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

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876 Chapter 27 Line integrals and multiple integralsInthree dimensions, ifF =P(x,y,z)i +Q(x,y,z)j +R(x,y,z)kisconservative, wecan writeSoF=∇φ= ∂φ∂x i + ∂φ∂y j + ∂φ∂z kP = ∂φ∂xQ = ∂φ∂yR = ∂φ∂zYou should refer back toExample 26.4 where thisidea was introduced.Example27.6 Thevector field visderivable from the potential φ = 2xy +zx. Find v.Solution If v isderivable from the potential φ, then v = ∇φ and sov=∇φ=(2y+z)i+2xj+xkThis vector field isconservative asiseasilyverified by findingcurl v. Infact,i jk∇×v=∂ ∂ ∂∂x ∂y ∂z∣2y+z2x x∣=0i−(1−1)j+(2−2)k= 0Indeed, recall from Example 26.10 thatcurl (grad φ) isidentically zero forany φ.When the vector field F is conservative there is an alternative method of evaluating theline integral ∫ F·ds which involves the use of the potential function φ. Consider theCfollowing example.Example27.7 Thetwo-dimensional vector fieldF = i +2j isconservative.(a) Findasuitablepotential function φ suchthat F = ∇φ = ∂φ∂x i + ∂φ∂y j.(b) Evaluate ∫ (3,2)F·ds along any convenient path.(0,0)(c) Find the value of φ at B(3, 2), and the value of φ at A(0, 0), and show that thedifferencebetweentheseisequaltothevalueofthelineintegralobtainedinpart(b).Solution (a) We are given that ∂φ = 1 so that φ = x + f (y), where f is an arbitrary function∂xofy. We are also given that ∂φ = 2 so that φ = 2y +g(x), wheregis an arbitrary∂yfunction ofx. Itiseasy toverify that φ =x +2yisasuitablepotential function.
27.5 Conservative fields and potential functions 877(b) Tofind thelineintegralweshallchoosethestraightlinepath,C,joining (0,0) and(3,2). This has equationy = 2 x. The lineintegral becomes3∫ (3,2)(0,0)F·ds====∫ (3,2)(0,0)∫ (3,2)(0,0)∫ (3,2)(0,0)∫ x=3x=0[ ] 7 3=3 x 0= 7(i +2j)·(dxi +dyj)dx+2dydx + 4 3 dx73 dxsince dy = 2 3 dxonC(c) The value of φ at B(3, 2) is 3 + 2(2) = 7. The value of φ at A(0, 0) is 0. Clearlythe difference between these values is 7, the same as the value of the line integralobtained inpart(b).Theresultobtained inthe previousexample isimportant:IfFisaconservative vectorfield suchthat F = ∇φ then, forany points Aand B,∫ BAF· ds = φ(B)−φ(A)27.5.2 ThelineintegralaroundaclosedloopAsecondimportantpropertyofconservativefieldsariseswhenthecurveCofintegrationforms a closed loop. Suppose we evaluate a line integral from A to B firstly along thecurveC 1and secondly alongC 2asshown inFigure 27.7.If the field,F, isconservative bothintegrals will yield the same answer, thatis∫ ∫F·ds= F·dsC 1C 2But∫ ∫F·dsfromAtoB =−F·dsfromBtoAConsequently,thelineintegralaroundtheclosedcurveAtoBandbacktoAmustequalzero.When the pathofintegration isaclosedcurve weuse the symbol∮
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ENGINEERINGMATHEMATICSA Foundation
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At Pearson, we have a simple missio
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PEARSONEDUCATIONLIMITEDEdinburghGat
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ContentsPrefaceAcknowledgementsxvii
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Contents ix8.3 Addition,subtraction
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Contents xiChapter17 Numericalinteg
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Contents xiiiChapter24 TheFouriertr
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Contents xvAppendixI Representing a
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PrefaceAudienceThis book has been w
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AcknowledgementsWearegratefultothef
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1 ReviewofalgebraictechniquesConten
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1.2 Laws of indices 3So6 2 6 3 =6 5
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1.2 Laws of indices 5(c)(d)x 9x 5 =
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1.2 Laws of indices 71.2.4 Multiple
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1.2 Laws of indices 9Similarly(2 1/
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1.3 Number bases 11Solutions1 (a) 8
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1.3 Number bases 13giving83=64+16+3
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1.3 Number bases 15Example1.16 Conv
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1.3 Number bases 17Table1.4isusedto
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1.3 Number bases 19The display show
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1.4 Polynomial equations 21Example1
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1.4 Polynomial equations 23We now i
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so thatβ=−13Bycomparing coeffici
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1.5.1 Properandimproperfractions1.5
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1.5 Algebraic fractions 29(b) Facto
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1.5 Algebraic fractions 31and4x +2
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1.6 Solution of inequalities 33(g)(
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1.6 Solution of inequalities 35Exam
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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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Page 831 and 832:
Page 833 and 834:
Page 835 and 836:
Page 837 and 838:
Page 839 and 840:
Page 841 and 842:
Page 843 and 844:
25 FunctionsofseveralvariablesConte
Page 845 and 846: 25.3 Partial derivatives 825DCAz =
Page 847 and 848: 25.3 Partial derivatives 827Enginee
Page 849 and 850: 25.4 Higher order derivatives 829
Page 851 and 852: 25.4 Higher order derivatives 831Ex
Page 853 and 854: 25.5 Partial differential equations
Page 855 and 856: 25.6 Taylor polynomials and Taylor
Page 861 and 862: 25.7 Maximum and minimum points of
Page 867 and 868: Review exercises 25 8474 Find allfi
Page 869 and 870: 26 VectorcalculusContents 26.1 Intr
Page 871 and 872: 26.3 The gradient of a scalar field
Page 873 and 874: (b) At (0,0,0), ∇φ =0i+3j+0k =3j
Page 875 and 876: 26.3 The gradient of a scalar field
Page 877 and 878: 26.4.1 Physicalinterpretationof ∇
Page 879 and 880: 26.5 The curl of a vector field 859
Page 881 and 882: 26.6 Combining the operators grad,
Page 883 and 884: 26.6 Combining the operators grad,
Page 885 and 886: Equation3Review exercises 26 865cur
Page 887 and 888: 27 Lineintegralsandmultipleintegral
Page 889 and 890: 27.2 Line integrals 869SolutionThe
Page 891 and 892: 27.3 Evaluation of line integrals i
Page 893 and 894: 27.4 Evaluation of line integrals i
Page 895: 27.5 Conservative fields and potent
Page 899 and 900: Substitutingy =x 2 and dy = 2xdx we
Page 901 and 902: 27.6 Double and triple integrals 88
Page 907 and 908: y10GDR27.6 Double and triple integr
Page 909 and 910: 27.7 Some simple volume and surface
Page 915 and 916: 27.8 The divergence theorem and Sto
Page 917 and 918: 27.8 The divergence theorem and Sto
Page 919 and 920: 27.9 Maxwell’s equations in integ
Page 921 and 922: Review exercises 27 901REVIEWEXERCI
Page 923 and 924: 28 ProbabilityContents 28.1 Introdu
Page 925 and 926: 28.2 Introducing probability 905Ifm
Page 927 and 928: 28.2 Introducing probability 907Eng
Page 929 and 930: 28.3 Mutually exclusive events: the
Page 931 and 932: 28.3 Mutually exclusive events: the
Page 933 and 934: 28.4 Complementary events 913EXERCI
Page 935 and 936: 28.5 Concepts from communication th
Page 937 and 938: 28.5 Concepts from communication th
Page 939 and 940: 28.6 Conditional probability: the m
Page 941 and 942: 28.6 Conditional probability: the m
Page 943 and 944: P(A ∩C) =P(C ∩A) =P(C)P(A|C)P(A
Page 945 and 946: 28.7 Independent events 9255 Compon
Page 947 and 948:
28.7 Independent events 927Solution
Page 949 and 950:
28.7 Independent events 929E 1,E 2a
Page 951 and 952:
Review exercises 28 931(d) A compon
Page 953 and 954:
29 Statisticsandprobabilitydistribu
Page 955 and 956:
29.3 Probability distributions -- d
Page 957 and 958:
29.4 Probability density functions
Page 959 and 960:
29.5 Mean value 939Example29.3 Find
Page 961 and 962:
29.6 Standard deviation 94129.6 STA
Page 963 and 964:
29.7 Expected value of a random var
Page 965 and 966:
29.7 Expected value of a random var
Page 967 and 968:
29.8 Standard deviation of a random
Page 969 and 970:
29.9 Permutations and combinations
Page 971 and 972:
29.9 Permutations and combinations
Page 973 and 974:
29.10 The binomial distribution 953
Page 975 and 976:
29.10 The binomial distribution 955
Page 977 and 978:
29.11 The Poisson distribution 957I
Page 979 and 980:
Table29.6Theprobabilitiesforbinomia
Page 981 and 982:
29.12 The uniform distribution 9615
Page 983 and 984:
29.14 The normal distribution 963So
Page 985 and 986:
29.14 The normal distribution 965N(
Page 987 and 988:
29.14 The normal distribution 967Ta
Page 989 and 990:
29.14 The normal distribution 969EX
Page 991 and 992:
29.15 Reliability engineering 971in
Page 993 and 994:
29.15 Reliability engineering 973En
Page 995 and 996:
29.15 Reliability engineering 975k
Page 997 and 998:
Review exercises 29 977Solutions1 0
Page 999 and 1000:
AppendicesContents I Representingac
Page 1001 and 1002:
II The Greek alphabet 981f(t)T0 t 0
Page 1003 and 1004:
Indexabsolute quantity88absorption
Page 1005 and 1006:
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
Page 1021 and 1022:
Index 1001solenoidal vectorfield 85
Page 1023 and 1024:
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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