Mathematics for Electrical Engineering and Computing
Contents
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
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
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
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
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
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
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