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15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTSwe are free to choose our constraints as we wish, let us define the expression inparentheses to be zero, giving the first equation in (15.55). Differentiating againwe findy p ′′ = k 1 y 1 ′′ + k 2 y 2 ′′ + ···+ k n y n ′′ +[k 1y ′ 1 ′ + k 2y ′ 2 ′ + ···+ k ny ′ n ′ ].Once more we can choose the expression in brackets to be zero, giving the secondequation in (15.55). We can repeat this procedure, choosing the correspondingexpression in each case to be zero. This yields the first n − 1 equations in (15.55).The mth derivative of y p <strong>for</strong> m
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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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Page 1026: FIRST-ORDER ORDINARY DIFFERENTIAL E
Page 1038: 15Higher-order ordinary differentia
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Page 1104: 15.4 EXERCISES15.3.6 Equations havi
Page 1108: 15.4 EXERCISES15.9 Find the general
Page 1112: 15.4 EXERCISES15.23 Prove that the
Page 1116: 15.5 HINTS AND ANSWERS15.36 Find th
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Page 1124: 16.1 SECOND-ORDER LINEAR ORDINARY D
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16.2 SERIES SOLUTIONS ABOUT AN ORDI
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16.2 SERIES SOLUTIONS ABOUT AN ORDI
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16.3 SERIES SOLUTIONS ABOUT A REGUL
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16.4 OBTAINING A SECOND SOLUTIONto
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16.4 OBTAINING A SECOND SOLUTIONwhi
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16.5 POLYNOMIAL SOLUTIONSis a posit
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16.6 EXERCISES(c) Determine the rad
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16.7 HINTS AND ANSWERS(c)Show that
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EIGENFUNCTION METHODS FOR DIFFERENT
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17.1 SETS OF FUNCTIONSwhere the d n
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17.2 ADJOINT, SELF-ADJOINT AND HERM
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17.3 PROPERTIES OF HERMITIAN OPERAT
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17.3 PROPERTIES OF HERMITIAN OPERAT
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17.4 STURM-LIOUVILLE EQUATIONScerta
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17.4 STURM-LIOUVILLE EQUATIONS(ii)
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17.5 SUPERPOSITION OF EIGENFUNCTION
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17.5 SUPERPOSITION OF EIGENFUNCTION
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17.7 EXERCISESWe note that if µ =
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17.7 EXERCISESwhere κ is a constan
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18Special functionsIn the previous
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18.1 LEGENDRE FUNCTIONS2P 01P 1−1
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18.1 LEGENDRE FUNCTIONS1Q 00.5−1
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18.1 LEGENDRE FUNCTIONSMutual ortho
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18.1 LEGENDRE FUNCTIONSEquation (18
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18.2 ASSOCIATED LEGENDRE FUNCTIONS1
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18.2 ASSOCIATED LEGENDRE FUNCTIONSw
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18.2 ASSOCIATED LEGENDRE FUNCTIONSS
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18.3 SPHERICAL HARMONICSbe derived
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18.4 CHEBYSHEV FUNCTIONSSince δ(Ω
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18.4 CHEBYSHEV FUNCTIONS1T 0T 2T 30
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18.4 CHEBYSHEV FUNCTIONSEvaluating
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18.4 CHEBYSHEV FUNCTIONSin which th
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18.5 BESSEL FUNCTIONSgenerality. Th
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18.5 BESSEL FUNCTIONSWe note that B
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18.5 BESSEL FUNCTIONSand hence that
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18.5 BESSEL FUNCTIONSTo determine t
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18.5 BESSEL FUNCTIONS◮Prove the e
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18.5 BESSEL FUNCTIONSin subsection
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18.6 SPHERICAL BESSEL FUNCTIONSwher
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18.7 LAGUERRE FUNCTIONSit has a reg
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18.7 LAGUERRE FUNCTIONS◮Prove tha
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18.8 ASSOCIATED LAGUERRE FUNCTIONSw
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18.8 ASSOCIATED LAGUERRE FUNCTIONS
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18.9 HERMITE FUNCTIONS105−1.5H 0H
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18.9 HERMITE FUNCTIONS◮Show thatI
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18.10 HYPERGEOMETRIC FUNCTIONSby ma
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18.10 HYPERGEOMETRIC FUNCTIONSF(a,
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18.11 CONFLUENT HYPERGEOMETRIC FUNC
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18.12 THE GAMMA FUNCTION AND RELATE
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18.13 EXERCISES√√Y0 0 = 1, Y 04
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18.13 EXERCISES[ You will find it c
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18.13 EXERCISES(a) use their series
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18.14 HINTS AND ANSWERS18.15 (a) Sh
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19.1 OPERATOR FORMALISMrepresent di
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19.1 OPERATOR FORMALISMspectrum of
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19.1 OPERATOR FORMALISMwhilstthatfo
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19.1 OPERATOR FORMALISMdefining ser
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19.2 PHYSICAL EXAMPLES OF OPERATORS
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19.3 EXERCISESthat would involve a
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19.3 EXERCISESNow evaluate the expe
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20Partial differential equations:ge
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20.1 IMPORTANT PARTIAL DIFFERENTIAL
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20.1 IMPORTANT PARTIAL DIFFERENTIAL
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20.3 GENERAL AND PARTICULAR SOLUTIO
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20.4 THE WAVE EQUATION20.4 The wave
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20.5 THE DIFFUSION EQUATIONterm is
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20.5 THE DIFFUSION EQUATIONwritten
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20.6 CHARACTERISTICS AND THE EXISTE
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20.7 UNIQUENESS OF SOLUTIONSEquatio
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20.8 EXERCISESWe also note that oft
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20.8 EXERCISES20.14 Solve∂ 2 u u
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20.9 HINTS AND ANSWERS20.25 The Kle
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21Partial differential equations:se
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21.1 SEPARATION OF VARIABLES: THE G
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21.2 SUPERPOSITION OF SEPARATED SOL
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21.3 SEPARATION OF VARIABLES IN POL
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21.4 INTEGRAL TRANSFORM METHODS21.4
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21.4 INTEGRAL TRANSFORM METHODS◮A
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21.5 INHOMOGENEOUS PROBLEMS - GREEN
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21.6 EXERCISESUsing plane polar coo
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21.6 EXERCISES(a) Evaluate dPl m(µ
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21.6 EXERCISES21.18 A sphere of rad
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21.7 HINTS AND ANSWERSin V and take
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22Calculus of variationsIn chapters
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22.2 SPECIAL CASESto these variatio
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22.2 SPECIAL CASESydsdydxxFigure 22
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22.3 SOME EXTENSIONSbzρ−ba(a) (b
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22.3 SOME EXTENSIONSy(x)+η(x)∆yy
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22.4 CONSTRAINED VARIATIONwhere k i
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22.5 PHYSICAL VARIATIONAL PRINCIPLE
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22.5 PHYSICAL VARIATIONAL PRINCIPLE
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22.6 GENERAL EIGENVALUE PROBLEMScon
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22.7 ESTIMATION OF EIGENVALUES AND
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22.8 ADJUSTMENT OF PARAMETERSIt is
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22.9 EXERCISES22.9 Exercises22.1 A
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22.9 EXERCISESpath of a small test
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22.10 HINTS AND ANSWERStotal energy
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23Integral equationsIt is not unusu
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23.3 OPERATOR NOTATION AND THE EXIS
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23.4 CLOSED-FORM SOLUTIONS23.4.1 Se
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23.4 CLOSED-FORM SOLUTIONS23.4.2 In
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23.4 CLOSED-FORM SOLUTIONSso we can
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23.5 NEUMANN SERIES23.5 Neumann ser
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23.6 FREDHOLM THEORYcommon ratio λ
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23.7 SCHMIDT-HILBERT THEORYLet us b
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23.8 EXERCISESthus Hermitian. In or
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23.8 EXERCISES(b) Obtain the eigenv
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23.9 HINTS AND ANSWERS23.9 Hints an
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24.1 FUNCTIONS OF A COMPLEX VARIABL
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24.2 THE CAUCHY-RIEMANN RELATIONS
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24.2 THE CAUCHY-RIEMANN RELATIONSSi
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24.3 POWER SERIES IN A COMPLEX VARI
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24.4 SOME ELEMENTARY FUNCTIONSreal-
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24.5 MULTIVALUED FUNCTIONS AND BRAN
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24.6 SINGULARITIES AND ZEROS OF COM
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24.7 CONFORMAL TRANSFORMATIONSThus
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24.7 CONFORMAL TRANSFORMATIONSpoint
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24.7 CONFORMAL TRANSFORMATIONSysw 5
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24.8 COMPLEX INTEGRALSyw 3 s w 3w =
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24.8 COMPLEX INTEGRALSyRtyyC 1 C 2R
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24.9 CAUCHY’S THEOREMnamely Cauch
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24.10 CAUCHY’S INTEGRAL FORMULAyC
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24.11 TAYLOR AND LAURENT SERIESFurt
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24.11 TAYLOR AND LAURENT SERIESof o
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24.11 TAYLOR AND LAURENT SERIESdeno
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24.12 RESIDUE THEOREMSuppose the fu
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24.13 DEFINITE INTEGRALS USING CONT
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24.14 EXERCISESWe have seen that
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24.14 EXERCISES24.14 Prove that, fo
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25Applications of complex variables
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25.1 COMPLEX POTENTIALSthe field pr
Page 1808:
25.1 COMPLEX POTENTIALSyQxPˆnFigur
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25.2 APPLICATIONS OF CONFORMAL TRAN
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25.3 LOCATION OF ZEROSφ =0yπ/αz
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25.3 LOCATION OF ZEROSpolynomials,
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25.4 SUMMATION OF SERIES◮By consi
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25.5 INVERSE LAPLACE TRANSFORMΓRΓ
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25.5 INVERSE LAPLACE TRANSFORMf(x)1
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25.6 STOKES’ EQUATION AND AIRY IN
Page 1840:
Page 1844:
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25.7 WKB METHODSthere exist many re
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25.7 WKB METHODSThis still requires
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25.7 WKB METHODSThe precise combina
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25.7 WKB METHODSfor some constant A
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25.7 WKB METHODSone function and th
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25.8 APPROXIMATIONS TO INTEGRALSFin
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25.8 APPROXIMATIONS TO INTEGRALSFro
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25.8 APPROXIMATIONS TO INTEGRALSany
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25.8 APPROXIMATIONS TO INTEGRALSto
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25.8 APPROXIMATIONS TO INTEGRALSwhi
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25.8 APPROXIMATIONS TO INTEGRALS(a)
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25.8 APPROXIMATIONS TO INTEGRALSare
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25.8 APPROXIMATIONS TO INTEGRALS◮
Page 1900:
25.9 EXERCISESimaginary z-axes, fin
Page 1904:
25.9 EXERCISES(b) Calculate F(s) on
Page 1908:
25.10 HINTS AND ANSWERSt = −i and
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26TensorsIt may seem obvious that t
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26.2 CHANGE OF BASISIn the second o
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26.3 CARTESIAN TENSORSx 2x ′ 1x
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26.4 FIRST- AND ZERO-ORDER CARTESIA
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26.5 SECOND- AND HIGHER-ORDER CARTE
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26.5 SECOND- AND HIGHER-ORDER CARTE
Page 1936:
26.7 THE QUOTIENT LAWAn operation t
Page 1940:
26.8 THE TENSORS δ ij AND ɛ ijkN
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26.8 THE TENSORS δ ij AND ɛ ijk
Page 1948:
26.9 ISOTROPIC TENSORSare independe
Page 1952:
26.10 IMPROPER ROTATIONS AND PSEUDO
Page 1956:
26.11 DUAL TENSORSformations, for w
Page 1960:
26.12 PHYSICAL APPLICATIONS OF TENS
Page 1964:
26.12 PHYSICAL APPLICATIONS OF TENS
Page 1968:
26.14 NON-CARTESIAN COORDINATESThe
Page 1972:
26.15 THE METRIC TENSORsecond-order
Page 1976:
26.15 THE METRIC TENSORwhere we hav
Page 1980:
26.16 GENERAL COORDINATE TRANSFORMA
Page 1984:
26.17 RELATIVE TENSORS◮Show that
Page 1988:
26.18 DERIVATIVES OF BASIS VECTORS
Page 1992:
26.18 DERIVATIVES OF BASIS VECTORS
Page 1996:
26.19 COVARIANT DIFFERENTIATIONcons
Page 2000:
26.20 VECTOR OPERATORS IN TENSOR FO
Page 2004:
26.20 VECTOR OPERATORS IN TENSOR FO
Page 2008:
26.21 ABSOLUTE DERIVATIVES ALONG CU
Page 2012:
26.23 EXERCISESWriting out the cova
Page 2016:
26.23 EXERCISES26.10 A symmetric se
Page 2020:
26.23 EXERCISES26.23 A fourth-order
Page 2024:
26.24 HINTS AND ANSWERSin the (mult
Page 2028:
27.1 ALGEBRAIC AND TRANSCENDENTAL E
Page 2032:
Page 2036:
Page 2040:
Page 2044:
27.2 CONVERGENCE OF ITERATION SCHEM
Page 2048:
27.3 SIMULTANEOUS LINEAR EQUATIONSv
Page 2052:
27.3 SIMULTANEOUS LINEAR EQUATIONSt
Page 2056:
. . .. . .27.3 SIMULTANEOUS LINEAR
Page 2060:
27.4 NUMERICAL INTEGRATION(a) (b) (
Page 2064:
27.4 NUMERICAL INTEGRATIONThis prov
Page 2068:
27.4 NUMERICAL INTEGRATION27.4.3 Ga
Page 2072:
27.4 NUMERICAL INTEGRATIONso, provi
Page 2076:
27.4 NUMERICAL INTEGRATIONfactor is
Page 2080:
27.4 NUMERICAL INTEGRATIONhas becom
Page 2084:
27.4 NUMERICAL INTEGRATIONwill have
Page 2088:
27.4 NUMERICAL INTEGRATIONy = f(x)y
Page 2092:
27.4 NUMERICAL INTEGRATIONIt will b
Page 2096:
27.5 FINITE DIFFERENCESmany values
Page 2100:
27.6 DIFFERENTIAL EQUATIONSx h y(ex
Page 2104:
27.6 DIFFERENTIAL EQUATIONSbut they
Page 2108:
27.6 DIFFERENTIAL EQUATIONSThe forw
Page 2112:
27.6 DIFFERENTIAL EQUATIONSWe assum
Page 2116:
27.7 HIGHER-ORDER EQUATIONSy1.00.80
Page 2120:
27.8 PARTIAL DIFFERENTIAL EQUATIONS
Page 2124:
27.9 EXERCISES27.9 Exercises27.1 Us
Page 2128:
27.9 EXERCISES(b) Try to repeat the
Page 2132:
27.9 EXERCISES27.21 Write a compute
Page 2136:
27.10 HINTS AND ANSWERS27.27 The Sc
Page 2140:
28Group theoryFor systems that have
Page 2144:
28.1 GROUPS28.1.1 Definition of a g
Page 2148:
28.1 GROUPS◮Using only the first
Page 2152:
28.1 GROUPSLMKFigure 28.2 Reflectio
Page 2156:
28.2 FINITE GROUPS28.2 Finite group
Page 2160:
28.2 FINITE GROUPS(a)1 5 7 111 1 5
Page 2164:
28.3 NON-ABELIAN GROUPSAs a first e
Page 2168:
28.3 NON-ABELIAN GROUPSI A B C D EI
Page 2172:
28.4 PERMUTATION GROUPSSuppose that
Page 2176:
28.5 MAPPINGS BETWEEN GROUPS28.5 Ma
Page 2180:
28.6 SUBGROUPS(a)I A B C D EI I A B
Page 2184:
28.7 SUBDIVIDING A GROUP(i) the set
Page 2188:
28.7 SUBDIVIDING A GROUPthis implie
Page 2192:
28.7 SUBDIVIDING A GROUP• Two cos
Page 2196:
28.7 SUBDIVIDING A GROUP(iii) In an
Page 2200:
28.8 EXERCISES28.4 Prove that the r
Page 2204:
28.8 EXERCISESSimilarly compute C 2
Page 2208:
28.9 HINTS AND ANSWERS≠For Φ 4 ,
Page 2212:
29.1 DIPOLE MOMENTS OF MOLECULESABA
Page 2216:
29.2 CHOOSING AN APPROPRIATE FORMAL
Page 2220:
Page 2224:
Page 2228:
29.3 EQUIVALENT REPRESENTATIONSresp
Page 2232:
29.4 REDUCIBILITY OF A REPRESENTATI
Page 2236:
29.4 REDUCIBILITY OF A REPRESENTATI
Page 2240:
29.5 THE ORTHOGONALITY THEOREM FOR
Page 2244:
29.6 CHARACTERS3m I A, B C, D, EA 1
Page 2248:
29.7 COUNTING IRREPS USING CHARACTE
Page 2252:
Page 2256:
Page 2260:
29.8 CONSTRUCTION OF A CHARACTER TA
Page 2264:
29.10 PRODUCT REPRESENTATIONSgive a
Page 2268:
29.11 PHYSICAL APPLICATIONS OF GROU
Page 2272:
Page 2276:
Page 2280:
Page 2284:
29.12 EXERCISESas the sum of two on
Page 2288:
29.12 EXERCISESUse this to show tha
Page 2292:
29.13 HINTS AND ANSWERS(a) Make an
Page 2296:
30ProbabilityAll scientists will kn
Page 2300:
30.1 VENN DIAGRAMSA42 6 3BS15Figure
Page 2304:
30.1 VENN DIAGRAMSgets beyond three
Page 2308:
30.2 PROBABILITYtimes then we expec
Page 2312:
30.2 PROBABILITYHowever, we may wri
Page 2316:
30.2 PROBABILITYace from a pack of
Page 2320:
30.2 PROBABILITYA 4A 3OA 1A 2BFigur
Page 2324:
30.3 PERMUTATIONS AND COMBINATIONSW
Page 2328:
30.3 PERMUTATIONS AND COMBINATIONSt
Page 2332:
30.3 PERMUTATIONS AND COMBINATIONSm
Page 2336:
30.4 RANDOM VARIABLES AND DISTRIBUT
Page 2340:
30.4 RANDOM VARIABLES AND DISTRIBUT
Page 2344:
30.5 PROPERTIES OF DISTRIBUTIONSIn
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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