Mathematical Methods for Physics and Engineering - Matematica.NET

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CONTENTS24.9 Cauchy’s theorem 84924.10 Cauchy’s integral formula 85124.11 Taylor and Laurent series 85324.12 Residue theorem 85824.13 Definite integrals using contour integration 86124.14 Exercises 86724.15 Hints and answers 87025 Applications of complex variables 87125.1 Complex potentials 87125.2 Applications of conformal transformations 87625.3 Location of zeros 87925.4 Summation of series 88225.5 Inverse Laplace transform 88425.6 Stokes’ equation and Airy integrals 88825.7 WKB methods 89525.8 Approximations to integrals 905Level lines and saddle points; steepest descents; stationary phase25.9 Exercises 92025.10 Hints and answers 92526 Tensors 92726.1 Some notation 92826.2 Change of basis 92926.3 Cartesian tensors 93026.4 First- and zero-order Cartesian tensors 93226.5 Second- and higher-order Cartesian tensors 93526.6 The algebra of tensors 93826.7 The quotient law 93926.8 The tensors δ ij and ɛ ijk 94126.9 Isotropic tensors 94426.10 Improper rotations and pseudotensors 94626.11 Dual tensors 94926.12 Physical applications of tensors 95026.13 Integral theorems for tensors 95426.14 Non-Cartesian coordinates 95526.15 The metric tensor 95726.16 General coordinate transformations and tensors 96026.17 Relative tensors 96326.18 Derivatives of basis vectors and Christoffel symbols 96526.19 Covariant differentiation 96826.20 Vector operators in tensor form 971xiv
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Page 6: Physicists. He is also a Director o
Page 10: cambridge university pressCambridge
Page 14: CONTENTS2.2 Integration 59Integrati
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Page 32: CONTENTS26.21 Absolute derivatives
Page 36: CONTENTS31 Statistics 122131.1 Expe
Page 42: Preface to the third editionAs is n
Page 46: PREFACE TO THE THIRD EDITIONstudent
Page 50: PREFACE TO THE SECOND EDITIONStatis
Page 54: PREFACE TO THE FIRST EDITIONgrounds
Page 60: 1Preliminary algebraThis opening ch
Page 64: 1.1 SIMPLE FUNCTIONS AND EQUATIONSI
Page 68: 1.1 SIMPLE FUNCTIONS AND EQUATIONS
Page 72: 1.1 SIMPLE FUNCTIONS AND EQUATIONS(
Page 76: 1.1 SIMPLE FUNCTIONS AND EQUATIONSW
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1.2 TRIGONOMETRIC IDENTITIESy ′yR
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1.2 TRIGONOMETRIC IDENTITIESConsequ
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1.3 COORDINATE GEOMETRYwith K and
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1.3 COORDINATE GEOMETRYyNP(x, y)OF(
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1.4 PARTIAL FRACTIONSin partial fra
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1.4 PARTIAL FRACTIONSThus any one o
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1.4 PARTIAL FRACTIONSA i but linear
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1.5 BINOMIAL EXPANSIONx = −1 give
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1.6 PROPERTIES OF BINOMIAL COEFFICI
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1.6 PROPERTIES OF BINOMIAL COEFFICI
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1.7 SOME PARTICULAR METHODS OF PROO
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1.8 EXERCISES(a) the sum of the sin
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1.9 HINTS AND ANSWERS1.27 Establish
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2Preliminary calculusThis chapter i
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2.1 DIFFERENTIATIONapproximate the
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2.1 DIFFERENTIATIONseparation is no
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2.1 DIFFERENTIATION◮Find the deri
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2.1 DIFFERENTIATIONThe pattern emer
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2.1 DIFFERENTIATIONgradient of the
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2.1 DIFFERENTIATIONf(x)C∆θρPQθ
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2.1 DIFFERENTIATION◮Show that the
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2.1 DIFFERENTIATIONand hence the di
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2.2 INTEGRATIONf(x)abxFigure 2.7An
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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
Page 1692:
23.7 SCHMIDT-HILBERT THEORYLet us b
Page 1696:
23.8 EXERCISESthus Hermitian. In or
Page 1700:
23.8 EXERCISES(b) Obtain the eigenv
Page 1704:
23.9 HINTS AND ANSWERS23.9 Hints an
Page 1708:
24.1 FUNCTIONS OF A COMPLEX VARIABL
Page 1712:
24.2 THE CAUCHY-RIEMANN RELATIONS
Page 1716:
24.2 THE CAUCHY-RIEMANN RELATIONSSi
Page 1720:
24.3 POWER SERIES IN A COMPLEX VARI
Page 1724:
24.4 SOME ELEMENTARY FUNCTIONSreal-
Page 1728:
24.5 MULTIVALUED FUNCTIONS AND BRAN
Page 1732:
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
Page 1748:
24.8 COMPLEX INTEGRALSyw 3 s w 3w =
Page 1752:
24.8 COMPLEX INTEGRALSyRtyyC 1 C 2R
Page 1756:
24.9 CAUCHY’S THEOREMnamely Cauch
Page 1760:
24.10 CAUCHY’S INTEGRAL FORMULAyC
Page 1764:
24.11 TAYLOR AND LAURENT SERIESFurt
Page 1768:
24.11 TAYLOR AND LAURENT SERIESof o
Page 1772:
24.11 TAYLOR AND LAURENT SERIESdeno
Page 1776:
24.12 RESIDUE THEOREMSuppose the fu
Page 1780:
24.13 DEFINITE INTEGRALS USING CONT
Page 1784:
Page 1788:
Page 1792:
24.14 EXERCISESWe have seen that
Page 1796:
24.14 EXERCISES24.14 Prove that, fo
Page 1800:
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
Page 1816:
25.3 LOCATION OF ZEROSφ =0yπ/αz
Page 1820:
25.3 LOCATION OF ZEROSpolynomials,
Page 1824:
25.4 SUMMATION OF SERIES◮By consi
Page 1828:
25.5 INVERSE LAPLACE TRANSFORMΓRΓ
Page 1832:
25.5 INVERSE LAPLACE TRANSFORMf(x)1
Page 1836:
25.6 STOKES’ EQUATION AND AIRY IN
Page 1840:
Page 1844:
Page 1848:
25.7 WKB METHODSthere exist many re
Page 1852:
25.7 WKB METHODSThis still requires
Page 1856:
25.7 WKB METHODSThe precise combina
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25.7 WKB METHODSfor some constant A
Page 1864:
25.7 WKB METHODSone function and th
Page 1868:
25.8 APPROXIMATIONS TO INTEGRALSFin
Page 1872:
25.8 APPROXIMATIONS TO INTEGRALSFro
Page 1876:
25.8 APPROXIMATIONS TO INTEGRALSany
Page 1880:
25.8 APPROXIMATIONS TO INTEGRALSto
Page 1884:
25.8 APPROXIMATIONS TO INTEGRALSwhi
Page 1888:
25.8 APPROXIMATIONS TO INTEGRALS(a)
Page 1892:
25.8 APPROXIMATIONS TO INTEGRALSare
Page 1896:
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
Page 1912:
26TensorsIt may seem obvious that t
Page 1916:
26.2 CHANGE OF BASISIn the second o
Page 1920:
26.3 CARTESIAN TENSORSx 2x ′ 1x
Page 1924:
26.4 FIRST- AND ZERO-ORDER CARTESIA
Page 1928:
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
Page 1944:
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:
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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