Mathematical Methods for Physics and Engineering - Matematica.NET

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1.8 EXERCISES(a) the sum of the sines of π/3 andπ/6,(b) the sine of the sum of π/3 andπ/4.1.8 The following exercises are based on the half-angle formulae.(a) Use the fact that sin(π/6) = 1/2 to prove that tan(π/12) = 2 − √ 3.(b) Use the result of (a) to show further that tan(π/24) = q(2 − q) whereq 2 =2+ √ 3.1.9 Find the real solutions of(a) 3 sin θ − 4cosθ =2,(b) 4 sin θ +3cosθ =6,(c) 12 sin θ − 5cosθ = −6.1.10 If s = sin(π/8), prove that8s 4 − 8s 2 +1=0,and hence show that s =[(2− √ 2)/4] 1/2 .1.11 Find all the solutions ofsin θ +sin4θ =sin2θ +sin3θthat lie in the range −π
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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
Page 82: PRELIMINARY ALGEBRAdrawn through R,
Page 86: PRELIMINARY ALGEBRAand use made of
Page 90: PRELIMINARY ALGEBRAwith the coordin
Page 94: PRELIMINARY ALGEBRAthe well-known r
Page 98: PRELIMINARY ALGEBRAnumerators on bo
Page 102: PRELIMINARY ALGEBRAWe illustrate th
Page 106: PRELIMINARY ALGEBRAIn this form, al
Page 110: PRELIMINARY ALGEBRAIn fact, the gen
Page 114: PRELIMINARY ALGEBRAThe first is a f
Page 118: PRELIMINARY ALGEBRAbe obvious, but
Page 122: PRELIMINARY ALGEBRAThis is precisel
Page 126: PRELIMINARY ALGEBRA◮The prime int
Page 130: PRELIMINARY ALGEBRA1.8 ExercisesPol
Page 136: 1.9 HINTS AND ANSWERS1.27 Establish
Page 140: 2Preliminary calculusThis chapter i
Page 144: 2.1 DIFFERENTIATIONapproximate the
Page 148: 2.1 DIFFERENTIATIONseparation is no
Page 152: 2.1 DIFFERENTIATION◮Find the deri
Page 156: 2.1 DIFFERENTIATIONThe pattern emer
Page 160: 2.1 DIFFERENTIATIONgradient of the
Page 164: 2.1 DIFFERENTIATIONf(x)C∆θρPQθ
Page 168: 2.1 DIFFERENTIATION◮Show that the
Page 172: 2.1 DIFFERENTIATIONand hence the di
Page 176: 2.2 INTEGRATIONf(x)abxFigure 2.7An
Page 180: 2.2 INTEGRATIONCombining (2.23) and
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2.2 INTEGRATIONfound near the end o
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2.2 INTEGRATION◮Evaluate the inte
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2.2 INTEGRATION◮Evaluate the inte
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2.2 INTEGRATIONRearranging this exp
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2.2 INTEGRATIONyCρ(φ + dφ)dAρ(
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2.2 INTEGRATIONf(x)mabxFigure 2.10T
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2.2 INTEGRATIONydsf(x)adxVbxSFigure
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2.3 EXERCISES2.10 The function y(x)
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2.3 EXERCISES2.26 Use the mean valu
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2.4 HINTS AND ANSWERS2.45 If J r is
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3Complex numbers andhyperbolic func
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3.2 MANIPULATION OF COMPLEX NUMBERS
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3.3 POLAR REPRESENTATION OF COMPLEX
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3.4 DE MOIVRE’S THEOREMIm zr 1 e
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3.4 DE MOIVRE’S THEOREM◮Find an
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3.5 COMPLEX LOGARITHMS AND COMPLEX
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3.6 APPLICATIONS TO DIFFERENTIATION
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3.7 HYPERBOLIC FUNCTIONS43cosh x21s
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3.7 HYPERBOLIC FUNCTIONS3.7.4 Solvi
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3.7 HYPERBOLIC FUNCTIONS4cosech −
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3.8 EXERCISES◮Evaluate (d/dx)sinh
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3.8 EXERCISES3.13 Prove that x 2m+1
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3.9 HINTS AND ANSWERS3.27 A closed
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4Series and limits4.1 SeriesMany ex
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4.2 SUMMATION OF SERIES4.2.1 Arithm
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4.2 SUMMATION OF SERIES◮Sum the s
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4.2 SUMMATION OF SERIES4.2.5 Series
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4.2 SUMMATION OF SERIESIntegrating
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4.3 CONVERGENCE OF INFINITE SERIES4
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4.3 CONVERGENCE OF INFINITE SERIESR
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4.3 CONVERGENCE OF INFINITE SERIESU
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4.4 OPERATIONS WITH SERIESis always
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4.5 POWER SERIES◮Determine the ra
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4.5 POWER SERIESQ(x) intoP (x) toob
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4.6 TAYLOR SERIESf(x)QRf(a)Pθhhf
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4.6 TAYLOR SERIESWe may follow a si
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4.7 EVALUATION OF LIMITSThese can a
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4.7 EVALUATION OF LIMITSTherefore w
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4.8 EXERCISES4.8 The N + 1 complex
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4.8 EXERCISES4.20 Identify the seri
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4.9 HINTS AND ANSWERSfind a closed-
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5Partial differentiationIn chapter
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5.2 THE TOTAL DIFFERENTIAL AND TOTA
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5.3 EXACT AND INEXACT DIFFERENTIALS
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5.4 USEFUL THEOREMS OF PARTIAL DIFF
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5.6 CHANGE OF VARIABLESyρφxFigure
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5.7 TAYLOR’S THEOREM FOR MANY-VAR
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5.8 STATIONARY VALUES OF MANY-VARIA
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5.8 STATIONARY VALUES OF MANY-VARIA
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5.9 STATIONARY VALUES UNDER CONSTRA
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5.10 ENVELOPESWe now have the gener
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5.10 ENVELOPESh is made arbitrarily
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5.11 THERMODYNAMIC RELATIONS◮Show
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5.13 EXERCISESconstant limits of in
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5.13 EXERCISES5.9 The function f(x,
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5.13 EXERCISESyORθθ2θxFigure 5.5
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5.14 HINTS AND ANSWERS5.33 If∫ 1I
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6Multiple integralsFor functions of
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6.1 DOUBLE INTEGRALSAn alternative
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6.3 APPLICATIONS OF MULTIPLE INTEGR
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6.4 CHANGE OF VARIABLES IN MULTIPLE
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6.5 EXERCISESand similarly for J yz
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6.5 EXERCISESThis is an example of
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6.6 HINTS AND ANSWERS(a) Let R be a
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7.2 ADDITION AND SUBTRACTION OF VEC
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7.3 MULTIPLICATION BY A SCALARλ aa
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7.4 BASIS VECTORS AND COMPONENTSThe
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7.6 MULTIPLICATION OF VECTORSbOθ b
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7.6 MULTIPLICATION OF VECTORSIf we
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7.6 MULTIPLICATION OF VECTORSPFθRr
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7.6 MULTIPLICATION OF VECTORSvPOφ
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7.7 EQUATIONS OF LINES, PLANES AND
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7.8 USING VECTORS TO FIND DISTANCES
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7.8 USING VECTORS TO FIND DISTANCES
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7.9 RECIPROCAL VECTORSthe line to t
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7.10 EXERCISES7.2 A unit cell of di
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7.10 EXERCISESabcdaFigure 7.17A fac
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7.10 EXERCISESV 0 cos ωtV 4V 1V 2R
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8Matrices and vector spacesIn the p
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8.1 VECTOR SPACESthe trivial case i
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8.1 VECTOR SPACESIn the above basis
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8.2 LINEAR OPERATORSwhere the equal
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8.3 MATRICES8.2.1 Properties of lin
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8.4 BASIC MATRIX ALGEBRANow, since
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8.4 BASIC MATRIX ALGEBRAexcept for
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8.5 FUNCTIONS OF MATRICESThe identi
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8.7 THE COMPLEX AND HERMITIAN CONJU
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8.9 THE DETERMINANT OF A MATRIXwhic
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8.9 THE DETERMINANT OF A MATRIX◮S
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8.10 THE INVERSE OF A MATRIX◮Eval
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8.10 THE INVERSE OF A MATRIX◮Find
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8.11 THE RANK OF A MATRIX8.11 The r
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8.12 SPECIAL TYPES OF SQUARE MATRIX
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8.12 SPECIAL TYPES OF SQUARE MATRIX
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8.13 EIGENVECTORS AND EIGENVALUESre
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8.13 EIGENVECTORS AND EIGENVALUESwr
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8.13 EIGENVECTORS AND EIGENVALUESBu
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8.13 EIGENVECTORS AND EIGENVALUESei
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8.14 DETERMINATION OF EIGENVALUES A
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8.15 CHANGE OF BASIS AND SIMILARITY
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8.16 DIAGONALISATION OF MATRICESort
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8.16 DIAGONALISATION OF MATRICES◮
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8.17 QUADRATIC AND HERMITIAN FORMSi
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8.17 QUADRATIC AND HERMITIAN FORMSa
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8.18 SIMULTANEOUS LINEAR EQUATIONSw
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8.18 SIMULTANEOUS LINEAR EQUATIONSa
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8.18 SIMULTANEOUS LINEAR EQUATIONSt
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8.18 SIMULTANEOUS LINEAR EQUATIONS(
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8.18 SIMULTANEOUS LINEAR EQUATIONS
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8.18 SIMULTANEOUS LINEAR EQUATIONSW
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8.19 EXERCISESwhere U and V are giv
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8.19 EXERCISES(b) Without assuming
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8.19 EXERCISES8.20 Demonstrate that
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8.19 EXERCISESis 2 and that an orth
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8.20 HINTS AND ANSWERS8.5 Use the (
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9.1 TYPICAL OSCILLATORY SYSTEMScorr
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9.1 TYPICAL OSCILLATORY SYSTEMScoor
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9.1 TYPICAL OSCILLATORY SYSTEMSmk
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9.2 SYMMETRY AND NORMAL MODESy 1 y
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9.2 SYMMETRY AND NORMAL MODES(a) ω
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9.3 RAYLEIGH-RITZ METHODand that th
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9.4 EXERCISES◮Estimate the eigenf
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9.4 EXERCISESthe figure and obtain
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9.5 HINTS AND ANSWERS1 2 3mMm(a) ω
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10.1 DIFFERENTIATION OF VECTORSa(u
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10.1 DIFFERENTIATION OF VECTORSin t
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10.2 INTEGRATION OF VECTORSNote tha
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10.3 SPACE CURVESThis parametric re
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10.3 SPACE CURVESso we finally obta
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10.5 SURFACESzT∂r∂uSu = c 1P∂
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10.6 SCALAR AND VECTOR FIELDSA norm
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10.7 VECTOR OPERATORS∇φaθQPdφd
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10.7 VECTOR OPERATORSz(0, 0,a)ˆn 0
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10.7 VECTOR OPERATORS10.7.3 Curl of
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10.8 VECTOR OPERATOR FORMULAE◮Sho
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10.9 CYLINDRICAL AND SPHERICAL POLA
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10.10 GENERAL CURVILINEAR COORDINAT
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10.10 GENERAL CURVILINEAR COORDINAT
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10.11 EXERCISES∇Φ =∇ · a =∇
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10.11 EXERCISES10.6 Prove that for
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10.11 EXERCISES(a) For cylindrical
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10.12 HINTS AND ANSWERS(a) Express
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11Line, surface and volume integral
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11.1 LINE INTEGRALSA similar proced
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11.1 LINE INTEGRALS◮Evaluate the
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11.2 CONNECTIVITY OF REGIONS(a) (b)
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11.3 GREEN’S THEOREM IN A PLANEy
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11.4 CONSERVATIVE FIELDS AND POTENT
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11.5 SURFACE INTEGRALSindependent o
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11.5 SURFACE INTEGRALSzkαdSSyxRdAF
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11.5 SURFACE INTEGRALSSzadSxaCadA =
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11.5 SURFACE INTEGRALS◮Find the v
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11.6 VOLUME INTEGRALSdSSVrOFigure 1
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11.7 INTEGRAL FORMS FOR grad, div A
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11.8 DIVERGENCE THEOREM AND RELATED
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11.9 STOKES’ THEOREM AND RELATED
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11.10 EXERCISESeverywhere except on
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11.10 EXERCISES11.12 Show that the
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11.10 EXERCISES11.24 Prove equation
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12Fourier seriesWe have already dis
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12.2 THE FOURIER COEFFICIENTSwe can
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12.3 SYMMETRY CONSIDERATIONSf(t)1
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12.4 DISCONTINUOUS FUNCTIONS(a)1(b)
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12.5 NON-PERIODIC FUNCTIONSf(x) =x
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12.7 COMPLEX FOURIER SERIESwhere th
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12.9 EXERCISESthe sine and cosine f
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12.9 EXERCISESDeduce the value of t
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12.10 HINTS AND ANSWERS0 1 0 1 0 1
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13Integral transformsIn the previou
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13.1 FOURIER TRANSFORMSand (13.3) b
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13.1 FOURIER TRANSFORMSis a wavefun
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13.1 FOURIER TRANSFORMSf(y)1−a−
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13.1 FOURIER TRANSFORMSThe derivati
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13.1 FOURIER TRANSFORMS˜fΩ2Ω(2π
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13.1 FOURIER TRANSFORMSIgnoring in
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13.1 FOURIER TRANSFORMSf(x)∗ g(y)
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13.1 FOURIER TRANSFORMSThe inverse
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13.1 FOURIER TRANSFORMSobtained sim
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13.2 LAPLACE TRANSFORMSA similar re
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13.2 LAPLACE TRANSFORMSf(t) ¯f(s)
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13.2 LAPLACE TRANSFORMSWe may now c
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13.3 CONCLUDING REMARKSThe properti
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13.4 EXERCISESDetermine the convolu
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13.4 EXERCISES(a) Find the Fourier
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13.4 EXERCISES(c) L [sinh at cos bt
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13.5 HINTS AND ANSWERS13.17 Ṽ (k)
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14.1 GENERAL FORM OF SOLUTIONthe ap
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14.2 FIRST-DEGREE FIRST-ORDER EQUAT
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14.3 HIGHER-DEGREE FIRST-ORDER EQUA
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14.3 HIGHER-DEGREE FIRST-ORDER EQUA
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14.4 EXERCISES14.5 By finding suita
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14.4 EXERCISES(c) Find an appropria
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14.5 HINTS AND ANSWERS14.31 Show th
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HIGHER-ORDER ORDINARY DIFFERENTIAL
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15.1 LINEAR EQUATIONS WITH CONSTANT
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15.2 LINEAR EQUATIONS WITH VARIABLE
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15.3 GENERAL ORDINARY DIFFERENTIAL
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15.3 GENERAL ORDINARY DIFFERENTIAL
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15.4 EXERCISES15.3.6 Equations havi
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15.4 EXERCISES15.9 Find the general
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15.4 EXERCISES15.23 Prove that the
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15.5 HINTS AND ANSWERS15.36 Find th
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16Series solutions of ordinarydiffe
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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
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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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27.2 CONVERGENCE OF ITERATION SCHEM
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27.3 SIMULTANEOUS LINEAR EQUATIONSv
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. . .. . .27.3 SIMULTANEOUS LINEAR
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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
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27.4 NUMERICAL INTEGRATIONhas becom
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27.4 NUMERICAL INTEGRATIONwill have
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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
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27.6 DIFFERENTIAL EQUATIONSbut they
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27.6 DIFFERENTIAL EQUATIONSThe forw
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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
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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
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27.10 HINTS AND ANSWERS27.27 The Sc
Page 2140:
28Group theoryFor systems that have
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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
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28.2 FINITE GROUPS28.2 Finite group
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28.2 FINITE GROUPS(a)1 5 7 111 1 5
Page 2164:
28.3 NON-ABELIAN GROUPSAs a first e
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28.3 NON-ABELIAN GROUPSI A B C D EI
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28.4 PERMUTATION GROUPSSuppose that
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28.5 MAPPINGS BETWEEN GROUPS28.5 Ma
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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
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28.7 SUBDIVIDING A GROUP• Two cos
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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 ,
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29.1 DIPOLE MOMENTS OF MOLECULESABA
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29.2 CHOOSING AN APPROPRIATE FORMAL
Page 2220:
Page 2224:
Page 2228:
29.3 EQUIVALENT REPRESENTATIONSresp
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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
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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
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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:
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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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