*Mathematics for Electrical Engineering and Computing* embraces many applications of modern mathematics, such as Boolean Algebra and Sets and Functions, and also teaches both discrete and continuous systems – particularly vital for Digital Signal Processing (DSP). In addition, as most modern engineers are required to study software, material suitable for Software Engineering – set theory, predicate and prepositional calculus, language and graph theory – is fully integrated into the book.

Excessive technical detail and language are avoided, recognising that the real requirement for practising engineers is the need to understand the *applications* of mathematics in everyday engineering contexts. Emphasis is given to an appreciation of the fundamental concepts behind the mathematics, for problem solving and undertaking critical analysis of results, whether using a calculator or a computer.

The text is backed up by numerous exercises and worked examples throughout, firmly rooted in engineering practice, ensuring that all mathematical theory introduced is directly relevant to real-world engineering. The book includes introductions to advanced topics such as Fourier analysis, vector calculus and random processes, also making this a suitable introductory text for second year undergraduates of electrical, electronic and computer engineering, undertaking engineering mathematics courses.

**Dr Attenborough** is a former Senior Lecturer in the School of Electrical, Electronic and Information Engineering at South Bank University. She is currently Technical Director of The Webbery – Internet development company, Co. Donegal, Ireland.

- Fundamental principles of mathematics introduced and applied in engineering practice, reinforced through over 300 examples directly relevant to real-world engineering

## Editorial Reviews

### Book Description – Mathematics for Electrical Engineering and Computing

A comprehensive mathematics textbook for all first year undergraduates of electrical, electronic, and computer engineering, with introductory material for students of software engineering

### From the Back Cover

*Mathematics for Electrical Engineering and Computing* embraces many applications of modern mathematics, such as Boolean Algebra and Sets and Functions, and also teaches both discrete and continuous systems – particularly vital for Digital Signal Processing (DSP). In addition, as most modern engineers are required to study software, material suitable for Software Engineering – set theory, predicate and prepositional calculus, language and graph theory – is fully integrated into the book. Excessive technical detail and language are avoided, recognising that the real requirement for practising engineers is the need to understand the *applications* of mathematics in everyday engineering contexts. Emphasis is given to an appreciation of the fundamental concepts behind the mathematics, for problem solving and undertaking critical analysis of results, whether using a calculator or a computer. The text is backed up by numerous exercises and worked examples throughout, firmly rooted in engineering practice, ensuring that all mathematical theory introduced is directly relevant to real-world engineering. The book includes introductions to advanced topics such as Fourier analysis, vector calculus and random processes, also making this a suitable introductory text for second year undergraduates of electrical, electronic and computer engineering, undertaking engineering mathematics courses. **Dr Attenborough** is a former Senior Lecturer in the School of Electrical, Electronic and Information Engineering at South Bank University. She is currently Technical Director of The Webbery – Internet development company, Co. Donegal, Ireland.

## Table of Contents

Preface. Acknowledgement.

**Sets, functions and calculus:** Sets and Functions. Functions and their graphs. Problem solving and the art of the convincing argument. Boolean algebra. Trigonometric functions and waves. Differentiation. Integration. The exponential function. Vectors. Complex numbers. Maxima and minima and sketching functions. Sequences and series.

**Systems:** Systems of linear equations, matrices and determinants. Ordinary differential equations and difference equations. Laplace and z transforms. Fourier series.

**Functions of more than one variable:** Functions of more than one variable. Vector calculus.

**Graph and Language Theory:** Graph Theory. Language Theory.

**Probability and Statistics:** Probability and Statistics.

Answers to Exercises. Index.

### Detailed Content – Mathematics for Electrical Engineering and Computing

Preface xi

Acknowledgements xii

Part 1 Sets, functions, and calculus

1 Sets and functions 3

1.1 Introduction 3

1.2 Sets 4

1.3 Operations on sets 5

1.4 Relations and functions 7

1.5 Combining functions 17

1.6 Summary 23

1.7 Exercises 24

2 Functions and their graphs 26

2.1 Introduction 26

2.2 The straight line: y = mx + c 26

2.3 The quadratic function: y = ax2 + bx + c 32

2.4 The function y = 1/x 33

2.5 The functions y = ax 33

2.6 Graph sketching using simple

transformations 35

2.7 The modulus function, y = |x| or

y = abs(x) 41

2.8 Symmetry of functions and their graphs 42

2.9 Solving inequalities 43

2.10 Using graphs to find an expression for the function

from experimental data 50

2.11 Summary 54

2.12 Exercises 55

3 Problem solving and the art of the convincing

argument 57

3.1 Introduction 57

3.2 Describing a problem in mathematical

language 59

3.3 Propositions and predicates 61

3.4 Operations on propositions and predicates 62

3.5 Equivalence 64

3.6 Implication 67

3.7 Making sweeping statements 70

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3.8 Other applications of predicates 72

3.9 Summary 73

3.10 Exercises 74

4 Boolean algebra 76

4.1 Introduction 76

4.2 Algebra 76

4.3 Boolean algebras 77

4.4 Digital circuits 81

4.5 Summary 86

4.6 Exercises 86

5 Trigonometric functions and waves 88

5.1 Introduction 88

5.2 Trigonometric functions and radians 88

5.3 Graphs and important properties 91

5.4 Wave functions of time and distance 97

5.5 Trigonometric identities 103

5.6 Superposition 107

5.7 Inverse trigonometric functions 109

5.8 Solving the trigonometric equations sin x = a,

cos x = a, tan x = a 110

5.9 Summary 111

5.10 Exercises 113

6 Differentiation 116

6.1 Introduction 116

6.2 The average rate of change and the gradient of a

chord 117

6.3 The derivative function 118

6.4 Some common derivatives 120

6.5 Finding the derivative of combinations of

functions 122

6.6 Applications of differentiation 128

6.7 Summary 130

6.9 Exercises 131

7 Integration 132

7.1 Introduction 132

7.2 Integration 132

7.3 Finding integrals 133

7.4 Applications of integration 145

7.5 The definite integral 147

7.6 The mean value and r.m.s. value 155

7.7 Numerical Methods of Integration 156

7.8 Summary 159

7.9 Exercises 160

8 The exponential function 162

8.1 Introduction 162

8.2 Exponential growth and decay 162

8.3 The exponential function y = et 166

8.4 The hyperbolic functions 173

8.5 More differentiation and integration 180

8.6 Summary 186

8.7 Exercises 187

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9 Vectors 188

9.1 Introduction 188

9.2 Vectors and vector quantities 189

9.3 Addition and subtraction of vectors 191

9.4 Magnitude and direction of a 2D vector – polar

co-ordinates 192

9.5 Application of vectors to represent waves

(phasors) 195

9.6 Multiplication of a vector by a scalar and unit

vectors 197

9.7 Basis vectors 198

9.8 Products of vectors 198

9.9 Vector equation of a line 202

9.10 Summary 203

9.12 Exercises 205

10 Complex numbers 206

10.1 Introduction 206

10.2 Phasor rotation by π/2 206

10.3 Complex numbers and operations 207

10.4 Solution of quadratic equations 212

10.5 Polar form of a complex number 215

10.6 Applications of complex numbers to AC linear

circuits 218

10.7 Circular motion 219

10.8 The importance of being exponential 226

10.9 Summary 232

10.10 Exercises 235

11 Maxima and minima and sketching functions 237

11.1 Introduction 237

11.2 Stationary points, local maxima and

minima 237

11.3 Graph sketching by analysing the function

behaviour 244

11.4 Summary 251

11.5 Exercises 252

12 Sequences and series 254

12.1 Introduction 254

12.2 Sequences and series definitions 254

12.3 Arithmetic progression 259

12.4 Geometric progression 262

12.5 Pascal’s triangle and the binomial series 267

12.6 Power series 272

12.7 Limits and convergence 282

12.8 Newton–Raphson method for solving

equations 283

12.9 Summary 287

12.10 Exercises 289

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Part 2 Systems

13 Systems of linear equations, matrices, and

determinants 295

13.1 Introduction 295

13.2 Matrices 295

13.3 Transformations 306

13.4 Systems of equations 314

13.5 Gauss elimination 324

13.6 The inverse and determinant of a 3 × 3

matrix 330

13.7 Eigenvectors and eigenvalues 335

13.8 Least squares data fitting 338

13.9 Summary 342

13.10 Exercises 343

14 Differential equations and difference equations 346

14.1 Introduction 346

14.2 Modelling simple systems 347

14.3 Ordinary differential equations 352

14.4 Solving first-order LTI systems 358

14.5 Solution of a second-order LTI systems 363

14.6 Solving systems of differential equations 372

14.7 Difference equations 376

14.8 Summary 378

14.9 Exercises 380

15 Laplace and z transforms 382

15.1 Introduction 382

15.2 The Laplace transform – definition 382

15.3 The unit step function and the (impulse) delta

function 384

15.4 Laplace transforms of simple functions and

properties of the transform 386

15.5 Solving linear differential equations with constant

coefficients 394

15.6 Laplace transforms and systems theory 397

15.7 z transforms 403

15.8 Solving linear difference equations with constant

coefficients using z transforms 408

15.9 z transforms and systems theory 411

15.10 Summary 414

15.11 Exercises 415

16 Fourier series 418

16.1 Introduction 418

16.2 Periodic Functions 418

16.3 Sine and cosine series 419

16.4 Fourier series of symmetric periodic

functions 424

16.5 Amplitude and phase representation of a Fourier

series 426

16.6 Fourier series in complex form 428

16.7 Summary 430

16.8 Exercises 431

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Part 3 Functions of more than one variable

17 Functions of more than one variable 435

17.1 Introduction 435

17.2 Functions of two variables – surfaces 435

17.3 Partial differentiation 436

17.4 Changing variables – the chain rule 438

17.5 The total derivative along a path 440

17.6 Higher-order partial derivatives 443

17.7 Summary 444

17.8 Exercises 445

18 Vector calculus 446

18.1 Introduction 446

18.2 The gradient of a scalar field 446

18.3 Differentiating vector fields 449

18.4 The scalar line integral 451

18.5 Surface integrals 454

18.6 Summary 456

18.7 Exercises 457

Part 4 Graph and language theory

19 Graph theory 461

19.1 Introduction 461

19.2 Definitions 461

19.3 Matrix representation of a graph 465

19.4 Trees 465

19.5 The shortest path problem 468

19.6 Networks and maximum flow 471

19.7 State transition diagrams 474

19.8 Summary 476

19.9 Exercises 477

20 Language theory 479

20.1 Introduction 479

20.2 Languages and grammars 480

20.3 Derivations and derivation trees 483

20.4 Extended Backus-Naur Form (EBNF) 485

20.5 Extensible markup language (XML) 487

20.6 Summary 489

20.7 Exercises 489

Part 5 Probability and statistics

21 Probability and statistics 493

21.1 Introduction 493

21.2 Population and sample, representation of data, mean,

variance and standard deviation 494

21.3 Random systems and probability 501

21.4 Addition law of probability 505

21.5 Repeated trials, outcomes, and

probabilities 508

21.6 Repeated trials and probability trees 508

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21.7 Conditional probability and probability

trees 511

21.8 Application of the probability laws to the probability

of failure of an electrical circuit 514

21.9 Statistical modelling 516

21.10 The normal distribution 517

21.11 The exponential distribution 521

21.12 The binomial distribution 524

21.13 The Poisson distribution 526

21.14 Summary 528

21.15 Exercises 531

Answers to exercises 533

Index 542

## Details – Mathematics for Electrical Engineering and Computing

No. of pages: 576Language: EnglishCopyright: © Newnes 2003Published: 30th June 2003Imprint: NewnesPaperback ISBN: 9780750658553eBook ISBN: 9780080473406

### Preface – Mathematics for Electrical Engineering and Computing

This book is based on my notes from lectures to students of electrical, electronic, and computer engineering at South Bank University. It presents a first year degree/diploma course in engineering mathematics with an emphasis on important concepts, such as algebraic structure, symmetries, linearity, and inverse problems, clearly presented in an accessible style. It encompasses the requirements, not only of students with a good maths grounding, but also of those who, with enthusiasm and motivation, can make up the necessary knowledge. Engineering applications are integrated at each opportunity. Situations where a computer should be used to perform calculations are indicated and ‘hand’ calculations are encouraged only in order to illustrate methods and important special cases. Algorithmic procedures are discussed with reference to their efficiency and convergence, with a presentation appropriate to someone new to computational methods. Developments in the fields of engineering, particularly the extensive use of computers and microprocessors, have changed the necessary subject emphasis within mathematics. This has meant incorporating areas such as Boolean algebra, graph and language theory, and logic into the content. A particular area of interest is digital signal processing, with applications as diverse as medical, control and structural engineering, non-destructive testing, and geophysics. An important consideration when writing this book was to give more prominence to the treatment of discrete functions (sequences), solutions of difference equations and z transforms, and also to contextualize the mathematics within a systems approach to engineering problems.

### Acknowledgements – Mathematics for Electrical Engineering and Computing

I should like to thank my former colleagues in the School of Electrical, Electronic and Computer Engineering at South Bank University who supported and encouraged me with my attempts to re-think approaches to the teaching of engineering mathematics. I should like to thank all the reviewers for their comments and the editorial and production staff at Elsevier Science. Many friends have helped out along the way, by discussing ideas or reading chapters. Above all Gabrielle Sinnadurai who checked the original manuscript of Engineering Mathematics Exposed, wrote the major part of the solutions manual and came to the rescue again by reading some of the new material in this publication. My partner Michael has given unstinting support throughout and without him I would never have found the energy

## About the Author

### Mary Attenborough

### Affiliations and Expertise

The Webbery – Internet development, Co. Donegal, Ireland.