modern engineering mathematics - gbv

Modern Engineering Mathematics

GLYN JAMES Coventry University

a n d

D A V I D B U R L E Y University of Sheffield

DICK C L E M E N T S University of Bristol

PHIL D Y K E Polytechnic South West

J O H N S E A R L University of Edinburgh

J E R R Y W R I G H T University of Bristol

ADDISON-WESLEY PUBLISHING COMPANY Wokingham, England • Reading, Massachusetts • Menlo Park, California • New York Don Mills, Ontario • Amsterdam • Bonn • Sydney • Singapore Tokyo • Madrid • San Juan • Milan • Paris • Mexico City • Seoul • Taipei

Contents

About the authors

Preface

Chapter 1

Numbers and Functions

1.1 Introduction

1.1.1 Numbers 1.1.2 Exercises (1-13) 1.1.3 Algebra and geometry 1.1.4 Exercises (14-20)

1.2 Numbers and accuracy

1.2.1 Representation of numbers 1.2.2 Estimating the effect of rounding errors 1.2.3 Exercises (21-34) 1.2.4 Computer arithmetic 1.2.5 Exercises (35-37)

1.3 Functions

1.3.1 Basic definitions 1.3.2 Inverse functions 1.3.3 Composite functions 1.3.4 Exercises (38-46)

1.4 Elementary functions

1.4.1 Polynomial functions 1.4.2 Rational functions 1.4.3 Exercises (47-58) 1.4.4 Circular functions 1.4.5 Polar coordinates 1.4.6 Exercises (59-71) 1.4.7 Exponential, logarithmic and hyperbolic functions 1.4.8 Exercises (72-82)

xix

1

1

2 7 8

18

19

19 22 27 28 29

29

29 32 36 38

39

39 48 56 58 71 73 74 79

CONTENTS

1.4.9 Other irrational functions 80 1.4.10 Exercises (83-90) 83

1.5 Numerical evaluation of functions 84

1.5.1 Tabulated functions and interpolation 85

1.5.2 Exercises (91-95) 89

1.6 Engineering application 90

1.7 Engineering application: a design problem 91

1.8 Review exercises (1-27) 94

Chapter 2 97

Complex Numbers

2.1 Introduction 97

2.2 Properties 99

2.2.1 The arithmetic of complex numbers 99 2.2.2 Graphical representation 101 2.2.3 Polar form of a complex number 103 2.2.4 Exercises (1-13) 106 2.2.5 Euler's formula 107 2.2.6 Exercises (14-23) 112

2.3 Powers of complex numbers 112

2.3.1 De Moivre's theorem 113 2.3.2 Exercises (24-31) 118

2.4 Loci in the complex plane 118

2.4.1 Straight lines and circles 119 2.4.2 Exercises (32-36) 123

2.5 Engineering application: alternating currents in electrical networks 124

2.5.1 Exercises (37-38) 126

2.6 Review exercises (1 30) 127

Chapter 3 131

Vector Algebra

3.1 Introduction 131

3.2 Basic definitions 133

3.2.1 Cartesian coordinates 133 3.2.2 Sealars and vectors 135 3.2.3 Addition of vectors 136

CONTENTS ix ™g : m :•:. iiii!;:s««;Mi# mmmmvm m m mm^mmm M:w:aasi!i :::. :: : :: >

3.2.4 Cartesian components 142 3.2.5 Complex numbers as vectors 146 3.2.6 Exercises (1-13) 148 3.2.7 The scalar product 149 3.2.8 Exercises (14-23) 155 3.2.9 The vector product 156 3.2.10 Exercises (24-31) 164 3.2.11 Triple products 165 3.2.12 Exercises (32-41) 168

3.3 The vector treatment of the geometry of lines and planes 169

3.3.1 Vector equation of a line 169 3.3.2 Vector equation of a plane 173 3.3.3 Exercises (42-56) 177

3.4 Engineering applications: spin-dryer Suspension 178

3.4.1 Point-particle model 178 3.4.2 Cylinder model 181


Chapter 4 187

Matrix Algebra


4.2 Definitions and properties 190

4.2.1 Definitions 192 4.2.2 Basic properties of matrices 195 4.2.3 Matrix multiplication 197 4.2.4 Exercises (1-12) 206

4.3 Determinants 209

4.3.1 Exercises (13-22) 218

4.4 The inverse matrix 219

4.4.1 Exercises (23-26) 221

4.5 Linear equations 222

4.5.1 Exercises (27-32) 226 4.5/2 The Solution of linear equations: elimination methods 228 4.5.3 Exercises (33-39) 239 4.5.4 The Solution of linear equations: iterative methods 240 4.5.5 Exercises (40-44) 244

4.6 Rank 245

4.6.1 Exercises (45-50) 250

CONTENTS Wimmmmyrnm mmmmmmm

4.7 Vector Spaces 251

4.7.1 Introduction 251 4.7.2 Linear independence 252 4.7.3 Transformations between bases 254 4.7.4 Exercises (51-54) 255

4.8 Engineering application: spring Systems 256

4.8.1 A two-particle System 256

4.8.2 An n-particle System 257

4.9 Engineering application: splines 260


Chapter 5 271

An Introduction to Discrete Mathematics


5.2 Set theory 272

5.2.1 Definitions and notation 273 5.2.2 Union and intersection 275 5.2.3 Exercises (1-8) 276 5.2.4 Algebra of sets 276 5.2.5 Exercises (9-17) 282

5.3 Switching and logic circuits 283

5.3.1 Switching circuits 283 5.3.2 Algebra of switching circuits 285 5.3.3 Exercises (18-29) 291 5.3.4 Logic circuits 292 5.3.5 Exercises (30-31) 297

5.4 Propositional logic and methods of proof 298

5.4.1 Propositions 298 5.4.2 Compound propositions 300 5.4.3 Algebra of Statements 303 5.4.4 Exercises (32-37) 307 5.4.5 Implications and proofs 307 5.4.6 Exercises (38-47) 313

5.5 Undirected graphs 313

5.5.1 A test for planarity 315 5.5.2 Exercises (48-51) 320 5.5.3 More about planar graphs 321 5.5.4 Exercises (52-58) 326

CONTENTS

5.5.5 Trees 327 5.5.6 Exercises (59-62) 331

5.6 An introduction to directed graphs 331

5.6.1 Networks and CPM 332

5.6.2 Exercises (63-65) 334

5.7 Engineering application: expert Systems 334

5.8 Engineering application: control 337


Chapter 6 345

Sequences , Series and Limits


6.2 Sequences and series 346

6.2.1 Sequences 346 6.2.2 Exercises (1-10) 349 6.2.3 Series 350 6.2.4 Exercises (11-16) 354

6.3 Limit of a sequence 354

6.3.1 Convergent sequences 355 6.3.2 Exercises (17-21) 360 6.3.3 Infinite series 360 6.3.4 Exercises (22-28) 364 6.3.5 Power series 365 6.3.6 Exercises (29-35) 369

6.4 Functions of a real variable 370

6.4.1 Limit of a function of a real variable 370 6.4.2 The order notation 373 6.4.3 Exercises (36-40) 374 6.4.4 Continuity of functions of a real variable 374 6.4.5 Numerical location of zeros 380 6.4.6 Exercises (41-48) 382

6.5 Engineering application: approximating functions 383


ChapterJ7 389

Differentiation and Integration


XÜ CONTENTS mMmmik

7.2 Differentiation 390

7.2.1 Basic ideas and definitions ' 391 7.2.2 Exercises (1-10) 398

7.3 Techniques of differentiation 399

7.3.1 Elementary results 399 7.3.2 Rules of differentiation 401 7.3.3 Exercises (11-14) 407 7.3.4 Parametric and implicit differentiation 407 7.3.5 Higher derivatives 411 7.3.6 Curvature of plane curves 414 7.3.7 Exercises (15-34) 416

7.4 Numerical differentiation 417

7.4.1 The chord approximation 418 7.4.2 Exercises (35-39) 420

7.5 Integration 421

7.5.1 Basic ideas and definitions 421 7.5.2 Exploitation of ideas 426 7.5.3 Exercises (40-48) 430 7.5.4 Definite and indefinite integrals 431 7.5.5 The Fundamental Theorem of Calculus 433 7.5.6 Exercises (49) 435

7.6 Techniques of integration 436

7.6.1 Integration as antiderivative 436 7.6.2 Exercises (50-55) 442 7.6.3 Further analytical methods of integration 443 7.6.4 Exercises (56-67) 450 7.6.5 Application of integration 451 7.6.6 Exercises (68-76) 458

7.7 Numerical evaluation of integrals 459

7.7.1 The trapezium rule 459

7.7.2 Exercises (77-82) 466

7.8 Engineering application: design of prismatic Channels 467


Chapter 8 475

Further Calculus of One Variable


8.2 Applications to optimization problems 476

CONTENTS xih m< mmmm ;;:: «; e sx: :v:^ ; ; ; ; ; ; : ; ; :: > :mmm • ••: • ::

8.2.1 Optimal values 476 8.2.2 Exercises (1-9) 484

8.3 Improper integrals 486

8.3.1 Integrand with an infinite discontinuity 487 8.3.2 Infinite integrals 489 8.3.3 Beta and gamma functions 489 8.3.4 Exercises (10-12) 491

8.4 Some theorems with applications to numerical problems 492

8.4.1 Rolle's theorem 492 8.4.2 Exercises (13-14) 494 8.4.3 The first mean value theorems 494 8.4.4 Exercises (15-19) 500

8.5 Taylor's theorem and related results 500

8.5.1 Taylor polynomials and Taylor's theorem 501 8.5.2 Taylor and Maclaurin series 503 8.5.3 L'Hopital's rule 508 8.5.4 Exercises (20-32) 510 8.5.5 The convergence of iterations 511 8.5.6 Optimization revisited 514 8.5.7 Exercises (33-37) 515 8.5.8 Numerical Integration 515 8.5.9 Exercises (38-42) 518

8.6 Engineering application: deflection of a built-in column 518

8.7 Calculus of vectors 521

8.7.1 Differentiation and Integration of vectors 522 8.7.2 Exercises (43-45) 524


Chapter 9 529

Functions of More than One Variable


9.2 Functions of two and three variables 530

9.2.1 Geometrical representation 531 9.2.2 Exercises (1-2) 533 9.2.3 Partial differentiation 533 9.2.4 Exercises (3-10) 538 9.2.5 The chain rule 539 9.2.6 Exercises (11-24) 544 9.2.7 Successive differentiation 545

CONTENTS mm:mmMmAmm\ :~M:mmmM'm<mmm>i<mm:A mmmmtmmmmwm

9.2.8 Exercises (25-33) 549 9.2.9 The total differential and small errors 550 9.2.10 Exercises (34-41) 553 9.2.11 Exact differentials 554 9.2.12 Exercises (42-44) 556

9.3 Applications of partial differentiation 556

9.3.1 Integrals dependent on a parameter 557 9.3.2 Exercises (45-50) 559 9.3.3 Tangent planes and normals to surfaces in three

dimensions 559 9.3.4 Exercises (51-53) 562

9.4 Taylor's theorem and its applications 562

9.4.1 Taylor's theorem 563 9.4.2 Optimization of unconstrained functions 565 9.4.3 Exercises (54-62) 570 9.4.4 Optimization of constrained functions 571 9.4.5 Exercises (63-68) 575

9.5 Topics in integration 576

9.5.1 Line integrals 576 9.5.2 Exercises (69-74) 582 9.5.3 Double integrals 582 9.5.4 Exercises (75-85) 588

9.6 Engineering application: streamlines in fluid dynamics 589

9.7 Engineering application: calculating the capacity of a reservoir 592


Chapter 10 597

Introduction to Ordinary Differential Equations


10.2 Engineering examples 598

10.2.1 The take-off run of an aircraft 598 10.2.2 Domestic hot-water supply 599 10.2.3 Hydro-electric power generation 601 10.2.4 Simple electrical circuits 602

10.3 The Classification of differential equations 604

10.3.1 Ordinary and partial differential equations 604 10.3.2 Independent and dependent variables 605 10.3.3 The order of a differential equation 605 10.3.4 Linear and nonlinear differential equations 606

CONTENTS xv

10.3.5 Homogeneous and nonhomogeneous equations 607 10.3.6 Exercises (1-15) 609

10.4 Solving differential equations 609

10.4.1 Solution by inspection 609 10.4.2 General and particular Solutions 610 10.4.3 Boundary and initial conditions 610 10.4.4 Analytical and numerical Solution 612 10.4.5 Exercises (16-18) 613

10.5 First-order ordinary differential equations 614

10.5.1 A geometrical perspective 615 10.5.2 Exercises (19-22) 618 10.5.3 Elementary analytical Solution methods: separable

equations 618 10.5.4 Exercises (23-27) 621 10.5.5 Elementary analytical Solution methods: exact equations 622 10.5.6 Exercises (28-33) 625 10.5.7 Further analytical Solution methods: linear equations 625 10.5.8 Exercises (34-37) 628

10.6 Numerical Solution of first-order ordinary differential equations 629

10.6.1 A simple Solution method: Euler's method 629 10.6.2 Analysing Euler's method 632 10.6.3 Using numerical methods to solve engineering problems 634 10.6.4 Exercises (38-42) 637 10.6.5 More accurate Solution methods: multistep methods 638 10.6.6 Local and global truncation errors 644 10.6.7 More accurate Solution methods: predictor-corrector

methods 646 10.6.8 More accurate Solution methods: Runge-Kut ta methods 650 10.6.9 Exercises (43-52) 653 10.6.10 Stiff equations 655 10.6.11 Computer Software libraries and the 'state of the art' 658

10.7 Engineering application: analysis of damper Performance 660


Chapter 11 669

Second- and Higher-Order Ordinary Differential Equations


11.2 Linear differential equations 670

11.2.1 Differential Operators 670 11.2.2 Linear differential equations 672 11.2.3 Exercises (1-6) 678

xvi CONTENTS

11.3 Linear constant-coefficient differential equations 679

11.3.1 Linear homogeneous constant-coefficient equations 679 11.3.2 Linear nonhomogeneous constant-coefficient equations 683 11.3.3 Exercises (7-14) 689

11.4 Engineering application: second-order linear constant-coefficient differential equations 691

11.4.1 Free oscillations of elastic Systems 691 11.4.2 Free oscillations of damped elastic Systems 695 11.4.3 Forced oscillations of elastic Systems 699 11.4.4 Oscillations in electrical circuits 703 11.4.5 Exercises (15-20) 705

11.5 Numerical Solution of second- and higher-order differential equations 706

11.5.1 Numerical Solution of linked first-order equations 707 11.5.2 State-space representation of higher-order Systems 711 11.5.3 Exercises (21-26) 715 11.5.4 Boundary-value problems 716 11.5.5 The method of shooting 718 11.5.6 Function approximation methods 720

11.6 Qualitative analysis of second-order differential equations 726

11.6.1 Phase-plane plots 726 11.6.2 Exercises (27-28) 731


Chapter 12 735

Fourier Series


12.2 Fourier series expansion 737

12.2.1 Periodic functions 737 12.2.2 Fourier's theorem 738 12.2.3 The Fourier coefficients 739 12.2.4 Functions of period 27t 742 12.2.5 Even and odd functions 748 12.2.6 Even and odd harmonics 752 12.2.7 Linearity property 754 12.2.8 Convergence of the Fourier series 756 12.2.9 Exercises (1-7) 760 12.2.10 Function of period T 761 12.2.11 Exercises (8-13) 763

CONTENTS xvü

12.3 Functions defined over a finite interval 764

12.3.1 Full-range series 764 12.3.2 Half-range cosine and sine series 766 12.3.3 Exercises (14-23) 770

12.4 Engineering application: harmonic analysis 771

12.4.1 Exercises (24-25) 775

12.5 Engineering application: analysis of a slider-crank mechanism 776


Chapter 13 783

Data Handling and Probability Theory


13.2 The raw material of statistics 784

13.2.1 Experiments and sampling 784 13.2.2 Histograms of data 785 13.2.3 Alternative types of plot 788 13.2.4 Exercises (1-4) 790

13.3 Probabüities of random events 790

13.3.1 Interpretations of probability 790 13.3.2 Sample space and events 791 13.3.3 Axioms of probability 792 13.3.4 Conditional probability 795 13.3.5 Independence 799 13.3.6 Exercises (5-20) 802

13.4 Random variables 803

13.4.1 Introduction and definition 803 13.4.2 Discrete random variables 804 13.4.3 Continuous random variables 806 13.4.4 Properties of density and distribution functions 807 13.4.5 Exercises (21-27) 810 13.4.6 Measures of location and dispersion 810 13.4.7 Expected values 814 13.4.8 Independence of random variables 815 13.4.9 Scaling and adding random variables 817 13.4.10 Measures from sample data 819 13.4.11 Exercises (28-41) 824

13.5 Important practical distributions 825

13.5.1 The binomial distribution 825 13.5.2 The Poisson distribution 828

xviii CONTENTS

13.5.3 The normal distribution 830 13.5.4 The central limit theorem 834 13.5.5 Normal approximation to the binomial 838 13.5.6 Random variables for Simulation 839 13.5.7 Exercises (42-56) 841

13.6 Engineering application: quality control 842

13.6.1 Attribute control charts 843 13.6.2 United States Standard attribute charts 845 13.6.3 Exercises (57-58) 846

13.7 Engineering application: clustering of rare events 846

13.7.1 Introduction 846 13.7.2 Survey of near-misses between aircraft 847 13.7.3 Exercises (59-60) 849


Appendix I 851

Tables

ALI Trigonometrie identities 851

AI.2 Derivatives and integrals 852

AI.3 Some useful Standard integrals 853

Ans wers to exercises 855

Index 887

modern engineering mathematics - gbv

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