Mathematical Methods for Physics and Engineering - Matematica.NET

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30.5 PROPERTIES OF DISTRIBUTIONSIn many circumstances, however, random variables do not depend on oneanother, i.e. they are independent. As an example, for a person drawn at randomfrom a population, we might expect height and IQ to be independent randomvariables. Let us suppose that X and Y are two random variables with probabilitydensity functions g(x) andh(y) respectively. In mathematical terms, X and Y areindependent RVs if their joint probability density function is given by f(x, y) =g(x)h(y). Thus, for independent RVs, if X and Y are both discrete thenPr(X = x i , Y = y j )=g(x i )h(y j )or, if X and Y are both continuous, thenPr(x
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Physicists. He is also a Director o
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cambridge university pressCambridge
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CONTENTS2.2 Integration 59Integrati
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CONTENTS7.7 Equations of lines, pla
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CONTENTS12.2 The Fourier coefficien
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CONTENTS18.6 Spherical Bessel funct
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CONTENTS24.9 Cauchy’s theorem 849
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CONTENTS29.6 Characters 1092Orthogo
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CONTENTSI am the very Model for a S
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PREFACE TO THE THIRD EDITIONthe phy
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Preface to the second editionSince
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Preface to the first editionA knowl
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PREFACE TO THE FIRST EDITIONsupport
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PRELIMINARY ALGEBRAforms an equatio
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PRELIMINARY ALGEBRAmany real roots
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PRELIMINARY ALGEBRAat a value of x
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PRELIMINARY ALGEBRAwhere f 1 (x) is
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PRELIMINARY ALGEBRAIn the case of a
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PRELIMINARY ALGEBRAdrawn through R,
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PRELIMINARY ALGEBRAand use made of
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PRELIMINARY ALGEBRAwith the coordin
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PRELIMINARY ALGEBRAthe well-known r
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PRELIMINARY ALGEBRAnumerators on bo
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PRELIMINARY ALGEBRAWe illustrate th
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PRELIMINARY ALGEBRAIn this form, al
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PRELIMINARY ALGEBRAIn fact, the gen
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PRELIMINARY ALGEBRAThe first is a f
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PRELIMINARY ALGEBRAbe obvious, but
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PRELIMINARY ALGEBRAThis is precisel
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PRELIMINARY ALGEBRA◮The prime int
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PRELIMINARY ALGEBRA1.8 ExercisesPol
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PRELIMINARY ALGEBRA1.16 Express the
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PRELIMINARY ALGEBRA1.11 Show that t
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PRELIMINARY CALCULUSf(x +∆x)AP∆
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PRELIMINARY CALCULUS◮Find from fi
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PRELIMINARY CALCULUSand using (2.6)
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PRELIMINARY CALCULUS◮Find dy/dx i
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PRELIMINARY CALCULUSf(x)QABCSFigure
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PRELIMINARY CALCULUSf(x)GxFigure 2.
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PRELIMINARY CALCULUSrelative to the
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PRELIMINARY CALCULUSf(x)a b cxFigur
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PRELIMINARY CALCULUSIn each case, a
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PRELIMINARY CALCULUSf(x)ax 1 x 2 x
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PRELIMINARY CALCULUSFrom the last t
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PRELIMINARY CALCULUS◮Evaluate the
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PRELIMINARY CALCULUSSincethe requir
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PRELIMINARY CALCULUSThe separation
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PRELIMINARY CALCULUS2.2.10 Infinite
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PRELIMINARY CALCULUS2.2.12 Integral
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PRELIMINARY CALCULUSf(x)y = f(x)∆
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PRELIMINARY CALCULUS◮Find the vol
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PRELIMINARY CALCULUSOcCρr +∆rrρ
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PRELIMINARY CALCULUS(c) [(x − a)/
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PRELIMINARY CALCULUSy2aπa2πaxFigu
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COMPLEX NUMBERS AND HYPERBOLIC FUNC
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SERIES AND LIMITSsome sort of relat
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SERIES AND LIMITSFor a series with
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SERIES AND LIMITSThe difference met
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SERIES AND LIMITS◮Sum the seriesN
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SERIES AND LIMITSAgain using the Ma
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SERIES AND LIMITSwhich is merely th
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SERIES AND LIMITS◮Given that the
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SERIES AND LIMITSThe divergence of
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SERIES AND LIMITSalthough in princi
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SERIES AND LIMITSr = − exp iθ. T
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SERIES AND LIMITS4.6 Taylor seriesT
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SERIES AND LIMITSx = a + h in the a
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SERIES AND LIMITSvalue of ξ that s
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SERIES AND LIMITS◮Evaluate the li
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SERIES AND LIMITSSummary of methods
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SERIES AND LIMITS4.15 Prove that∞
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SERIES AND LIMITSsin 3x(a) limx→0
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SERIES AND LIMITS4.15 Divide the se
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PARTIAL DIFFERENTIATIONto x and y r
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PARTIAL DIFFERENTIATIONcan be obtai
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PARTIAL DIFFERENTIATIONit exact. Co
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PARTIAL DIFFERENTIATIONFrom equatio
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PARTIAL DIFFERENTIATIONThus, from (
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PARTIAL DIFFERENTIATIONtheorem then
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PARTIAL DIFFERENTIATIONTo establish
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PARTIAL DIFFERENTIATIONmaximum0.40.
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PARTIAL DIFFERENTIATIONvaried. Howe
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PARTIAL DIFFERENTIATION◮Find the
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PARTIAL DIFFERENTIATION◮A system
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PARTIAL DIFFERENTIATIONP 1PP 2yf(x,
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PARTIAL DIFFERENTIATION5.11 Thermod
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PARTIAL DIFFERENTIATIONAlthough the
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PARTIAL DIFFERENTIATION(a) Find all
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PARTIAL DIFFERENTIATIONthe horizont
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PARTIAL DIFFERENTIATIONBy consideri
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PARTIAL DIFFERENTIATION5.19 The cos
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MULTIPLE INTEGRALSydSdxdA = dxdyRVd
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MULTIPLE INTEGRALSy1dyRx + y =100dx
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MULTIPLE INTEGRALSzcdV = dx dy dzdz
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MULTIPLE INTEGRALSzz =2yz = x 2 + y
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MULTIPLE INTEGRALSza√a2 − z 2dz
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MULTIPLE INTEGRALSaθdCFigure 6.8Su
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MULTIPLE INTEGRALSyu =constantv =co
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MULTIPLE INTEGRALS◮Evaluate the d
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MULTIPLE INTEGRALSzRTu = c 1v = c 2
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MULTIPLE INTEGRALSwhich agrees with
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MULTIPLE INTEGRALS6.6 The function(
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MULTIPLE INTEGRALSover the ellipsoi
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7Vector algebraThis chapter introdu
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VECTOR ALGEBRAabcb + cbcab + ca +(b
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VECTOR ALGEBRACEAGFDacBbOFigure 7.6
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VECTOR ALGEBRAkaja z ka y ja x iiFi
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VECTOR ALGEBRAFrom (7.15) we see th
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VECTOR ALGEBRAa × bθbaFigure 7.9s
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VECTOR ALGEBRAis the forward direct
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VECTOR ALGEBRA◮Find the volume V
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VECTOR ALGEBRAˆnARadrOFigure 7.13
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VECTOR ALGEBRAPp − apdAbθaOFigur
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VECTOR ALGEBRAQbqˆnPpaOFigure 7.16
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VECTOR ALGEBRAnot coplanar. Moreove
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VECTOR ALGEBRA7.12 The plane P 1 co
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VECTOR ALGEBRA7.22 In subsection 7.
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VECTOR ALGEBRAof vector plots for p
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MATRICES AND VECTOR SPACESa discuss
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MATRICES AND VECTOR SPACESWe reiter
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MATRICES AND VECTOR SPACES8.1.3 Som
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MATRICES AND VECTOR SPACESmay be th
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MATRICES AND VECTOR SPACESIn a simi
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MATRICES AND VECTOR SPACES◮The ma
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MATRICES AND VECTOR SPACESThese are
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MATRICES AND VECTOR SPACES◮Find t
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MATRICES AND VECTOR SPACESthe right
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MATRICES AND VECTOR SPACESdetermina
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MATRICES AND VECTOR SPACESIt follow
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MATRICES AND VECTOR SPACESequivalen
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MATRICES AND VECTOR SPACESand may b
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MATRICES AND VECTOR SPACESmay be sh
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MATRICES AND VECTOR SPACESClearly r
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MATRICES AND VECTOR SPACESHence 〈
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MATRICES AND VECTOR SPACESWe also s
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MATRICES AND VECTOR SPACESa result
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MATRICES AND VECTOR SPACESHence λ
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MATRICES AND VECTOR SPACES8.14 Dete
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MATRICES AND VECTOR SPACES◮Constr
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MATRICES AND VECTOR SPACESComparing
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MATRICES AND VECTOR SPACESthat is,
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MATRICES AND VECTOR SPACES| exp A|.
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MATRICES AND VECTOR SPACESalso. Ano
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MATRICES AND VECTOR SPACES8.17.2 Qu
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MATRICES AND VECTOR SPACESIf a vect
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MATRICES AND VECTOR SPACES◮Show t
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MATRICES AND VECTOR SPACESThis set
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MATRICES AND VECTOR SPACESthe uniqu
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MATRICES AND VECTOR SPACESthe numbe
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MATRICES AND VECTOR SPACESnon-zero
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MATRICES AND VECTOR SPACESUsing the
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MATRICES AND VECTOR SPACES8.3 Using
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MATRICES AND VECTOR SPACES(b) find
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MATRICES AND VECTOR SPACES8.26 Show
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MATRICES AND VECTOR SPACES8.40 Find
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9Normal modesAny student of the phy
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NORMAL MODESP P Pθ 1θ 2θ 2lθ 1
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NORMAL MODESfrequency corresponds t
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NORMAL MODESThe final and most comp
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NORMAL MODESThe potential matrix is
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NORMAL MODESneous equations for α
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NORMAL MODESbe shown that they do p
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NORMAL MODESunder gravity. At time
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NORMAL MODES9.8 (It is recommended
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10Vector calculusIn chapter 7 we di
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VECTOR CALCULUSyê φjê ρρiφxFi
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VECTOR CALCULUSThe order of the fac
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VECTOR CALCULUSzCˆnPˆtˆbr(u)OyxF
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VECTOR CALCULUSTherefore, rememberi
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VECTOR CALCULUSFinally, we note tha
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VECTOR CALCULUStotal derivative, th
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VECTOR CALCULUSmathematical point o
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VECTOR CALCULUS◮For the function
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VECTOR CALCULUSIn addition to these
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VECTOR CALCULUS∇(φ + ψ) =∇φ
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VECTOR CALCULUSa is a vector field,
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VECTOR CALCULUSρ, φ, z, wherex =
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VECTOR CALCULUS∇Φ = ∂Φ∂ρ
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VECTOR CALCULUSand r ≥ 0, 0 ≤
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VECTOR CALCULUS10.10 General curvil
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VECTOR CALCULUSFor orthogonal coord
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VECTOR CALCULUS◮Prove the express
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VECTOR CALCULUS10.3 The general equ
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VECTOR CALCULUSUse this formula to
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VECTOR CALCULUS10.21 Paraboloidal c
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VECTOR CALCULUS10.23 The tangent ve
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LINE, SURFACE AND VOLUME INTEGRALSE
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LINE, SURFACE AND VOLUME INTEGRALSy
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LINE, SURFACE AND VOLUME INTEGRALSi
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LINE, SURFACE AND VOLUME INTEGRALSw
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LINE, SURFACE AND VOLUME INTEGRALSS
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LINE, SURFACE AND VOLUME INTEGRALSw
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LINE, SURFACE AND VOLUME INTEGRALSd
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LINE, SURFACE AND VOLUME INTEGRALSI
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LINE, SURFACE AND VOLUME INTEGRALS1
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LINE, SURFACE AND VOLUME INTEGRALSz
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LINE, SURFACE AND VOLUME INTEGRALSS
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LINE, SURFACE AND VOLUME INTEGRALSi
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FOURIER SERIESf(x)xLLFigure 12.1 An
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FOURIER SERIESapply for r = 0 as we
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FOURIER SERIESare not used as often
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FOURIER SERIES(a)0L(b)0L2L(c)0L2L(d
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FOURIER SERIESconverge to the corre
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FOURIER SERIES12.8 Parseval’s the
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FOURIER SERIESbe better for numeric
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FOURIER SERIES12.21 Find the comple
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FOURIER SERIES12.21 c n =[(−1) n
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INTEGRAL TRANSFORMSc(ω)expiωt−
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INTEGRAL TRANSFORMS◮Find the Four
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INTEGRAL TRANSFORMSYyk ′k0θx−Y
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INTEGRAL TRANSFORMSequals zero. Thi
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INTEGRAL TRANSFORMS◮Prove relatio
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INTEGRAL TRANSFORMS(i) Differentiat
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INTEGRAL TRANSFORMSg(y)(a)(b)(c)(d)
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INTEGRAL TRANSFORMSgiven by∫1 ∞
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INTEGRAL TRANSFORMS◮Prove the Wie
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INTEGRAL TRANSFORMStwo- or three-di
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INTEGRAL TRANSFORMS(iii) Once again
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INTEGRAL TRANSFORMS◮Find the Lapl
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INTEGRAL TRANSFORMSFigure 13.7text)
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INTEGRAL TRANSFORMS13.4 Exercises13
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INTEGRAL TRANSFORMS13.10 In many ap
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INTEGRAL TRANSFORMS13.18 The equiva
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INTEGRAL TRANSFORMS13.27 The functi
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14First-order ordinary differential
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FIRST-ORDER ORDINARY DIFFERENTIAL E
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15Higher-order ordinary differentia
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HIGHER-ORDER ORDINARY DIFFERENTIAL
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SERIES SOLUTIONS OF ORDINARY DIFFER
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17Eigenfunction methods fordifferen
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EIGENFUNCTION METHODS FOR DIFFERENT
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SPECIAL FUNCTIONSwhich on collectin
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SPECIAL FUNCTIONSwhere P l (x) is a
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SPECIAL FUNCTIONSwhich reduces to(x
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SPECIAL FUNCTIONS◮Prove the expre
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SPECIAL FUNCTIONSr and r ′ must b
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SPECIAL FUNCTIONSin (18.3) and (18.
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SPECIAL FUNCTIONSto be zero, since
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SPECIAL FUNCTIONSGenerating functio
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SPECIAL FUNCTIONSorthonormal set, i
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SPECIAL FUNCTIONSand has three regu
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SPECIAL FUNCTIONS4U 2U 32U 0U 1−1
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SPECIAL FUNCTIONSThe normalisation,
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SPECIAL FUNCTIONSUsing (18.65) and
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SPECIAL FUNCTIONSthe form of a Frob
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SPECIAL FUNCTIONS1.51J 0J 1J 20.52
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SPECIAL FUNCTIONS10.5Y 0Y 1 Y22 4 6
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SPECIAL FUNCTIONSevaluated using l
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SPECIAL FUNCTIONSFinally, subtracti
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SPECIAL FUNCTIONSUsing de Moivre’
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SPECIAL FUNCTIONS◮Show that the l
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SPECIAL FUNCTIONS105L 2L 3L 0L 11 2
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SPECIAL FUNCTIONSThe above orthogon
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SPECIAL FUNCTIONSIn particular, we
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SPECIAL FUNCTIONSwhere, in the seco
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SPECIAL FUNCTIONS18.9.1 Properties
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SPECIAL FUNCTIONSDifferentiating th
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SPECIAL FUNCTIONSgamma function. §
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SPECIAL FUNCTIONSwhere in the secon
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SPECIAL FUNCTIONSsecond solution to
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SPECIAL FUNCTIONSThe gamma function
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SPECIAL FUNCTIONSIf we let x = n +
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SPECIAL FUNCTIONSwhich is the requi
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SPECIAL FUNCTIONSand hence that the
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SPECIAL FUNCTIONSDeduce the value o
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SPECIAL FUNCTIONS18.24 The solution
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19Quantum operatorsAlthough the pre
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QUANTUM OPERATORSis to produce a sc
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QUANTUM OPERATORSIf A| a n 〉 = a|
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QUANTUM OPERATORSSimple identities
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QUANTUM OPERATORSlater algebraic co
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QUANTUM OPERATORSRHS gives(−i) 2
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QUANTUM OPERATORSwith[L 2 ,L z]=[L2
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QUANTUM OPERATORSoperate repeatedly
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QUANTUM OPERATORS19.2.2 Uncertainty
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QUANTUM OPERATORShence formally an
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QUANTUM OPERATORSan arbitrary compl
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QUANTUM OPERATORSThe proof, which m
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QUANTUM OPERATORSFor a particle of
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QUANTUM OPERATORS19.4 Hints and ans
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PDES: GENERAL AND PARTICULAR SOLUTI
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PDES: SEPARATION OF VARIABLES AND O
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CALCULUS OF VARIATIONSyabxFigure 22
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CALCULUS OF VARIATIONSB(b, y(b))dsd
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CALCULUS OF VARIATIONSwe can use (2
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CALCULUS OF VARIATIONS22.3.1 Severa
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CALCULUS OF VARIATIONSAx = x 0xyBFi
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CALCULUS OF VARIATIONS−ayOaxFigur
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CALCULUS OF VARIATIONSyBn 2xθ 1θ
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CALCULUS OF VARIATIONSUsing (22.13)
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CALCULUS OF VARIATIONS◮Show that
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CALCULUS OF VARIATIONSy(x)1(c)0.80.
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CALCULUS OF VARIATIONSoperator H is
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CALCULUS OF VARIATIONS22.8 Derive t
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CALCULUS OF VARIATIONS22.23 For the
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CALCULUS OF VARIATIONS22.5 (a) ∂x
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INTEGRAL EQUATIONSWe shall illustra
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INTEGRAL EQUATIONSinhomogeneous Fre
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INTEGRAL EQUATIONSThese two simulta
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INTEGRAL EQUATIONSThus, using (23.1
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INTEGRAL EQUATIONSSubstituting (23.
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INTEGRAL EQUATIONSwe may write the
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INTEGRAL EQUATIONSNeumann series, w
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INTEGRAL EQUATIONSsides of (23.51)
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INTEGRAL EQUATIONS23.5 Solve for φ
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INTEGRAL EQUATIONSBy examining the
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24Complex variablesThroughout this
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COMPLEX VARIABLESWe then find that[
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COMPLEX VARIABLESFor f to be differ
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COMPLEX VARIABLESwhere i and j are
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COMPLEX VARIABLESwhich is an altern
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COMPLEX VARIABLESderived from them
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COMPLEX VARIABLESy Cy yrθxrθxxC
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COMPLEX VARIABLESwhere a is a finit
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COMPLEX VARIABLESyz 1z 2sC ′ 1w 1
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COMPLEX VARIABLESysi Pw = g(z)Q R S
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COMPLEX VARIABLESysibw 3w = g(z)φ
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COMPLEX VARIABLESyBC 2C 1xC 3AFigur
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COMPLEX VARIABLESmust be made in te
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COMPLEX VARIABLESyBC 1RC 2AxFigure
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COMPLEX VARIABLEScontour C and z 0
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COMPLEX VARIABLESwhere a n is given
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COMPLEX VARIABLESyRz 0C 1C 2xFigure
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COMPLEX VARIABLESDifferentiating bo
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COMPLEX VARIABLESCC ′C(a)(b)Figur
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COMPLEX VARIABLESformula (24.56) wi
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COMPLEX VARIABLESyΓγ−RORxFigure
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COMPLEX VARIABLESyΓγABCDxFigure 2
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COMPLEX VARIABLES24.7 Find the real
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COMPLEX VARIABLES24.22 The equation
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APPLICATIONS OF COMPLEX VARIABLESyx
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APPLICATIONS OF COMPLEX VARIABLESis
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APPLICATIONS OF COMPLEX VARIABLES
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APPLICATIONS OF COMPLEX VARIABLESde
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APPLICATIONS OF COMPLEX VARIABLESwh
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APPLICATIONS OF COMPLEX VARIABLESyY
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APPLICATIONS OF COMPLEX VARIABLESIm
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APPLICATIONS OF COMPLEX VARIABLESId
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APPLICATIONS OF COMPLEX VARIABLES25
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APPLICATIONS OF COMPLEX VARIABLESfo
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APPLICATIONS OF COMPLEX VARIABLESIm
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APPLICATIONS OF COMPLEX VARIABLESx-
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APPLICATIONS OF COMPLEX VARIABLESwh
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APPLICATIONS OF COMPLEX VARIABLES
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APPLICATIONS OF COMPLEX VARIABLESTh
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APPLICATIONS OF COMPLEX VARIABLESex
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APPLICATIONS OF COMPLEX VARIABLESFi
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APPLICATIONS OF COMPLEX VARIABLESan
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APPLICATIONS OF COMPLEX VARIABLESsi
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APPLICATIONS OF COMPLEX VARIABLESFi
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APPLICATIONS OF COMPLEX VARIABLESfr
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APPLICATIONS OF COMPLEX VARIABLESst
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APPLICATIONS OF COMPLEX VARIABLESvv
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APPLICATIONS OF COMPLEX VARIABLESAV
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APPLICATIONS OF COMPLEX VARIABLESTh
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TENSORS26.1 Some notationBefore pro
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TENSORSScalars behave differently u
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TENSORS26.4 First- and zero-order C
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TENSORSIn fact any scalar product o
Page 1930:
TENSORSanother, without reference t
Page 1934:
TENSORSPhysical examples involving
Page 1938:
TENSORSdoes this imply that the A p
Page 1942:
TENSORSLet us begin, however, by no
Page 1946:
TENSORSA useful application of (26.
Page 1950:
TENSORSRotate by π/2 about the Ox
Page 1954:
TENSORSbut since |L| = ±1 we may r
Page 1958:
TENSORS◮Using (26.40), show that
Page 1962:
TENSORS(iii) referred to these axes
Page 1966:
TENSORSFurther, Poisson’s ratio i
Page 1970:
TENSORScontrary is specifically sta
Page 1974:
TENSORS◮Calculate the elements g
Page 1978:
TENSORSThus, by inverting the matri
Page 1982:
TENSORSwhere the elements L ij are
Page 1986:
TENSORSu i to another u ′i , we m
Page 1990:
TENSORS◮Using (26.76), deduce the
Page 1994:
TENSORS26.19 Covariant differentiat
Page 1998:
TENSORSand sov i ; i = ∂vρ∂ρ
Page 2002:
TENSORSIn order to compare the resu
Page 2006:
TENSORSand so the covariant compone
Page 2010:
TENSORScomponents of a second-order
Page 2014:
TENSORS26.3 In section 26.3 the tra
Page 2018:
TENSORS(b) Find the principal axes
Page 2022:
TENSORS26.28 A curve r(t) is parame
Page 2026:
27Numerical methodsIt happens frequ
Page 2030:
NUMERICAL METHODSf(x)141210 f(x) =x
Page 2034:
NUMERICAL METHODSn x n f(x n )1 1.7
Page 2038:
NUMERICAL METHODSn A n f(A n ) B n
Page 2042:
NUMERICAL METHODSOf the four method
Page 2046:
NUMERICAL METHODSn x n+1 ɛ n1 8.5
Page 2050:
NUMERICAL METHODSappreciate how thi
Page 2054:
NUMERICAL METHODSn x 1 x 2 x 31 2 2
Page 2058:
NUMERICAL METHODS◮Solve the follo
Page 2062:
NUMERICAL METHODSother exact expres
Page 2066:
NUMERICAL METHODSThe difference bet
Page 2070:
NUMERICAL METHODSand orthogonal ove
Page 2074:
NUMERICAL METHODSGauss-Legendre int
Page 2078:
NUMERICAL METHODSGauss-Laguerre and
Page 2082:
NUMERICAL METHODSaveraged:t = 1 nn
Page 2086:
NUMERICAL METHODSsampling, in both
Page 2090:
NUMERICAL METHODSh(ξ 1 ) >ξ 2 . T
Page 2094:
NUMERICAL METHODSon (0, 1) and then
Page 2098:
NUMERICAL METHODSderivatives beyond
Page 2102:
NUMERICAL METHODSx y(estim.) y(exac
Page 2106:
NUMERICAL METHODSx y(estim.) y(exac
Page 2110:
NUMERICAL METHODSSteps (ii) and (ii
Page 2114:
NUMERICAL METHODS(ii) To order h 4
Page 2118:
NUMERICAL METHODSThese can then be
Page 2122:
NUMERICAL METHODSOur final example
Page 2126:
NUMERICAL METHODS27.8 A possible ru
Page 2130:
NUMERICAL METHODS(b) Substitute the
Page 2134:
NUMERICAL METHODS27.25 Laplace’s
Page 2138:
NUMERICAL METHODSV =80−∞40.541.
Page 2142:
GROUP THEORYNHMHHH(a)(b)HFigure 28.
Page 2146:
GROUP THEORYif matrices are involve
Page 2150:
GROUP THEORYand setting X = I ′ g
Page 2154:
GROUP THEORYIt is clear that cyclic
Page 2158:
GROUP THEORY1 3 5 71 1 3 5 73 3 1 7
Page 2162:
GROUP THEORY1 i −1 −i1 1 i −1
Page 2166:
GROUP THEORYI R R ′ K L MI I R R
Page 2170:
GROUP THEORYThe multiplication tabl
Page 2174:
GROUP THEORYeach number appears onc
Page 2178:
GROUP THEORYThree immediate consequ
Page 2182:
GROUP THEORY(a)I A BI I A BA A B IB
Page 2186:
GROUP THEORYthan they are like any
Page 2190:
GROUP THEORY(iii) Transitivity: X
Page 2194:
GROUP THEORYif H is a normal subgro
Page 2198:
GROUP THEORYmathematical details, a
Page 2202:
GROUP THEORY(a) Identify the distin
Page 2206:
GROUP THEORYm 1 (π)m 2 (π)m 3 (π
Page 2210:
29Representation theoryAs indicated
Page 2214:
REPRESENTATION THEORYFinally, for o
Page 2218:
REPRESENTATION THEORY◮For the gro
Page 2222:
REPRESENTATION THEORYcolumn matrice
Page 2226:
REPRESENTATION THEORYone comprises
Page 2230:
REPRESENTATION THEORYa representati
Page 2234:
REPRESENTATION THEORYof a particula
Page 2238:
REPRESENTATION THEORYof simple pair
Page 2242:
REPRESENTATION THEORY(d) No explici
Page 2246:
REPRESENTATION THEORY29.6.1 Orthogo
Page 2250:
REPRESENTATION THEORYconjugacy clas
Page 2254:
REPRESENTATION THEORY(a)I A BI I A
Page 2258:
REPRESENTATION THEORYwhere the λ i
Page 2262:
REPRESENTATION THEORY4mm I Q R, R
Page 2266:
REPRESENTATION THEORYbe non-zero th
Page 2270:
REPRESENTATION THEORY1y42x3Figure 2
Page 2274:
REPRESENTATION THEORY(i) Under I al
Page 2278:
REPRESENTATION THEORYy 3x 3y 1y2x 1
Page 2282:
REPRESENTATION THEORYMMMFigure 29.6
Page 2286:
REPRESENTATION THEORY(d) Complete t
Page 2290:
REPRESENTATION THEORYthe correspond
Page 2294: REPRESENTATION THEORY(because of or
Page 2298: PROBABILITYAiiiiiiBSivFigure 30.1A
Page 2302: PROBABILITYA1B24756 3C8SFigure 30.4
Page 2306: PROBABILITYSince A ∩ X = X we mus
Page 2310: PROBABILITYThis is particularly use
Page 2314: PROBABILITY◮Find the probability
Page 2318: PROBABILITYTwo events A and B are s
Page 2322: PROBABILITY30.2.3 Bayes’ theoremI
Page 2326: PROBABILITYIn calculating the numbe
Page 2330: PROBABILITYAnother useful result th
Page 2334: PROBABILITYparticles can be distrib
Page 2338: PROBABILITYf(x)2pF(x)1p12 p 12 3 4
Page 2342: PROBABILITY◮A random variable X h
Page 2348: 30.5 PROPERTIES OF DISTRIBUTIONSInt
Page 2352: 30.5 PROPERTIES OF DISTRIBUTIONS|x
Page 2356: 30.5 PROPERTIES OF DISTRIBUTIONSWe
Page 2360: 30.6 FUNCTIONS OF RANDOM VARIABLESf
Page 2364: 30.6 FUNCTIONS OF RANDOM VARIABLESY
Page 2368: 30.6 FUNCTIONS OF RANDOM VARIABLESw
Page 2372: 30.7 GENERATING FUNCTIONSvariance o
Page 2376: 30.7 GENERATING FUNCTIONSand differ
Page 2380: 30.7 GENERATING FUNCTIONSi.e. the P
Page 2384: 30.7 GENERATING FUNCTIONSThe MGF wi
Page 2388: 30.7 GENERATING FUNCTIONSprobabilit
Page 2392: 30.7 GENERATING FUNCTIONSComparing
Page 2396:
30.8 IMPORTANT DISCRETE DISTRIBUTIO
Page 2400:
Page 2404:
Page 2408:
Page 2412:
Page 2416:
30.9 IMPORTANT CONTINUOUS DISTRIBUT
Page 2420:
Page 2424:
Page 2428:
Page 2432:
Page 2436:
Page 2440:
Page 2444:
Page 2448:
30.10 THE CENTRAL LIMIT THEOREMand
Page 2452:
30.11 JOINT DISTRIBUTIONSconsult on
Page 2456:
30.12 PROPERTIES OF JOINT DISTRIBUT
Page 2460:
Page 2464:
Page 2468:
30.13 GENERATING FUNCTIONS FOR JOIN
Page 2472:
30.15 IMPORTANT JOINT DISTRIBUTIONS
Page 2476:
30.15 IMPORTANT JOINT DISTRIBUTIONS
Page 2480:
30.16 EXERCISEStivariate Gaussian.
Page 2484:
30.16 EXERCISES30.11 A boy is selec
Page 2488:
30.16 EXERCISES30.18 A particle is
Page 2492:
30.16 EXERCISESaccording to one of
Page 2496:
30.17 HINTS AND ANSWERSconstraint
Page 2500:
31StatisticsIn this chapter, we tur
Page 2504:
31.2 SAMPLE STATISTICS188.7 204.7 1
Page 2508:
31.2 SAMPLE STATISTICSand the sampl
Page 2512:
31.2 SAMPLE STATISTICSmoments of th
Page 2516:
31.3 ESTIMATORS AND SAMPLING DISTRI
Page 2520:
Page 2524:
Page 2528:
Page 2532:
Page 2536:
Page 2540:
Page 2544:
31.4 SOME BASIC ESTIMATORSâ 2a 2(a
Page 2548:
31.4 SOME BASIC ESTIMATORSexact exp
Page 2552:
31.4 SOME BASIC ESTIMATORSwhere s 4
Page 2556:
31.4 SOME BASIC ESTIMATORSthe form(
Page 2560:
31.4 SOME BASIC ESTIMATORS(known) c
Page 2564:
31.4 SOME BASIC ESTIMATORSSince the
Page 2568:
31.5 MAXIMUM-LIKELIHOOD METHODSubst
Page 2572:
31.5 MAXIMUM-LIKELIHOOD METHODL(x;
Page 2576:
31.5 MAXIMUM-LIKELIHOOD METHOD◮In
Page 2580:
31.5 MAXIMUM-LIKELIHOOD METHOD31.5.
Page 2584:
31.5 MAXIMUM-LIKELIHOOD METHOD31.5.
Page 2588:
31.5 MAXIMUM-LIKELIHOOD METHODwhere
Page 2592:
31.5 MAXIMUM-LIKELIHOOD METHODL(x;
Page 2596:
31.5 MAXIMUM-LIKELIHOOD METHODBy su
Page 2600:
31.6 THE METHOD OF LEAST SQUARESThe
Page 2604:
31.6 THE METHOD OF LEAST SQUARESwhe
Page 2608:
31.6 THE METHOD OF LEAST SQUARESy76
Page 2612:
31.7 HYPOTHESIS TESTINGhowever, suc
Page 2616:
31.7 HYPOTHESIS TESTINGP (t|H 0 )α
Page 2620:
31.7 HYPOTHESIS TESTING◮Ten indep
Page 2624:
31.7 HYPOTHESIS TESTING◮Ten indep
Page 2628:
31.7 HYPOTHESIS TESTINGThe sum of s
Page 2632:
31.7 HYPOTHESIS TESTINGP (t|H 0 )0.
Page 2636:
31.7 HYPOTHESIS TESTINGdistribution
Page 2640:
31.7 HYPOTHESIS TESTINGλ(u)0.100.0
Page 2644:
31.7 HYPOTHESIS TESTINGWe now turn
Page 2648:
31.7 HYPOTHESIS TESTINGC n1 ,n 2(F)
Page 2652:
31.7 HYPOTHESIS TESTINGIn the last
Page 2656:
31.8 EXERCISES31.6 Prove that the s
Page 2660:
31.8 EXERCISES31.13 A similar techn
Page 2664:
31.9 HINTS AND ANSWERS31.9 Hints an
Page 2668:
IndexWhere the discussion of a topi
Page 2672:
INDEXrecurrence relations, 611-612s
Page 2676:
INDEXcomplement, 1121probability fo
Page 2680:
INDEXin spherical polars, 362Stoke
Page 2684:
INDEXin cylindrical polars, 360in s
Page 2688:
INDEXdiscontinuous functions, 420-4
Page 2692:
INDEXnomenclature, 1102non-Abelian,
Page 2696:
INDEXtriple, see multiple integrals
Page 2700:
INDEXlevel lines, 905, 906Levi-Civi
Page 2704:
INDEXMonte Carlo methods, of integr
Page 2708:
INDEXorthogonal transformations, 93
Page 2712:
INDEXstandard deviation σ, 1146var
Page 2716:
INDEXwave equation, 714-716, 737, 7
Page 2720:
INDEXsymmetric tensors, 938symmetry
Page 2724:
INDEXvolume integrals, 396and diver
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