Centre of gravity of a solid of revolution 6498.9.4 Mean values . 6498.9.5 Root mean square values 6508.9.6 Arclength and surface area 6508.9.7 Exercises (117-125) 656

8.10 Numerical evaluation of integrals 657

8.10.1 The trapezium rule 6578.10.2 Simpson's rule 6638.10.3 Exercises (126-131) \ • 666

8.11 Engineering application: design of prismatic channels 667

8.12 Engineering application: harmonic analysis of periodic functions 669


Chapter 9 Further Calculus

9.1 Introduction

9.2 Improper integrals

9.2.1 Integrand with an infinite discontinuity9.2.2 Infinite integrals9.2.3 Exercise (1)

9.3 Some theorems with applications to numerical methods

9.3.1 Rolle's theorem and the first mean value theorems9.3.2 Convergence of iterative schemes9,3.3 Exercises (2-7)

9.4 Taylor's theorem and related results

9.4.1 Taylor polynomials and Taylor's theorem9.4.2 Taylor and Maclaurin series9.4.3 L'Hopital's rule9.4.4 Exercises (8-20)9.4.5 Interpolation revisited9.4.6 Exercises (21-23)9A7 The convergence of iterations revisited9.4.8 Newton-Raphson procedure9.4.9 Optimization revisited9.4.10 Exercises (24-27)9.4.11 Numerical integration9.4.12 Exercises (28-31)

680

680

681684685

686

686689693

693

693696701702

: 703704705706709709709711

mBHI-

m.BfflBSi

iHm

WI

9.5 Calculus of vectors

9.5.1 Differentiation and integration of vectors9.5.2 Exercises (32-36)

9.6 Functions of several variables

9.6.1 Representation of functions of two variables9.6.2 Partial derivatives9.6.3 Directional derivatives9.6.4 Exercises (37-46)9.6.5 The chain rule ;9.6.6 Exercises (47-55)9.6.7 Successive differentiation9.6.8 Exercises (56-64)9.6.9 The total differential and small errors9.6.10 Exercises (65-72)9.6.11 Exact differentials9.6.12 Exercises (73-75)

9.7 Taylor's theorem for functions of two variables

9.7.1 Taylor's theorem9.7.2 Optimization of unconstrained functions9.7.3 Exercises (76-84)9.7.4 Optimization of constrained functions9.7.5 Exercises (85-90)

9.8 Engineering application: deflection of a built-in column

9.9 Engineering application: streamlines in fluid dynamics

9.10 Review exercises (1-35)

CONTENTS XV

712

712714

715

715717721724725729729733733736737739

739

740743748749753

754

756

759

Chapter 10 Introduction to Ordinary Differential Equations

10.1 Introduction

10.2 Engineering examples

10.2.1 The take-off run of an aircraft10.2.2 Domestic hot-water supply10.2.3 Hydro-electric power generation10.2.4 Simple electrical circuits

765

765

765767768769

10.3 The classification of differential equations

10.3.1 Ordinary and partial differential equations10.3.2 Independent and dependent variables

770

771771

XVI CONTENTS

10.4

10.5

10.6

10.7

10.8

10.9

10.3.3 The order of a differential equation10.3.4 Linear and nonlinear differential equations10.3.5 Homogeneous and nonhomogeneous equations10.3.6 Exercises (1-2)

Solving differential equations

10.4.1 Solution by inspection10.4.2 General and particular solutions10.4.3 Boundary and initial conditions10.4.4 Analytical and numerical solution10.4.5 Exercises (3-6)

First-order ordinary differential equations

10.5.1 A geometrical perspective10.5.2 Exercises (7-10)10.5.3 Solution of separable differential equations10.5.4 Exercises (11-17)

dx f x ~\10.5.5 Solution of differential equations of —- = fl - form

10.5.6 Exercises (18-22)10.5.7 Solution of exact differential equations10.5.8 Exercises (23-30)10.5.9 Solution of linear differential equations10.5.10 Solution of the Bernoulli differential equations10.5.11 Exercises (31-38)

Numerical solution of first-order ordinary differential equations

10.6.1 A simple solution method: Euler's method10.6.2 Analysing Euler's method10.6.3 Using numerical methods to solve engineering problems10.6.4 Exercises (39-45)

Engineering application: analysis of damper performance

Linear differential equations

10.8.1 Differential operators10.8.2 Linear differential equations10.8.3 Exercises (46-54)

Linear constant-coefficient differential equations

10.9.1 Linear homogeneous constant-coefficient equations10.9.2 Exercises (55-61)10.9.3 Linear nonhomogeneous constant-coefficient equations10.9.4 Exercises (62-65)

772773774775

776

776111118781782

783

783786786788

789

791791794795799801

802

803805808810

811

816

816818824

826

826831832838

CONTENTS XVII

10.10 Engineering application: second-order linear constant-coefficientdifferential equations • , 839

10.10.1 Free oscillations of elastic systems 83910.10.2 Free oscillations of damped elastic systems 84310.10.3 Forced oscillations of elastic systems 84610.10.4 Oscillations in electrical circuits ' 85010.10.5 Exercises (66-73) 851

10.11 Numerical solution of seconds arid higher-orderdifferential equations 853

10.11.1 Numerical solution of coupled first-order equations 85310.11.2 State-space representation of higher-order systems 85610.11.3 Exercises (74-79) : 859

10.12 Qualitative analysis of second-order differential equations

10.12.1 Phase-plane plots10.12.2 Exercises (80-81)

861861865


Chapter 11 Introduction to Laplace Transforms


1.1.2 The Laplace transform 876

11.2.1 Definition and notation 87611.2.2 Transforms of simple functions 87811.2.3 Existence of the Laplace transform ; 88111.2.4 Properties of the Laplace transform 88311.2.5 Table of Laplace transforms 89111.2.6 Exercises (1-3) 89211.2.7 The inverse transform 89211.2.8 Evaluation of inverse transforms 89311.2.9 Inversion using the first shift theorem 89511.2.10 Exercise (4) 897

11.3 Solution of differential equations 897

11.3.1 Transforms of derivatives 89711.3.2 Transforms of integrals 89911.3.3 Ordinary differential equations 90011.3.4 Exercise (5) 90611.3.5 Simultaneous differential equations 90711.3.6 Exercise (6) • 909

xviii CONTENTS

11.4 Engineering applications; electrical circuits and; mechanical vibrations

11.4.1 Electrical circuits11.4.2 Mechanical vibrations :11.4.3 Exercises (7-12) ;

910910915919


Chapter 12 Introduction to Fourier Series


12.2 Fourier series expansion

12.2.1 Periodic functions : :

12.2.2 Fourier's theorem ,12.2.3 The Fourier coefficients12.2.4 Functions of period:2rr • ' ,'12.2.5 Even and odd functions . ';.12.2.6 Even and odd harmonics ;

12.2.7 Linearity property12.2.8 Convergence of the Fourier series12.2.9 Exercises (1-7)12.2.10 Functions of period T12.2.11 Exercises (8-13)

926926927928931938942944946949951953

12.3 Functions defined over a finite interval

12.3.1 Full-range series12.3.2 Half-range cosine and sine series12.3.3 Exercises (14-23)

954954956960

12.4 Differentiation and integration of Fourier series

12.4.1 Integration of:a Fourier series12.4.2 Differentiation of a Fourier series12.4.3 Exercises (24-26)

961961964965

12.5 Engineering application: analysis of aslider-crank mechanism 966

12.6 Review exercises (1-21.) 969

CONTENTS Xix

Chapter 13 Data Handling and Probability Theory

13.1 Introduction" 974

13.2 The raw material of statistics

13.2.1 Experiments and sampling13.2.2 Histograms of data13.2.3 Alternative types of plot13.2.4 Exercises (1-5)

975975975978980

13.3 Probabilities of random events

13.3.1 Interpretations, of probability13.3.2 Sample space and events13.3.3 Axioms, of probability13.3.4 Conditional probability13.3.5 Independence13.3.6 Exercises (6-23)

980

980981982984988991

13.4 Random variables

13.4.1 Introduction and definition13.4.2 Discrete random variables13.4.3 Continuous random variables13.4.4 • Properties of density and distribution functions13.4.5 Exercises (24-31)13.4.6 •• Measures of location and dispersion13.4.7 Expected values13.4.8 Independence of random variables13.4.9 Scaling and adding random variables13.4.10 Measures from sample data13.4.11 Exercises (32-48)

992

992993994995998998

10021003100410071011

13.5 Important practical distributions

13.5.1 The binomial distribution13.5.2 The Poisson distribution •.13.5.3 The normal distribution13.5.4 The central limit theorem13.5.5 Normal approximation to the binomial13.5.6 Random variables for simulation13.5.7 Exercises (49-65) ••'

1013

1013101510181021102410261027

13.6 Engineering application: quality control

13.6-1 Attribute control charts13.6.2 United States standard attribute charts13.6.3 Exercises (66-67)

1029

102910311032

XX CONTENTS

13.7 Engineering application: clustering of rare events

13.7.1 Introduction13.7.2 Survey of near-misses between aircraft13.7.3 Exercises (68-69)

1032103210331035

13.8 Review exercises (1-1.3) 1035

Appendix I Tables

Al. l Some useful resultsAI.2 Trigonometric identitiesAI.3 Derivatives and integralsAI.4 Some useful standard integrals

1038

1038104110421043

Answers to Exercises 1044

Index 1082

modern engineering mathematics

Documents