Mathematics for Electrical Engineering and Computing

562 Pages·2003·3.66 MB·English

by  Mary P Attenborough PhD

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Mathematics for Electrical
Engineering and Computing
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Mathematics for
Electrical Engineering
and Computing
Mary Attenborough
AMSTERDAM BOSTON LONDON HEIDELBERG NEWYORK
OXFORD PARIS SANDIEGO SANFRANCISCO
SINGAPORE SYDNEY TOKYO
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Newnes
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Firstpublished2003
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Contents
Preface xi
Acknowledgements xii
Part1 Sets,functions,andcalculus
1 Setsandfunctions 3
1.1 Introduction 3
1.2 Sets 4
1.3 Operationsonsets 5
1.4 Relationsandfunctions 7
1.5 Combiningfunctions 17
1.6 Summary 23
1.7 Exercises 24
2 Functionsandtheirgraphs 26
2.1 Introduction 26
2.2 Thestraightline:y =mx+c 26
2.3 Thequadraticfunction:y =ax2+bx+c 32
2.4 Thefunctiony =1/x 33
2.5 Thefunctionsy =ax 33
2.6 Graphsketchingusingsimple
transformations 35
2.7 Themodulusfunction,y =|x|or
y =abs(x) 41
2.8 Symmetryoffunctionsandtheirgraphs 42
2.9 Solvinginequalities 43
2.10 Usinggraphstofindanexpressionforthefunction
fromexperimentaldata 50
2.11 Summary 54
2.12 Exercises 55
3 Problemsolvingandtheartoftheconvincing
argument 57
3.1 Introduction 57
3.2 Describingaprobleminmathematical
language 59
3.3 Propositionsandpredicates 61
3.4 Operationsonpropositionsandpredicates 62
3.5 Equivalence 64
3.6 Implication 67
3.7 Makingsweepingstatements 70
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vi Contents
3.8 Otherapplicationsofpredicates 72
3.9 Summary 73
3.10 Exercises 74
4 Booleanalgebra 76
4.1 Introduction 76
4.2 Algebra 76
4.3 Booleanalgebras 77
4.4 Digitalcircuits 81
4.5 Summary 86
4.6 Exercises 86
5 Trigonometricfunctionsandwaves 88
5.1 Introduction 88
5.2 Trigonometricfunctionsandradians 88
5.3 Graphsandimportantproperties 91
5.4 Wavefunctionsoftimeanddistance 97
5.5 Trigonometricidentities 103
5.6 Superposition 107
5.7 Inversetrigonometricfunctions 109
5.8 Solvingthetrigonometricequationssinx =a,
cosx =a,tanx =a 110
5.9 Summary 111
5.10 Exercises 113
6 Differentiation 116
6.1 Introduction 116
6.2 Theaveragerateofchangeandthegradientofa
chord 117
6.3 Thederivativefunction 118
6.4 Somecommonderivatives 120
6.5 Findingthederivativeofcombinationsof
functions 122
6.6 Applicationsofdifferentiation 128
6.7 Summary 130
6.9 Exercises 131
7 Integration 132
7.1 Introduction 132
7.2 Integration 132
7.3 Findingintegrals 133
7.4 Applicationsofintegration 145
7.5 Thedefiniteintegral 147
7.6 Themeanvalueandr.m.s.value 155
7.7 NumericalMethodsofIntegration 156
7.8 Summary 159
7.9 Exercises 160
8 Theexponentialfunction 162
8.1 Introduction 162
8.2 Exponentialgrowthanddecay 162
8.3 Theexponentialfunctiony =et 166
8.4 Thehyperbolicfunctions 173
8.5 Moredifferentiationandintegration 180
8.6 Summary 186
8.7 Exercises 187
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Contents vii
9 Vectors 188
9.1 Introduction 188
9.2 Vectorsandvectorquantities 189
9.3 Additionandsubtractionofvectors 191
9.4 Magnitudeanddirectionofa2Dvector–polar
co-ordinates 192
9.5 Applicationofvectorstorepresentwaves
(phasors) 195
9.6 Multiplicationofavectorbyascalarandunit
vectors 197
9.7 Basisvectors 198
9.8 Productsofvectors 198
9.9 Vectorequationofaline 202
9.10 Summary 203
9.12 Exercises 205
10 Complexnumbers 206
10.1 Introduction 206
10.2 Phasorrotationbyπ/2 206
10.3 Complexnumbersandoperations 207
10.4 Solutionofquadraticequations 212
10.5 Polarformofacomplexnumber 215
10.6 ApplicationsofcomplexnumberstoAClinear
circuits 218
10.7 Circularmotion 219
10.8 Theimportanceofbeingexponential 226
10.9 Summary 232
10.10 Exercises 235
11 Maximaandminimaandsketchingfunctions 237
11.1 Introduction 237
11.2 Stationarypoints,localmaximaand
minima 237
11.3 Graphsketchingbyanalysingthefunction
behaviour 244
11.4 Summary 251
11.5 Exercises 252
12 Sequencesandseries 254
12.1 Introduction 254
12.2 Sequencesandseriesdefinitions 254
12.3 Arithmeticprogression 259
12.4 Geometricprogression 262
12.5 Pascal’striangleandthebinomialseries 267
12.6 Powerseries 272
12.7 Limitsandconvergence 282
12.8 Newton–Raphsonmethodforsolving
equations 283
12.9 Summary 287
12.10 Exercises 289
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viii Contents
Part2 Systems
13 Systemsoflinearequations,matrices,and
determinants 295
13.1 Introduction 295
13.2 Matrices 295
13.3 Transformations 306
13.4 Systemsofequations 314
13.5 Gausselimination 324
13.6 Theinverseanddeterminantofa3×3
matrix 330
13.7 Eigenvectorsandeigenvalues 335
13.8 Leastsquaresdatafitting 338
13.9 Summary 342
13.10 Exercises 343
14 Differentialequationsanddifferenceequations 346
14.1 Introduction 346
14.2 Modellingsimplesystems 347
14.3 Ordinarydifferentialequations 352
14.4 Solvingfirst-orderLTIsystems 358
14.5 Solutionofasecond-orderLTIsystems 363
14.6 Solvingsystemsofdifferentialequations 372
14.7 Differenceequations 376
14.8 Summary 378
14.9 Exercises 380
15 Laplaceandztransforms 382
15.1 Introduction 382
15.2 TheLaplacetransform–definition 382
15.3 Theunitstepfunctionandthe(impulse)delta
function 384
15.4 Laplacetransformsofsimplefunctionsand
propertiesofthetransform 386
15.5 Solvinglineardifferentialequationswithconstant
coefficients 394
15.6 Laplacetransformsandsystemstheory 397
15.7 ztransforms 403
15.8 Solvinglineardifferenceequationswithconstant
coefficientsusingztransforms 408
15.9 ztransformsandsystemstheory 411
15.10 Summary 414
15.11 Exercises 415
16 Fourierseries 418
16.1 Introduction 418
16.2 PeriodicFunctions 418
16.3 Sineandcosineseries 419
16.4 Fourierseriesofsymmetricperiodic
functions 424
16.5 AmplitudeandphaserepresentationofaFourier
series 426
16.6 Fourierseriesincomplexform 428
16.7 Summary 430
16.8 Exercises 431
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Contents ix
Part3 Functionsofmorethanonevariable
17 Functionsofmorethanonevariable 435
17.1 Introduction 435
17.2 Functionsoftwovariables–surfaces 435
17.3 Partialdifferentiation 436
17.4 Changingvariables–thechainrule 438
17.5 Thetotalderivativealongapath 440
17.6 Higher-orderpartialderivatives 443
17.7 Summary 444
17.8 Exercises 445
18 Vectorcalculus 446
18.1 Introduction 446
18.2 Thegradientofascalarfield 446
18.3 Differentiatingvectorfields 449
18.4 Thescalarlineintegral 451
18.5 Surfaceintegrals 454
18.6 Summary 456
18.7 Exercises 457
Part4 Graphandlanguagetheory
19 Graphtheory 461
19.1 Introduction 461
19.2 Definitions 461
19.3 Matrixrepresentationofagraph 465
19.4 Trees 465
19.5 Theshortestpathproblem 468
19.6 Networksandmaximumflow 471
19.7 Statetransitiondiagrams 474
19.8 Summary 476
19.9 Exercises 477
20 Languagetheory 479
20.1 Introduction 479
20.2 Languagesandgrammars 480
20.3 Derivationsandderivationtrees 483
20.4 ExtendedBackus-NaurForm(EBNF) 485
20.5 Extensiblemarkuplanguage(XML) 487
20.6 Summary 489
20.7 Exercises 489
Part5 Probabilityandstatistics
21 Probabilityandstatistics 493
21.1 Introduction 493
21.2 Populationandsample,representationofdata,mean,
varianceandstandarddeviation 494
21.3 Randomsystemsandprobability 501
21.4 Additionlawofprobability 505
21.5 Repeatedtrials,outcomes,and
probabilities 508
21.6 Repeatedtrialsandprobabilitytrees 508
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