Mathematical Methods for Physics and Engineering - Matematica.NET

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SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONSpower series of the <strong>for</strong>m (see section 24.11)∞∑y(z) = a n z n . (16.9)n=0Moreover, it may be shown that such a power series converges <strong>for</strong> |z|
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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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Page 1078: HIGHER-ORDER ORDINARY DIFFERENTIAL
Page 1122: SERIES SOLUTIONS OF ORDINARY DIFFER
Page 1126: SERIES SOLUTIONS OF ORDINARY DIFFER
Page 1132: 16.2 SERIES SOLUTIONS ABOUT AN ORDI
Page 1136: 16.3 SERIES SOLUTIONS ABOUT A REGUL
Page 1148: 16.4 OBTAINING A SECOND SOLUTIONto
Page 1152: 16.4 OBTAINING A SECOND SOLUTIONwhi
Page 1156: 16.5 POLYNOMIAL SOLUTIONSis a posit
Page 1160: 16.6 EXERCISES(c) Determine the rad
Page 1164: 16.7 HINTS AND ANSWERS(c)Show that
Page 1168: EIGENFUNCTION METHODS FOR DIFFERENT
Page 1172: 17.1 SETS OF FUNCTIONSwhere the d n
Page 1176: 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
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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
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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◮
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25.9 EXERCISESimaginary z-axes, fin
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25.9 EXERCISES(b) Calculate F(s) on
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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
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26.7 THE QUOTIENT LAWAn operation t
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26.8 THE TENSORS δ ij AND ɛ ijkN
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26.8 THE TENSORS δ ij AND ɛ ijk
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26.9 ISOTROPIC TENSORSare independe
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26.10 IMPROPER ROTATIONS AND PSEUDO
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26.11 DUAL TENSORSformations, for w
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26.12 PHYSICAL APPLICATIONS OF TENS
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26.12 PHYSICAL APPLICATIONS OF TENS
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26.14 NON-CARTESIAN COORDINATESThe
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26.15 THE METRIC TENSORsecond-order
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26.15 THE METRIC TENSORwhere we hav
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26.16 GENERAL COORDINATE TRANSFORMA
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26.17 RELATIVE TENSORS◮Show that
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26.18 DERIVATIVES OF BASIS VECTORS
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26.18 DERIVATIVES OF BASIS VECTORS
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26.19 COVARIANT DIFFERENTIATIONcons
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26.20 VECTOR OPERATORS IN TENSOR FO
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26.20 VECTOR OPERATORS IN TENSOR FO
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26.21 ABSOLUTE DERIVATIVES ALONG CU
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26.23 EXERCISESWriting out the cova
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26.23 EXERCISES26.10 A symmetric se
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26.23 EXERCISES26.23 A fourth-order
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26.24 HINTS AND ANSWERSin the (mult
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27.1 ALGEBRAIC AND TRANSCENDENTAL E
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Page 2036:
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Page 2044:
27.2 CONVERGENCE OF ITERATION SCHEM
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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
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27.4 NUMERICAL INTEGRATION27.4.3 Ga
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27.4 NUMERICAL INTEGRATIONso, provi
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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
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27.4 NUMERICAL INTEGRATIONIt will b
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27.5 FINITE DIFFERENCESmany values
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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
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27.7 HIGHER-ORDER EQUATIONSy1.00.80
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27.8 PARTIAL DIFFERENTIAL EQUATIONS
Page 2124:
27.9 EXERCISES27.9 Exercises27.1 Us
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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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