082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

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viii ContentsChapter4 Coordinatesystems 1544.1 Introduction 1544.2 Cartesiancoordinate system – two dimensions 1544.3 Cartesiancoordinate system – three dimensions 1574.4 Polar coordinates 1594.5 Some simple polar curves 1634.6 Cylindricalpolar coordinates 1664.7 Spherical polar coordinates 170Review exercises 4 173Chapter5 Discretemathematics 1755.1 Introduction 1755.2 Set theory 1755.3 Logic 1835.4 Boolean algebra 185Review exercises 5 197Chapter6 Sequencesandseries 2006.1 Introduction 2006.2 Sequences 2016.3 Series 2096.4 The binomial theorem 2146.5 Power series 2186.6 Sequences arising from the iterativesolutionof non-linear equations 219Review exercises 6 222Chapter7 Vectors 2247.1 Introduction 2247.2 Vectors and scalars: basic concepts 2247.3 Cartesiancomponents 2327.4 Scalar fields and vector fields 2407.5 The scalar product 2417.6 The vector product 2467.7 Vectors ofndimensions 253Review exercises 7 255Chapter8 Matrixalgebra 2578.1 Introduction 2578.2 Basic definitions 258
Contents ix8.3 Addition,subtraction and multiplication 2598.4 Using matrices in the translationand rotation of vectors 2678.5 Some special matrices 2718.6 The inverse of a 2 × 2 matrix 2748.7 Determinants 2788.8 The inverse of a 3 × 3 matrix 2818.9 Applicationto the solution of simultaneous equations 2838.10 Gaussian elimination 2868.11 Eigenvaluesand eigenvectors 2948.12 Analysis of electricalnetworks 3078.13 Iterative techniquesfor the solution of simultaneous equations 3128.14 Computer solutions of matrix problems 319Review exercises 8 321Chapter9 Complexnumbers 3249.1 Introduction 3249.2 Complex numbers 3259.3 Operations with complex numbers 3289.4 Graphical representationof complex numbers 3329.5 Polar form of a complex number 3339.6 Vectors and complex numbers 3369.7 The exponential form of a complex number 3379.8 Phasors 3409.9 De Moivre’s theorem 3449.10 Loci and regions of the complex plane 351Review exercises 9 354Chapter10 Differentiation 35610.1 Introduction 35610.2 Graphical approach to differentiation 35710.3 Limits and continuity 35810.4 Rate of change at a specific point 36210.5 Rate of change at a general point 36410.6 Existence of derivatives 37010.7 Common derivatives 37210.8 Differentiationas a linear operator 375Review exercises 10 385Chapter11 Techniquesofdifferentiation 38611.1 Introduction 386
Page 1 and 2: ENGINEERINGMATHEMATICSA Foundation
Page 3 and 4: At Pearson, we have a simple missio
Page 5 and 6: PEARSONEDUCATIONLIMITEDEdinburghGat
Page 7: ContentsPrefaceAcknowledgementsxvii
Page 11 and 12: Contents xiChapter17 Numericalinteg
Page 13 and 14: Contents xiiiChapter24 TheFouriertr
Page 15 and 16: Contents xvAppendixI Representing a
Page 17 and 18: PrefaceAudienceThis book has been w
Page 19 and 20: AcknowledgementsWearegratefultothef
Page 21 and 22: 1 ReviewofalgebraictechniquesConten
Page 23 and 24: 1.2 Laws of indices 3So6 2 6 3 =6 5
Page 25 and 26: 1.2 Laws of indices 5(c)(d)x 9x 5 =
Page 27 and 28: 1.2 Laws of indices 71.2.4 Multiple
Page 29 and 30: 1.2 Laws of indices 9Similarly(2 1/
Page 31 and 32: 1.3 Number bases 11Solutions1 (a) 8
Page 33 and 34: 1.3 Number bases 13giving83=64+16+3
Page 35 and 36: 1.3 Number bases 15Example1.16 Conv
Page 37 and 38: 1.3 Number bases 17Table1.4isusedto
Page 39 and 40: 1.3 Number bases 19The display show
Page 41 and 42: 1.4 Polynomial equations 21Example1
Page 43 and 44: 1.4 Polynomial equations 23We now i
Page 45 and 46: so thatβ=−13Bycomparing coeffici
Page 47 and 48: 1.5.1 Properandimproperfractions1.5
Page 49 and 50: 1.5 Algebraic fractions 29(b) Facto
Page 51 and 52: 1.5 Algebraic fractions 31and4x +2
Page 53 and 54: 1.6 Solution of inequalities 33(g)(
Page 55 and 56: 1.6 Solution of inequalities 35Exam
Page 57 and 58: 1.6 Solution of inequalities 37Case
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1.7 Partial fractions 391.7 PARTIAL
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1.7 Partial fractions 41Thus we hav
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1.7 Partial fractions 43Solution Th
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1.7 Partial fractions 45EXERCISES1.
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1.8 Summation notation 47Engineerin
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1.8 Summation notation 49thecircuit
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Review exercises 1 513 Removethe br
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Review exercises 1 53[ ((d) 3 z −
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2.2 Numbers and intervals 55in orde
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2.3 Basic concepts of functions 57f
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2.3 Basic concepts of functions 59s
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2.3 Basic concepts of functions 61b
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2.3 Basic concepts of functions 63E
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2.3 Basic concepts of functions 65f
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2.3 Basic concepts of functions 67T
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2.3 Basic concepts of functions 691
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2.4 Review of some common engineeri
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2.4.2 Rationalfunctions2.4 Review o
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Table2.2Values ofa x fora = 0.5, 2
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2.4.4 Logarithmfunctions2.4 Review
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Review exercises 2 113REVIEWEXERCIS
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3 ThetrigonometricfunctionsContents
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Note thattanθ = BCAB = BCAC × ACA
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3.3 The trigonometric ratios 119yyB
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3.4 The sine, cosine and tangent fu
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3.5 The sincxfunction 123EXERCISES3
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3.6 Trigonometric identities 125ors
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3.6 Trigonometric identities 127Exa
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3.6 Trigonometric identities 129Exa
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3.7 Modelling waves using sint and
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3.8 Trigonometric equations 145sin
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3.8 Trigonometric equations 147tan
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Usingascientific calculator wehavez
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Review exercises 3 1516 Simplify th
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Review exercises 3 153(d) 1.581sin(
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4.2 Cartesian coordinate system --
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4.3 Cartesian coordinate system --
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4.4 Polar coordinates 1594.4 POLARC
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4.4 Polar coordinates 161yOurxFigur
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4.5 Some simple polar curves 163EXE
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4.5 Some simple polar curves 165uAn
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4.6 Cylindrical polar coordinates 1
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4.6 Cylindrical polar coordinates 1
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4.7 Spherical polar coordinates 171
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Review exercises 4 173zzFeeding lin
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5 DiscretemathematicsContents 5.1 I
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5.2 Set theory 177Example5.1 Useset
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5.2 Set theory 179EMNM fNfFigure5.2
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5.2 Set theory 181someelementsinA.W
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5.3 Logic 183CDCD(a)(b)CDCD(c)(d)(e
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5.4 Boolean algebra 185ABA . BFigur
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5.4 Boolean algebra 187Table5.12Tru
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5.4 Boolean algebra 189However, we
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5.4 Boolean algebra 191S0Stage0A0 B
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5.4 Boolean algebra 193V DDPMOSQ1DD
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5.4 Boolean algebra 195(d) A ·B·(
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Review exercises 5 19712 (a) A·B+A
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Review exercises 5 1997 (a) (A·B)
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6.2 Sequences 2016.2 SEQUENCESAsequ
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6.2 Sequences 203sin t1x[k]1- 3p-2p
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6.2 Sequences 205Ageneralgeometricp
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6.2 Sequences 207can sensibly letk
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6.3 Series 20913 ±12814 315 016 (a
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6.3.2 Sumofafinitegeometricseries6.
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6.3 Series 213whereS ∞is known as
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6.4 The binomial theorem 215Example
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6.4 The binomial theorem 217EXERCIS
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6.6 Sequences arising from the iter
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6.6 Sequences arising from the iter
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Review exercises 6 22315 (a) Write
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7.2 Vectors and scalars: basic conc
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yjOirFigure7.19Thex--y plane withpo
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7.3 Cartesian components 235Solutio
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7.3 Cartesian components 237Enginee
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7.3 Cartesian components 2393 Ifa=4
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7.5 The scalar product 241OEFigure7
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7.5 The scalar product 243Example7.
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----------7.5 The scalar product 24
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7.6 The vector product 247a 3 bubPl
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7.6 The vector product 249The k com
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7.6 The vector product 251Engineeri
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7.7 Vectors ofndimensions 253andH =
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Review exercises 7 255EXERCISES7.71
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8 MatrixalgebraContents 8.1 Introdu
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8.3 Addition, subtraction and multi
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Example8.8 IfB =( ) 123andC=456⎛
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8.4 Using matrices in the translati
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8.4 Using matrices in the translati
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8.5 Some special matrices 271V new=
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Example8.18 IfA =Solution We haveA
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8.6.1 FindingtheinverseofamatrixFor
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8.6 The inverse of a 2 × 2 matrix
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8.7 Determinants 279In addition to
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8.8 The inverse of a 3 × 3 matrix
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8.9 Application to the solution of
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8.9 Application to the solution of
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Solution Firstconsider the equation
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8.10 Gaussian elimination 289anythi
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Eliminating the unwanted values int
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⎛R 1R 2→R 2−6R 1⎝R 3→R 3
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8.11 Eigenvalues and eigenvectors 2
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Itfollows thatso that(2−λ)(4−
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8.12 Analysis of electrical network
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8.13 Iterative techniques for the s
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8.14 Computer solutions of matrix p
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Review exercises 8 321V = [3;−6;4
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Review exercises 8 3234 (a) −8⎛
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9.2 Complex numbers 3259.2 COMPLEXN
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9.2 Complex numbers 3279.2.1 Thecom
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Similarly, wefind, by equating imag
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9.3 Operations with complex numbers
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9.5 Polar form of a complex number
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9.5.1 Multiplicationanddivisioninpo
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9.7 The exponential form of a compl
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9.7 The exponential form of a compl
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9.8 Phasors 341example, in the case
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9.8 Phasors 343Therefore,Ṽ S=ṼR
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which isclearly true,and the theore
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9.9 De Moivre’s theorem 347y25p
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9.9 De Moivre’s theorem 349Now, w
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9.10 Loci and regions of the comple
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9.10 Loci and regions of the comple
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Review exercises 9 3559 Sketch the
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10.2 Graphical approach to differen
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10.3 Limits and continuity 359f (t)
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10.3 Limits and continuity 361Theli
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10.4 Rate of change at a specific p
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10.5 Rate of change at a general po
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10.6 Existence of derivatives 371x
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10.7 Common derivatives 373Table10.
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10.8 Differentiation as a linear op
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Review exercises 10 385REVIEWEXERCI
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11.2 Rules of differentiation 387Th
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11.2 Rules of differentiation 389Th
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(b) Ify = ln(1 −t),theny ′ =
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11.3 Parametric, implicit and logar
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11.4 Higher derivatives 401Example1
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11.4 Higher derivatives 403yCByCAyB
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Review exercises 11 405Solutions1 (
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12.2 Maximum points and minimum poi
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12.3 Points of inflexion 415Solutio
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12.3 Points of inflexion 417y ′ =
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12.4 The Newton--Raphson method for
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12.4 The Newton--Raphson method for
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12.5 Differentiation of vectors 423
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Solution (a) Ifa = 3t 2 i +cos2tj,
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13.2 Elementary integration 42913.2
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13.2 Elementary integration 431Tabl
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Solution (a)sinx+cosx(j)2(k) 2t −
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13.2 Elementary integration 435Engi
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13.2 Elementary integration 4374.00
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Therefore,∫∫ 1sinmtsinnt dt = {
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13.2 Elementary integration 4412 (a
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13.3 Definite and indefinite integr
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Review exercises 13 4536716438 (a)
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Review exercises 13 45522 Calculate
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14 TechniquesofintegrationContents
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14.2 Integration by parts 459Usingt
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Usingintegration by parts we have
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14.3 Integration by substitution 46
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14.3 Integration by substitution 46
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14.4 Integration using partial frac
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Review exercises 14 469∫ π/3(b)
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15 ApplicationsofintegrationContent
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15.2 Average value of a function 47
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15.3 Root mean square value of a fu
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15.3 Root mean square value of a fu
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Review exercises 15 479(d) 0.7071(e
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[Example16.1 Show that f (x) =xandg
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Solution (a) We use the trigonometr
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16.3 Improper integrals 485iC 1 C 2
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∫ 2either of the integrals failst
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16.4 Integral properties of the del
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16.5 Integration of piecewise conti
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16.6 Integration of vectors 49316.6
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Review exercises 16 49512 Given14 G
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17.2 Trapezium rule 497y{ {y 0 y ny
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17.2 Trapezium rule 499The differen
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yABCDE17.3 Simpson’s rule 501quad
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17.3 Simpson’s rule 503Table17.6T
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Review Exercises 17 505EXERCISES17.
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18 Taylorpolynomials,Taylorseriesan
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18.2 Linearization using first-orde
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18.2 Linearization using first-orde
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18.3 Second-order Taylor polynomial
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18.3 Second-order Taylor polynomial
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18.4 Taylor polynomials of the nth
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18.4. Taylor polynomials of the nth
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18.5 Taylor’s formula and the rem
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We know |x| < 1, and so 4x51518.5 T
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18.6 Taylor and Maclaurin series 52
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Review exercises 18 5337 (a) 0, 2,
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19.2 Basic definitions 535In engine
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19.2 Basic definitions 537Note also
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19.2 Basic definitions 539fromwhich
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19.3 First-order equations: simple
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19.4 First-order linear equations:
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and so, upon integrating,so that(co
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19.5 Second-order linear equations
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19.6 Constant coefficient equations
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thatis,β = 332The required particu
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19.7 Series solution of differentia
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19.8 Bessel’s equation and Bessel
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fromwhicha 5=− 124 a 3 = 1192 a 1
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n 5 2n 5 3n5 419.8 Bessel’s equat
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Review exercises 19 601quencies, kn
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20 OrdinarydifferentialequationsIIC
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20.2 Analogue simulation 605y iy i
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20.3 Higher order equations 607High
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20.4 State-space models 609Solution
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20.4 State-space models 611statevar
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20.4 State-space models 613R a L aT
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20.5 Numerical methods 615Combining
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20.6 Euler’s method 617and we can
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20.6 Euler’s method 619Whenx = 1,
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20.7 Improved Euler method 621Table
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20.8 Runge--Kutta method of order 4
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20.8.1 Higherorderequations20.8 Run
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21 TheLaplacetransformContents 21.1
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21.3 Laplace transforms of some com
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21.4 Properties of the Laplace tran
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(b) Usethe first shift theorem with
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21.5 Laplace transform of derivativ
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21.5 Laplace transform of derivativ
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21.6 Inverse Laplace transforms 639
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21.7 Using partial fractions to fin
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21.8 Finding the inverse Laplace tr
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21.8 Finding the inverse Laplace tr
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21.9 The convolution theorem 64721.
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21.9 Solving linear constant coeffi
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21.10 Solving linear constant coeff
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X(s)= s3 −3s 2 −4s+6s 2 (s 2
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21.11 Transfer functions 659(c) x
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21.11 Transfer functions 661R(s)G(s
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21.11 Transfer functions 663R(s) +-
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21.11 Transfer functions 665Fuel va
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21.11 Transfer functions 667u a (t)
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21.12 Poles, zeros and thesplane 66
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21.13 Laplace transforms of some sp
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21.13.2 Periodicfunctions21.13 Lapl
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Review exercises 21 6792 Find the L
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22 Differenceequationsandtheztransf
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22.2 Basic definitions 68322.2.2 Th
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22.2 Basic definitions 685an inhomo
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22.3 Rewriting difference equations
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22.4 Block diagram representation o
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22.4 Block diagram representation o
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22.5 Design of a discrete-time cont
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22.6 Numerical solution of differen
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22.6 Numerical solution of differen
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22.7 Definition of the z transform
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22.7 Definition of the z transform
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22.8 Sampling a continuous signal 7
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22.9 The relationship between the z
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22.9 The relationship between the z
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22.10 Properties of the z transform
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22.11 Inversion of z transforms 715
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22.11 Inversion of z transforms 717
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22.12 The z transform and differenc
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Review exercises 22 721(e) q[k +3]
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23.2 Periodic waveforms 72323.2 PER
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23.2 Periodic waveforms 725f(t)1f(t
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23.3 Odd and even functions 727f(t)
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23.3 Odd and even functions 729f(t)
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23.3 Odd and even functions 731the
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23.5 Fourier series 733Table23.1Som
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23.5 Fourier series 735point out th
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23.5 Fourier series 737but cos(−n
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23.5 Fourier series 739Example23.16
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23.5 Fourier series 741y4- 3π -- -
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23.5 Fourier series 743We conclude
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23.6 Half-range series 745EXERCISES
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23.6 Half-range series 747Half-rang
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23.8 Complex notation 749Engineerin
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23.9 Frequency response of a linear
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23.9 Frequency response of a linear
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̸̸̸Review exercises 23 755The va
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24 TheFouriertransformContents 24.1
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24.2 The Fourier transform -- defin
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24.3 Some properties of the Fourier
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∫ 2[ ] eSolution (a) F(ω)=3 e
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24.3 Some properties of the Fourier
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24.4 Spectra 767uF(v)u2-4p -3p -2p
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The t−ω duality principle 769f(t
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24.6 Fourier transforms of some spe
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24.7 The relationship between the F
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24.8 Convolution and correlation 77
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and theirproduct isF(ω)G(ω) =1(1+
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24.9 The discrete Fourier transform
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24.9 The discrete Fourier transform
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24.10 Derivation of the d.f.t. 787T
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(Solution ConsiderF˜ ω + 2π )for
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24.11 Using the d.f.t.to estimate a
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24.13 Some properties of the d.f.t.
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24.14 The discrete cosine transform
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24.15 Discrete convolution and corr
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25 FunctionsofseveralvariablesConte
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25.3 Partial derivatives 825DCAz =
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25.3 Partial derivatives 827Enginee
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25.4 Higher order derivatives 829
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25.4 Higher order derivatives 831Ex
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25.5 Partial differential equations
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25.6 Taylor polynomials and Taylor
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25.7 Maximum and minimum points of
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Review exercises 25 8474 Find allfi
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26 VectorcalculusContents 26.1 Intr
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26.3 The gradient of a scalar field
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(b) At (0,0,0), ∇φ =0i+3j+0k =3j
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26.3 The gradient of a scalar field
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26.4.1 Physicalinterpretationof ∇
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26.5 The curl of a vector field 859
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26.6 Combining the operators grad,
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26.6 Combining the operators grad,
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Equation3Review exercises 26 865cur
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27 Lineintegralsandmultipleintegral
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27.2 Line integrals 869SolutionThe
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27.3 Evaluation of line integrals i
Page 893 and 894:
27.4 Evaluation of line integrals i
Page 895 and 896:
27.5 Conservative fields and potent
Page 897 and 898:
27.5 Conservative fields and potent
Page 899 and 900:
Substitutingy =x 2 and dy = 2xdx we
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27.6 Double and triple integrals 88
Page 903 and 904:
Page 905 and 906:
Page 907 and 908:
y10GDR27.6 Double and triple integr
Page 909 and 910:
27.7 Some simple volume and surface
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Page 913 and 914:
Page 915 and 916:
27.8 The divergence theorem and Sto
Page 917 and 918:
27.8 The divergence theorem and Sto
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27.9 Maxwell’s equations in integ
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Page 923 and 924:
28 ProbabilityContents 28.1 Introdu
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28.2 Introducing probability 905Ifm
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28.2 Introducing probability 907Eng
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28.3 Mutually exclusive events: the
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28.3 Mutually exclusive events: the
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28.4 Complementary events 913EXERCI
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28.5 Concepts from communication th
Page 937 and 938:
28.5 Concepts from communication th
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28.6 Conditional probability: the m
Page 941 and 942:
28.6 Conditional probability: the m
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P(A ∩C) =P(C ∩A) =P(C)P(A|C)P(A
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28.7 Independent events 9255 Compon
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28.7 Independent events 927Solution
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28.7 Independent events 929E 1,E 2a
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Review exercises 28 931(d) A compon
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29 Statisticsandprobabilitydistribu
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29.3 Probability distributions -- d
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29.4 Probability density functions
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29.5 Mean value 939Example29.3 Find
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29.6 Standard deviation 94129.6 STA
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29.7 Expected value of a random var
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29.7 Expected value of a random var
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29.8 Standard deviation of a random
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29.9 Permutations and combinations
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29.9 Permutations and combinations
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29.10 The binomial distribution 953
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29.10 The binomial distribution 955
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29.11 The Poisson distribution 957I
Page 979 and 980:
Table29.6Theprobabilitiesforbinomia
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29.12 The uniform distribution 9615
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29.14 The normal distribution 963So
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29.14 The normal distribution 965N(
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29.14 The normal distribution 967Ta
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29.14 The normal distribution 969EX
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29.15 Reliability engineering 971in
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29.15 Reliability engineering 973En
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29.15 Reliability engineering 975k
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Review exercises 29 977Solutions1 0
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AppendicesContents I Representingac
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II The Greek alphabet 981f(t)T0 t 0
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Indexabsolute quantity88absorption
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Index 985chord 357--8circuital law
Page 1007 and 1008:
Index 987cycle ofsint 131cycles ofl
Page 1009 and 1010:
Index 989eigenvalues andeigenvector
Page 1011 and 1012:
Index 991spectra 766--8t-w duality
Page 1013 and 1014:
Index 993characteristic impedance o
Page 1015 and 1016:
Index 995exclusive OR gate 189NAND
Page 1017 and 1018:
Index 997second-order 829--30, 833,
Page 1019 and 1020:
Index 999remainder term,Taylor’s
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Index 1001solenoidal vectorfield 85
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Index 1003calculus 849--66curl859--
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082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

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