Mathematical Methods for Physics and Engineering - Matematica.NET

More documents

Recommendations

Info

CONTENTS9.3 Rayleigh–Ritz method 3279.4 Exercises 3299.5 Hints and answers 33210 Vector calculus 33410.1 Differentiation of vectors 334Composite vector expressions; differential of a vector10.2 Integration of vectors 33910.3 Space curves 34010.4 Vector functions of several arguments 34410.5 Surfaces 34510.6 Scalar and vector fields 34710.7 Vector operators 347Gradient of a scalar field; divergence of a vector field; curl of a vector field10.8 Vector operator formulae 354Vector operators acting on sums and products; combinations of grad, div andcurl10.9 Cylindrical and spherical polar coordinates 35710.10 General curvilinear coordinates 36410.11 Exercises 36910.12 Hints and answers 37511 Line, surface and volume integrals 37711.1 Line integrals 377Evaluating line integrals; physical examples; line integrals with respect to ascalar11.2 Connectivity of regions 38311.3 Green’s theorem in a plane 38411.4 Conservative fields and potentials 38711.5 Surface integrals 389Evaluating surface integrals; vector areas of surfaces; physical examples11.6 Volume integrals 396Volumes of three-dimensional regions11.7 Integral forms for grad, div and curl 39811.8 Divergence theorem and related theorems 401Green’s theorems; other related integral theorems; physical applications11.9 Stokes’ theorem and related theorems 406Related integral theorems; physical applications11.10 Exercises 40911.11 Hints and answers 41412 Fourier series 41512.1 The Dirichlet conditions 415ix
Page 2: This page intentionally left blank
Page 6: Physicists. He is also a Director o
Page 10: cambridge university pressCambridge
Page 14: CONTENTS2.2 Integration 59Integrati
Page 18: CONTENTS7.7 Equations of lines, pla
Page 24: CONTENTS15.3 General ordinary diffe
Page 28: CONTENTS21.5 Inhomogeneous problems
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
Page 80:
1.2 TRIGONOMETRIC IDENTITIESy ′yR
Page 84:
1.2 TRIGONOMETRIC IDENTITIESConsequ
Page 88:
1.3 COORDINATE GEOMETRYwith K and
Page 92:
1.3 COORDINATE GEOMETRYyNP(x, y)OF(
Page 96:
1.4 PARTIAL FRACTIONSin partial fra
Page 100:
1.4 PARTIAL FRACTIONSThus any one o
Page 104:
1.4 PARTIAL FRACTIONSA i but linear
Page 108:
1.5 BINOMIAL EXPANSIONx = −1 give
Page 112:
1.6 PROPERTIES OF BINOMIAL COEFFICI
Page 116:
1.6 PROPERTIES OF BINOMIAL COEFFICI
Page 120:
1.7 SOME PARTICULAR METHODS OF PROO
Page 124:
Page 128:
Page 132:
1.8 EXERCISES(a) the sum of the sin
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
Page 184:
2.2 INTEGRATIONfound near the end o
Page 188:
2.2 INTEGRATION◮Evaluate the inte
Page 192:
2.2 INTEGRATION◮Evaluate the inte
Page 196:
2.2 INTEGRATIONRearranging this exp
Page 200:
2.2 INTEGRATIONyCρ(φ + dφ)dAρ(
Page 204:
2.2 INTEGRATIONf(x)mabxFigure 2.10T
Page 208:
2.2 INTEGRATIONydsf(x)adxVbxSFigure
Page 212:
2.3 EXERCISES2.10 The function y(x)
Page 216:
2.3 EXERCISES2.26 Use the mean valu
Page 220:
2.4 HINTS AND ANSWERS2.45 If J r is
Page 224:
3Complex numbers andhyperbolic func
Page 228:
3.2 MANIPULATION OF COMPLEX NUMBERS
Page 232:
Page 236:
Page 240:
Page 244:
3.3 POLAR REPRESENTATION OF COMPLEX
Page 248:
3.4 DE MOIVRE’S THEOREMIm zr 1 e
Page 252:
3.4 DE MOIVRE’S THEOREM◮Find an
Page 256:
3.5 COMPLEX LOGARITHMS AND COMPLEX
Page 260:
3.6 APPLICATIONS TO DIFFERENTIATION
Page 264:
3.7 HYPERBOLIC FUNCTIONS43cosh x21s
Page 268:
3.7 HYPERBOLIC FUNCTIONS3.7.4 Solvi
Page 272:
3.7 HYPERBOLIC FUNCTIONS4cosech −
Page 276:
3.8 EXERCISES◮Evaluate (d/dx)sinh
Page 280:
3.8 EXERCISES3.13 Prove that x 2m+1
Page 284:
3.9 HINTS AND ANSWERS3.27 A closed
Page 288:
4Series and limits4.1 SeriesMany ex
Page 292:
4.2 SUMMATION OF SERIES4.2.1 Arithm
Page 296:
4.2 SUMMATION OF SERIES◮Sum the s
Page 300:
4.2 SUMMATION OF SERIES4.2.5 Series
Page 304:
4.2 SUMMATION OF SERIESIntegrating
Page 308:
4.3 CONVERGENCE OF INFINITE SERIES4
Page 312:
4.3 CONVERGENCE OF INFINITE SERIESR
Page 316:
4.3 CONVERGENCE OF INFINITE SERIESU
Page 320:
4.4 OPERATIONS WITH SERIESis always
Page 324:
4.5 POWER SERIES◮Determine the ra
Page 328:
4.5 POWER SERIESQ(x) intoP (x) toob
Page 332:
4.6 TAYLOR SERIESf(x)QRf(a)Pθhhf
Page 336:
4.6 TAYLOR SERIESWe may follow a si
Page 340:
4.7 EVALUATION OF LIMITSThese can a
Page 344:
4.7 EVALUATION OF LIMITSTherefore w
Page 348:
4.8 EXERCISES4.8 The N + 1 complex
Page 352:
4.8 EXERCISES4.20 Identify the seri
Page 356:
4.9 HINTS AND ANSWERSfind a closed-
Page 360:
5Partial differentiationIn chapter
Page 364:
5.2 THE TOTAL DIFFERENTIAL AND TOTA
Page 368:
5.3 EXACT AND INEXACT DIFFERENTIALS
Page 372:
5.4 USEFUL THEOREMS OF PARTIAL DIFF
Page 376:
5.6 CHANGE OF VARIABLESyρφxFigure
Page 380:
5.7 TAYLOR’S THEOREM FOR MANY-VAR
Page 384:
5.8 STATIONARY VALUES OF MANY-VARIA
Page 388:
5.8 STATIONARY VALUES OF MANY-VARIA
Page 392:
5.9 STATIONARY VALUES UNDER CONSTRA
Page 396:
Page 400:
Page 404:
5.10 ENVELOPESWe now have the gener
Page 408:
5.10 ENVELOPESh is made arbitrarily
Page 412:
5.11 THERMODYNAMIC RELATIONS◮Show
Page 416:
5.13 EXERCISESconstant limits of in
Page 420:
5.13 EXERCISES5.9 The function f(x,
Page 424:
5.13 EXERCISESyORθθ2θxFigure 5.5
Page 428:
5.14 HINTS AND ANSWERS5.33 If∫ 1I
Page 432:
6Multiple integralsFor functions of
Page 436:
6.1 DOUBLE INTEGRALSAn alternative
Page 440:
6.3 APPLICATIONS OF MULTIPLE INTEGR
Page 444:
Page 448:
Page 452:
Page 456:
6.4 CHANGE OF VARIABLES IN MULTIPLE
Page 460:
Page 464:
Page 468:
Page 472:
6.5 EXERCISESand similarly for J yz
Page 476:
6.5 EXERCISESThis is an example of
Page 480:
6.6 HINTS AND ANSWERS(a) Let R be a
Page 484:
7.2 ADDITION AND SUBTRACTION OF VEC
Page 488:
7.3 MULTIPLICATION BY A SCALARλ aa
Page 492:
7.4 BASIS VECTORS AND COMPONENTSThe
Page 496:
7.6 MULTIPLICATION OF VECTORSbOθ b
Page 500:
7.6 MULTIPLICATION OF VECTORSIf we
Page 504:
7.6 MULTIPLICATION OF VECTORSPFθRr
Page 508:
7.6 MULTIPLICATION OF VECTORSvPOφ
Page 512:
7.7 EQUATIONS OF LINES, PLANES AND
Page 516:
7.8 USING VECTORS TO FIND DISTANCES
Page 520:
7.8 USING VECTORS TO FIND DISTANCES
Page 524:
7.9 RECIPROCAL VECTORSthe line to t
Page 528:
7.10 EXERCISES7.2 A unit cell of di
Page 532:
7.10 EXERCISESabcdaFigure 7.17A fac
Page 536:
7.10 EXERCISESV 0 cos ωtV 4V 1V 2R
Page 540:
8Matrices and vector spacesIn the p
Page 544:
8.1 VECTOR SPACESthe trivial case i
Page 548:
8.1 VECTOR SPACESIn the above basis
Page 552:
8.2 LINEAR OPERATORSwhere the equal
Page 556:
8.3 MATRICES8.2.1 Properties of lin
Page 560:
8.4 BASIC MATRIX ALGEBRANow, since
Page 564:
8.4 BASIC MATRIX ALGEBRAexcept for
Page 568:
8.5 FUNCTIONS OF MATRICESThe identi
Page 572:
8.7 THE COMPLEX AND HERMITIAN CONJU
Page 576:
8.9 THE DETERMINANT OF A MATRIXwhic
Page 580:
8.9 THE DETERMINANT OF A MATRIX◮S
Page 584:
8.10 THE INVERSE OF A MATRIX◮Eval
Page 588:
8.10 THE INVERSE OF A MATRIX◮Find
Page 592:
8.11 THE RANK OF A MATRIX8.11 The r
Page 596:
8.12 SPECIAL TYPES OF SQUARE MATRIX
Page 600:
8.12 SPECIAL TYPES OF SQUARE MATRIX
Page 604:
8.13 EIGENVECTORS AND EIGENVALUESre
Page 608:
8.13 EIGENVECTORS AND EIGENVALUESwr
Page 612:
8.13 EIGENVECTORS AND EIGENVALUESBu
Page 616:
8.13 EIGENVECTORS AND EIGENVALUESei
Page 620:
8.14 DETERMINATION OF EIGENVALUES A
Page 624:
8.15 CHANGE OF BASIS AND SIMILARITY
Page 628:
8.16 DIAGONALISATION OF MATRICESort
Page 632:
8.16 DIAGONALISATION OF MATRICES◮
Page 636:
8.17 QUADRATIC AND HERMITIAN FORMSi
Page 640:
8.17 QUADRATIC AND HERMITIAN FORMSa
Page 644:
8.18 SIMULTANEOUS LINEAR EQUATIONSw
Page 648:
8.18 SIMULTANEOUS LINEAR EQUATIONSa
Page 652:
8.18 SIMULTANEOUS LINEAR EQUATIONSt
Page 656:
Page 660:
8.18 SIMULTANEOUS LINEAR EQUATIONS(
Page 664:
8.18 SIMULTANEOUS LINEAR EQUATIONS
Page 668:
8.18 SIMULTANEOUS LINEAR EQUATIONSW
Page 672:
8.19 EXERCISESwhere U and V are giv
Page 676:
8.19 EXERCISES(b) Without assuming
Page 680:
8.19 EXERCISES8.20 Demonstrate that
Page 684:
8.19 EXERCISESis 2 and that an orth
Page 688:
8.20 HINTS AND ANSWERS8.5 Use the (
Page 692:
9.1 TYPICAL OSCILLATORY SYSTEMScorr
Page 696:
9.1 TYPICAL OSCILLATORY SYSTEMScoor
Page 700:
9.1 TYPICAL OSCILLATORY SYSTEMSmk
Page 704:
9.2 SYMMETRY AND NORMAL MODESy 1 y
Page 708:
9.2 SYMMETRY AND NORMAL MODES(a) ω
Page 712:
9.3 RAYLEIGH-RITZ METHODand that th
Page 716:
9.4 EXERCISES◮Estimate the eigenf
Page 720:
9.4 EXERCISESthe figure and obtain
Page 724:
9.5 HINTS AND ANSWERS1 2 3mMm(a) ω
Page 728:
10.1 DIFFERENTIATION OF VECTORSa(u
Page 732:
10.1 DIFFERENTIATION OF VECTORSin t
Page 736:
10.2 INTEGRATION OF VECTORSNote tha
Page 740:
10.3 SPACE CURVESThis parametric re
Page 744:
10.3 SPACE CURVESso we finally obta
Page 748:
10.5 SURFACESzT∂r∂uSu = c 1P∂
Page 752:
10.6 SCALAR AND VECTOR FIELDSA norm
Page 756:
10.7 VECTOR OPERATORS∇φaθQPdφd
Page 760:
10.7 VECTOR OPERATORSz(0, 0,a)ˆn 0
Page 764:
10.7 VECTOR OPERATORS10.7.3 Curl of
Page 768:
10.8 VECTOR OPERATOR FORMULAE◮Sho
Page 772:
10.9 CYLINDRICAL AND SPHERICAL POLA
Page 776:
Page 780:
Page 784:
Page 788:
10.10 GENERAL CURVILINEAR COORDINAT
Page 792:
10.10 GENERAL CURVILINEAR COORDINAT
Page 796:
10.11 EXERCISES∇Φ =∇ · a =∇
Page 800:
10.11 EXERCISES10.6 Prove that for
Page 804:
10.11 EXERCISES(a) For cylindrical
Page 808:
10.12 HINTS AND ANSWERS(a) Express
Page 812:
11Line, surface and volume integral
Page 816:
11.1 LINE INTEGRALSA similar proced
Page 820:
11.1 LINE INTEGRALS◮Evaluate the
Page 824:
11.2 CONNECTIVITY OF REGIONS(a) (b)
Page 828:
11.3 GREEN’S THEOREM IN A PLANEy
Page 832:
11.4 CONSERVATIVE FIELDS AND POTENT
Page 836:
11.5 SURFACE INTEGRALSindependent o
Page 840:
11.5 SURFACE INTEGRALSzkαdSSyxRdAF
Page 844:
11.5 SURFACE INTEGRALSSzadSxaCadA =
Page 848:
11.5 SURFACE INTEGRALS◮Find the v
Page 852:
11.6 VOLUME INTEGRALSdSSVrOFigure 1
Page 856:
11.7 INTEGRAL FORMS FOR grad, div A
Page 860:
11.8 DIVERGENCE THEOREM AND RELATED
Page 864:
Page 868:
Page 872:
11.9 STOKES’ THEOREM AND RELATED
Page 876:
11.10 EXERCISESeverywhere except on
Page 880:
11.10 EXERCISES11.12 Show that the
Page 884:
11.10 EXERCISES11.24 Prove equation
Page 888:
12Fourier seriesWe have already dis
Page 892:
12.2 THE FOURIER COEFFICIENTSwe can
Page 896:
12.3 SYMMETRY CONSIDERATIONSf(t)1
Page 900:
12.4 DISCONTINUOUS FUNCTIONS(a)1(b)
Page 904:
12.5 NON-PERIODIC FUNCTIONSf(x) =x
Page 908:
12.7 COMPLEX FOURIER SERIESwhere th
Page 912:
12.9 EXERCISESthe sine and cosine f
Page 916:
12.9 EXERCISESDeduce the value of t
Page 920:
12.10 HINTS AND ANSWERS0 1 0 1 0 1
Page 924:
13Integral transformsIn the previou
Page 928:
13.1 FOURIER TRANSFORMSand (13.3) b
Page 932:
13.1 FOURIER TRANSFORMSis a wavefun
Page 936:
13.1 FOURIER TRANSFORMSf(y)1−a−
Page 940:
13.1 FOURIER TRANSFORMSThe derivati
Page 944:
13.1 FOURIER TRANSFORMS˜fΩ2Ω(2π
Page 948:
13.1 FOURIER TRANSFORMSIgnoring in
Page 952:
13.1 FOURIER TRANSFORMSf(x)∗ g(y)
Page 956:
13.1 FOURIER TRANSFORMSThe inverse
Page 960:
13.1 FOURIER TRANSFORMSobtained sim
Page 964:
13.2 LAPLACE TRANSFORMSA similar re
Page 968:
13.2 LAPLACE TRANSFORMSf(t) ¯f(s)
Page 972:
13.2 LAPLACE TRANSFORMSWe may now c
Page 976:
13.3 CONCLUDING REMARKSThe properti
Page 980:
13.4 EXERCISESDetermine the convolu
Page 984:
13.4 EXERCISES(a) Find the Fourier
Page 988:
13.4 EXERCISES(c) L [sinh at cos bt
Page 992:
13.5 HINTS AND ANSWERS13.17 Ṽ (k)
Page 996:
14.1 GENERAL FORM OF SOLUTIONthe ap
Page 1000:
14.2 FIRST-DEGREE FIRST-ORDER EQUAT
Page 1004:
Page 1008:
Page 1012:
Page 1016:
Page 1020:
14.3 HIGHER-DEGREE FIRST-ORDER EQUA
Page 1024:
14.3 HIGHER-DEGREE FIRST-ORDER EQUA
Page 1028:
14.4 EXERCISES14.5 By finding suita
Page 1032:
14.4 EXERCISES(c) Find an appropria
Page 1036:
14.5 HINTS AND ANSWERS14.31 Show th
Page 1040:
HIGHER-ORDER ORDINARY DIFFERENTIAL
Page 1044:
15.1 LINEAR EQUATIONS WITH CONSTANT
Page 1048:
Page 1052:
Page 1056:
Page 1060:
Page 1064:
15.2 LINEAR EQUATIONS WITH VARIABLE
Page 1068:
Page 1072:
Page 1076:
Page 1080:
Page 1084:
Page 1088:
Page 1092:
Page 1096:
15.3 GENERAL ORDINARY DIFFERENTIAL
Page 1100:
15.3 GENERAL ORDINARY DIFFERENTIAL
Page 1104:
15.4 EXERCISES15.3.6 Equations havi
Page 1108:
15.4 EXERCISES15.9 Find the general
Page 1112:
15.4 EXERCISES15.23 Prove that the
Page 1116:
15.5 HINTS AND ANSWERS15.36 Find th
Page 1120:
16Series solutions of ordinarydiffe
Page 1124:
16.1 SECOND-ORDER LINEAR ORDINARY D
Page 1128:
16.2 SERIES SOLUTIONS ABOUT AN ORDI
Page 1132:
16.2 SERIES SOLUTIONS ABOUT AN ORDI
Page 1136:
16.3 SERIES SOLUTIONS ABOUT A REGUL
Page 1140:
Page 1144:
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
Page 1180:
17.3 PROPERTIES OF HERMITIAN OPERAT
Page 1184:
17.3 PROPERTIES OF HERMITIAN OPERAT
Page 1188:
17.4 STURM-LIOUVILLE EQUATIONScerta
Page 1192:
17.4 STURM-LIOUVILLE EQUATIONS(ii)
Page 1196:
17.5 SUPERPOSITION OF EIGENFUNCTION
Page 1200:
17.5 SUPERPOSITION OF EIGENFUNCTION
Page 1204:
17.7 EXERCISESWe note that if µ =
Page 1208:
17.7 EXERCISESwhere κ is a constan
Page 1212:
18Special functionsIn the previous
Page 1216:
18.1 LEGENDRE FUNCTIONS2P 01P 1−1
Page 1220:
18.1 LEGENDRE FUNCTIONS1Q 00.5−1
Page 1224:
18.1 LEGENDRE FUNCTIONSMutual ortho
Page 1228:
18.1 LEGENDRE FUNCTIONSEquation (18
Page 1232:
18.2 ASSOCIATED LEGENDRE FUNCTIONS1
Page 1236:
18.2 ASSOCIATED LEGENDRE FUNCTIONSw
Page 1240:
18.2 ASSOCIATED LEGENDRE FUNCTIONSS
Page 1244:
18.3 SPHERICAL HARMONICSbe derived
Page 1248:
18.4 CHEBYSHEV FUNCTIONSSince δ(Ω
Page 1252:
18.4 CHEBYSHEV FUNCTIONS1T 0T 2T 30
Page 1256:
18.4 CHEBYSHEV FUNCTIONSEvaluating
Page 1260:
18.4 CHEBYSHEV FUNCTIONSin which th
Page 1264:
18.5 BESSEL FUNCTIONSgenerality. Th
Page 1268:
18.5 BESSEL FUNCTIONSWe note that B
Page 1272:
18.5 BESSEL FUNCTIONSand hence that
Page 1276:
18.5 BESSEL FUNCTIONSTo determine t
Page 1280:
18.5 BESSEL FUNCTIONS◮Prove the e
Page 1284:
18.5 BESSEL FUNCTIONSin subsection
Page 1288:
18.6 SPHERICAL BESSEL FUNCTIONSwher
Page 1292:
18.7 LAGUERRE FUNCTIONSit has a reg
Page 1296:
18.7 LAGUERRE FUNCTIONS◮Prove tha
Page 1300:
18.8 ASSOCIATED LAGUERRE FUNCTIONSw
Page 1304:
18.8 ASSOCIATED LAGUERRE FUNCTIONS
Page 1308:
18.9 HERMITE FUNCTIONS105−1.5H 0H
Page 1312:
18.9 HERMITE FUNCTIONS◮Show thatI
Page 1316:
18.10 HYPERGEOMETRIC FUNCTIONSby ma
Page 1320:
18.10 HYPERGEOMETRIC FUNCTIONSF(a,
Page 1324:
18.11 CONFLUENT HYPERGEOMETRIC FUNC
Page 1328:
18.12 THE GAMMA FUNCTION AND RELATE
Page 1332:
Page 1336:
Page 1340:
18.13 EXERCISES√√Y0 0 = 1, Y 04
Page 1344:
18.13 EXERCISES[ You will find it c
Page 1348:
18.13 EXERCISES(a) use their series
Page 1352:
18.14 HINTS AND ANSWERS18.15 (a) Sh
Page 1356:
19.1 OPERATOR FORMALISMrepresent di
Page 1360:
19.1 OPERATOR FORMALISMspectrum of
Page 1364:
19.1 OPERATOR FORMALISMwhilstthatfo
Page 1368:
19.1 OPERATOR FORMALISMdefining ser
Page 1372:
19.2 PHYSICAL EXAMPLES OF OPERATORS
Page 1376:
Page 1380:
Page 1384:
Page 1388:
Page 1392:
Page 1396:
Page 1400:
19.3 EXERCISESthat would involve a
Page 1404:
19.3 EXERCISESNow evaluate the expe
Page 1408:
20Partial differential equations:ge
Page 1412:
20.1 IMPORTANT PARTIAL DIFFERENTIAL
Page 1416:
20.1 IMPORTANT PARTIAL DIFFERENTIAL
Page 1420:
20.3 GENERAL AND PARTICULAR SOLUTIO
Page 1424:
Page 1428:
Page 1432:
Page 1436:
Page 1440:
Page 1444:
20.4 THE WAVE EQUATION20.4 The wave
Page 1448:
20.5 THE DIFFUSION EQUATIONterm is
Page 1452:
20.5 THE DIFFUSION EQUATIONwritten
Page 1456:
20.6 CHARACTERISTICS AND THE EXISTE
Page 1460:
Page 1464:
Page 1468:
20.7 UNIQUENESS OF SOLUTIONSEquatio
Page 1472:
20.8 EXERCISESWe also note that oft
Page 1476:
20.8 EXERCISES20.14 Solve∂ 2 u u
Page 1480:
20.9 HINTS AND ANSWERS20.25 The Kle
Page 1484:
21Partial differential equations:se
Page 1488:
21.1 SEPARATION OF VARIABLES: THE G
Page 1492:
21.2 SUPERPOSITION OF SEPARATED SOL
Page 1496:
Page 1500:
Page 1504:
Page 1508:
21.3 SEPARATION OF VARIABLES IN POL
Page 1512:
Page 1516:
Page 1520:
Page 1524:
Page 1528:
Page 1532:
Page 1536:
Page 1540:
Page 1544:
Page 1548:
Page 1552:
21.4 INTEGRAL TRANSFORM METHODS21.4
Page 1556:
21.4 INTEGRAL TRANSFORM METHODS◮A
Page 1560:
21.5 INHOMOGENEOUS PROBLEMS - GREEN
Page 1564:
Page 1568:
Page 1572:
Page 1576:
Page 1580:
Page 1584:
Page 1588:
Page 1592:
21.6 EXERCISESUsing plane polar coo
Page 1596:
21.6 EXERCISES(a) Evaluate dPl m(µ
Page 1600:
21.6 EXERCISES21.18 A sphere of rad
Page 1604:
21.7 HINTS AND ANSWERSin V and take
Page 1608:
22Calculus of variationsIn chapters
Page 1612:
22.2 SPECIAL CASESto these variatio
Page 1616:
22.2 SPECIAL CASESydsdydxxFigure 22
Page 1620:
22.3 SOME EXTENSIONSbzρ−ba(a) (b
Page 1624:
22.3 SOME EXTENSIONSy(x)+η(x)∆yy
Page 1628:
22.4 CONSTRAINED VARIATIONwhere k i
Page 1632:
22.5 PHYSICAL VARIATIONAL PRINCIPLE
Page 1636:
22.5 PHYSICAL VARIATIONAL PRINCIPLE
Page 1640:
22.6 GENERAL EIGENVALUE PROBLEMScon
Page 1644:
22.7 ESTIMATION OF EIGENVALUES AND
Page 1648:
22.8 ADJUSTMENT OF PARAMETERSIt is
Page 1652:
22.9 EXERCISES22.9 Exercises22.1 A
Page 1656:
22.9 EXERCISESpath of a small test
Page 1660:
22.10 HINTS AND ANSWERStotal energy
Page 1664:
23Integral equationsIt is not unusu
Page 1668:
23.3 OPERATOR NOTATION AND THE EXIS
Page 1672:
23.4 CLOSED-FORM SOLUTIONS23.4.1 Se
Page 1676:
23.4 CLOSED-FORM SOLUTIONS23.4.2 In
Page 1680:
23.4 CLOSED-FORM SOLUTIONSso we can
Page 1684:
23.5 NEUMANN SERIES23.5 Neumann ser
Page 1688:
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
Page 1736:
24.7 CONFORMAL TRANSFORMATIONSThus
Page 1740:
24.7 CONFORMAL TRANSFORMATIONSpoint
Page 1744:
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
Page 1804:
25.1 COMPLEX POTENTIALSthe field pr
Page 1808:
25.1 COMPLEX POTENTIALSyQxPˆnFigur
Page 1812:
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
Page 1860:
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
Page 1932:
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
show all

Mathematical Methods for Physics and Engineering - Matematica.NET

Create successful ePaper yourself

Delete template?

Save as template?