Mathematics for Electrical Engineering and Computing (eBook)

Cover 1
Frontmatter 2
Half Title Page 2
Title Page 4
Copyright 5
Contents 6
Preface 12
Acknowledgements 13
Part 1: Sets, functions, and calculus 14
1. Sets and functions 16
1.1 Introduction 16
1.2 Sets 17
1.3 Operations on sets 18
1.4 Relations and functions 20
1.5 Combining functions 30
1.6 Summary 36
1.7 Exercises 37
2. Functions and their graphs 39
2.1 Introduction 39
2.2 The straight line: y=mx+c 39
2.3 The quadratic function: y=ax²+bx+c 45
2.4 The function y=1/x 46
2.5 The functions y=ax 46
2.6 Graph sketching using simple transformations 48
2.7 The modulus function, y=|x| or y=abs(x) 54
2.8 Symmetry of functions and their graphs 55
2.9 Solving inequalities 56
2.10 Using graphs to find an expression for the function from experimental data 63
2.11 Summary 67
2.12 Exercises 68
3. Problem solving and the art of the convincing argument 70
3.1 Introduction 70
3.2 Describing a problem in mathematical language 72
3.3 Propositions and predicates 74
3.4 Operations on propositions and predicates 75
3.5 Equivalence 77
3.6 Implication 80
3.7 Making sweeping statements 83
3.8 Other applications of predicates 85
3.9 Summary 86
3.10 Exercises 87
4. Boolean algebra 89
4.1 Introduction 89
4.2 Algebra 89
4.3 Boolean algebras 90
4.4 Digital circuits 94
4.5 Summary 99
4.6 Exercises 99
5. Trigonometric functions and waves 101
5.1 Introduction 101
5.2 Trigonometric functions and radians 101
5.3 Graphs and important properties 104
5.4 Wave functions of time and distance 110
5.5 Trigonometric identities 116
5.6 Superposition 120
5.7 Inverse trigonometric functions 122
5.8 Solving the trigonometric equations sin x=1, cos x=a, tan x=a 123
5.9 Summary 124
5.10 Exercises 126
6. Differentiation 129
6.1 Introduction 129
6.2 The average rate of change and the gradient of a chord 130
6.3 The derivative function 131
6.4 Some common derivatives 133
6.5 Finding the derivative of combinations of functions 135
6.6 Applications of differentiation 141
6.7 Summary 143
6.9 Exercises 144
7. Integration 145
7.1 Introduction 145
7.2 Integration 145
7.3 Finding integrals 146
7.4 Applications of integration 158
7.5 The definite integral 160
7.6 The mean value and r.m.s. value 168
7.7 Numerical Methods of Integration 169
7.8 Summary 172
7.9 Exercises 173
8. The exponential function 175
8.1 Introduction 175
8.2 Exponential growth and decay 175
8.3 The exponential function y=et 179
8.4 The hyperbolic functions 186
8.5 More differentiation and integration 193
8.6 Summary 199
8.7 Exercises 200
9. Vectors 201
9.1 Introduction 201
9.2 Vectors and vector quantities 202
9.3 Addition and subtraction of vectors 204
9.4 Magnitude and direction of a 2D vector — polar co-ordinates 205
9.5 Application of vectors to represent waves (phasors) 208
9.6 Multiplication of a vector by a scalar and unit vectors 210
9.7 Basis vectors 211
9.8 Products of vectors 211
9.9 Vector equation of a line 215
9.10 Summary 216
9.12 Exercises 218
10. Complex numbers 219
10.1 Introduction 219
10.2 Phasor rotation by p/2 219
10.3 Complex numbers and operations 220
10.4 Solution of quadratic equations 225
10.5 Polar form of a complex number 228
10.6 Applications of complex numbers to AC linear circuits 231
10.7 Circular motion 232
10.8 The importance of being exponential 239
10.9 Summary 245
10.10 Exercises 248
11. Maxima and minima and sketching functions 250
11.1 Introduction 250
11.2 Stationary points, local maxima and minima 250
11.3 Graph sketching by analysing the function behaviour 257
11.4 Summary 264
11.5 Exercises 265
12. Sequences and series 267
12.1 Introduction 267
12.2 Sequences and series definitions 267
12.3 Arithmetic progression 272
12.4 Geometric progression 275
12.5 Pascal's triangle and the binomial series 280
12.6 Power series 285
12.7 Limits and convergence 295
12.8 Newton–Raphson method for solving equations 296
12.9 Summary 300
12.10 Exercises 302
Part 2: Systems 306
13. Systems of linear equations, matrices, and determinants 308
13.1 Introduction 308
13.2 Matrices 308
13.3 Transformations 319
13.4 Systems of equations 327
13.5 Gauss elimination 337
13.6 The inverse and determinant of a 3 x 3 matrix 343
13.7 Eigenvectors and eigenvalues 348
13.8 Least squares data fitting 351
13.9 Summary 355
13.10 Exercises 356
14. Differential equations and difference equations 359
14.1 Introduction 359
14.2 Modelling simple systems 360
14.3 Ordinary differential equations 365
14.4 Solving first-order LTI systems 371
14.5 Solution of a second-order LTI systems 376
14.6 Solving systems of differential equations 385
14.7 Difference equations 389
14.8 Summary 391
14.9 Exercises 393
15. Laplace and z transforms 395
15.1 Introduction 395
15.2 The Laplace transform — definition 395
15.3 The unit step function and the (impulse) delta function 397
15.4 Laplace transforms of simple functions and properties of the transform 399
15.5 Solving linear differential equations with constant coefficients 407
15.6 Laplace transforms and systems theory 410
15.7 z transforms 416
15.8 Solving linear difference equations with constant coefficients using z tranforms 421
15.9 z transforms and systems theory 424
15.10 Summary 427
15.11 Exercises 428
16. Fourier series 431
16.1 Introduction 431
16.2 Periodic Functions 431
16.3 Sine and cosine series 432
16.4 Fourier series of symmetric periodic functions 437
16.5 Amplitude and phase representation of a Fourier series 439
16.6 Fourier series in complex form 441
16.7 Summary 443
16.8 Exercises 444
Part 3: Functions of more than one variable 446
17. Function of more than one variable 448
17.1 Introduction 448
17.2 Functions of two variables — surfaces 448
17.3 Partial differentiation 449
17.4 Changing variables — the chain rule 451
17.5 The total derivative along a path 453
17.6 Higher-order partial derivatives 456
17.7 Summary 457
17.8 Exercises 458
18. Vector calculus 459
18.1 Introduction 459
18.2 The gradient of a scalar field 459
18.3 Differentiating vector fields 462
18.4 The scalar line integral 464
18.5 Surface integrals 467
18.6 Summary 469
18.7 Exercises 470
Part 4: Graph and language theory 472
19. Graph theory 474
19.1 Introduction 474
19.2 Definitions 474
19.3 Matrix representation of a graph 478
19.4 Trees 478
19.5 The shortest path problem 481
19.6 Networks and maximum flow 484
19.7 State transition diagrams 487
19.8 Summary 489
19.9 Exercises 490
20. Language theory 492
20.1 Introduction 492
20.2 Languages and grammars 493
20.3 Derivations and derivation trees 496
20.4 Extended Backus-Naur Form (EBNF) 498
20.5 Extensible markup language (XML) 500
20.6 Summary 502
20.7 Exercises 502
Part 5: Probability and statistics 504
21. Probability and statistics 506
21.1 Introduction 506
21.2 Population and sample, representation of data, mean, variance and standard deviation 507
21.3 Random systems and probability 514
21.4 Addition law of probability 518
21.5 Repeated trials, outcomes, and probabilities 521
21.6 Repeated trials and probability trees 521
21.7 Conditional probability and probability trees 524
21.8 Application of the probability laws to the probability of failure of an electrical circuit 527
21.9 Statistical modelling 529
21.10 The normal distribution 530
21.11 The exponential distribution 534
21.12 The binomial distribution 537
21.13 The Poisson distribution 539
21.14 Summary 541
21.15 Exercises 544
Answers to exercises 546
Index 555