Mathematical Methods for Physics and Engineering - Matematica.NET

More documents

Recommendations

Info

FOURIER SERIES12.21 c n =[(−1) n sinh π]/[π(1 + n 2 )]. Having set x = 0, separate out the n =0termand note that (−1) n =(−1) −n .12.23 (π 2 − 8)/16.12.25 (b) All a n and α n are zero; b n =2(−1) n+1 /(nπ) andβ n =4/(nπ). You will needthe result quoted in exercise 12.19.432
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 22:
CONTENTS12.2 The Fourier coefficien
Page 26:
CONTENTS18.6 Spherical Bessel funct
Page 30:
CONTENTS24.9 Cauchy’s theorem 849
Page 34:
CONTENTS29.6 Characters 1092Orthogo
Page 40:
CONTENTSI am the very Model for a S
Page 44:
PREFACE TO THE THIRD EDITIONthe phy
Page 48:
Preface to the second editionSince
Page 52:
Preface to the first editionA knowl
Page 56:
PREFACE TO THE FIRST EDITIONsupport
Page 62:
PRELIMINARY ALGEBRAforms an equatio
Page 66:
PRELIMINARY ALGEBRAmany real roots
Page 70:
PRELIMINARY ALGEBRAat a value of x
Page 74:
PRELIMINARY ALGEBRAwhere f 1 (x) is
Page 78:
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 134:
PRELIMINARY ALGEBRA1.16 Express the
Page 138:
PRELIMINARY ALGEBRA1.11 Show that t
Page 142:
PRELIMINARY CALCULUSf(x +∆x)AP∆
Page 146:
PRELIMINARY CALCULUS◮Find from fi
Page 150:
PRELIMINARY CALCULUSand using (2.6)
Page 154:
PRELIMINARY CALCULUS◮Find dy/dx i
Page 158:
PRELIMINARY CALCULUSf(x)QABCSFigure
Page 162:
PRELIMINARY CALCULUSf(x)GxFigure 2.
Page 166:
PRELIMINARY CALCULUSrelative to the
Page 170:
PRELIMINARY CALCULUSf(x)a b cxFigur
Page 174:
PRELIMINARY CALCULUSIn each case, a
Page 178:
PRELIMINARY CALCULUSf(x)ax 1 x 2 x
Page 182:
PRELIMINARY CALCULUSFrom the last t
Page 186:
PRELIMINARY CALCULUS◮Evaluate the
Page 190:
PRELIMINARY CALCULUSSincethe requir
Page 194:
PRELIMINARY CALCULUSThe separation
Page 198:
PRELIMINARY CALCULUS2.2.10 Infinite
Page 202:
PRELIMINARY CALCULUS2.2.12 Integral
Page 206:
PRELIMINARY CALCULUSf(x)y = f(x)∆
Page 210:
PRELIMINARY CALCULUS◮Find the vol
Page 214:
PRELIMINARY CALCULUSOcCρr +∆rrρ
Page 218:
PRELIMINARY CALCULUS(c) [(x − a)/
Page 222:
PRELIMINARY CALCULUSy2aπa2πaxFigu
Page 226:
COMPLEX NUMBERS AND HYPERBOLIC FUNC
Page 230:
Page 234:
Page 238:
Page 242:
Page 246:
Page 250:
Page 254:
Page 258:
Page 262:
Page 266:
Page 270:
Page 274:
Page 278:
Page 282:
Page 286:
Page 290:
SERIES AND LIMITSsome sort of relat
Page 294:
SERIES AND LIMITSFor a series with
Page 298:
SERIES AND LIMITSThe difference met
Page 302:
SERIES AND LIMITS◮Sum the seriesN
Page 306:
SERIES AND LIMITSAgain using the Ma
Page 310:
SERIES AND LIMITSwhich is merely th
Page 314:
SERIES AND LIMITS◮Given that the
Page 318:
SERIES AND LIMITSThe divergence of
Page 322:
SERIES AND LIMITSalthough in princi
Page 326:
SERIES AND LIMITSr = − exp iθ. T
Page 330:
SERIES AND LIMITS4.6 Taylor seriesT
Page 334:
SERIES AND LIMITSx = a + h in the a
Page 338:
SERIES AND LIMITSvalue of ξ that s
Page 342:
SERIES AND LIMITS◮Evaluate the li
Page 346:
SERIES AND LIMITSSummary of methods
Page 350:
SERIES AND LIMITS4.15 Prove that∞
Page 354:
SERIES AND LIMITSsin 3x(a) limx→0
Page 358:
SERIES AND LIMITS4.15 Divide the se
Page 362:
PARTIAL DIFFERENTIATIONto x and y r
Page 366:
PARTIAL DIFFERENTIATIONcan be obtai
Page 370:
PARTIAL DIFFERENTIATIONit exact. Co
Page 374:
PARTIAL DIFFERENTIATIONFrom equatio
Page 378:
PARTIAL DIFFERENTIATIONThus, from (
Page 382:
PARTIAL DIFFERENTIATIONtheorem then
Page 386:
PARTIAL DIFFERENTIATIONTo establish
Page 390:
PARTIAL DIFFERENTIATIONmaximum0.40.
Page 394:
PARTIAL DIFFERENTIATIONvaried. Howe
Page 398:
PARTIAL DIFFERENTIATION◮Find the
Page 402:
PARTIAL DIFFERENTIATION◮A system
Page 406:
PARTIAL DIFFERENTIATIONP 1PP 2yf(x,
Page 410:
PARTIAL DIFFERENTIATION5.11 Thermod
Page 414:
PARTIAL DIFFERENTIATIONAlthough the
Page 418:
PARTIAL DIFFERENTIATION(a) Find all
Page 422:
PARTIAL DIFFERENTIATIONthe horizont
Page 426:
PARTIAL DIFFERENTIATIONBy consideri
Page 430:
PARTIAL DIFFERENTIATION5.19 The cos
Page 434:
MULTIPLE INTEGRALSydSdxdA = dxdyRVd
Page 438:
MULTIPLE INTEGRALSy1dyRx + y =100dx
Page 442:
MULTIPLE INTEGRALSzcdV = dx dy dzdz
Page 446:
MULTIPLE INTEGRALSzz =2yz = x 2 + y
Page 450:
MULTIPLE INTEGRALSza√a2 − z 2dz
Page 454:
MULTIPLE INTEGRALSaθdCFigure 6.8Su
Page 458:
MULTIPLE INTEGRALSyu =constantv =co
Page 462:
MULTIPLE INTEGRALS◮Evaluate the d
Page 466:
MULTIPLE INTEGRALSzRTu = c 1v = c 2
Page 470:
MULTIPLE INTEGRALSwhich agrees with
Page 474:
MULTIPLE INTEGRALS6.6 The function(
Page 478:
MULTIPLE INTEGRALSover the ellipsoi
Page 482:
7Vector algebraThis chapter introdu
Page 486:
VECTOR ALGEBRAabcb + cbcab + ca +(b
Page 490:
VECTOR ALGEBRACEAGFDacBbOFigure 7.6
Page 494:
VECTOR ALGEBRAkaja z ka y ja x iiFi
Page 498:
VECTOR ALGEBRAFrom (7.15) we see th
Page 502:
VECTOR ALGEBRAa × bθbaFigure 7.9s
Page 506:
VECTOR ALGEBRAis the forward direct
Page 510:
VECTOR ALGEBRA◮Find the volume V
Page 514:
VECTOR ALGEBRAˆnARadrOFigure 7.13
Page 518:
VECTOR ALGEBRAPp − apdAbθaOFigur
Page 522:
VECTOR ALGEBRAQbqˆnPpaOFigure 7.16
Page 526:
VECTOR ALGEBRAnot coplanar. Moreove
Page 530:
VECTOR ALGEBRA7.12 The plane P 1 co
Page 534:
VECTOR ALGEBRA7.22 In subsection 7.
Page 538:
VECTOR ALGEBRAof vector plots for p
Page 542:
MATRICES AND VECTOR SPACESa discuss
Page 546:
MATRICES AND VECTOR SPACESWe reiter
Page 550:
MATRICES AND VECTOR SPACES8.1.3 Som
Page 554:
MATRICES AND VECTOR SPACESmay be th
Page 558:
MATRICES AND VECTOR SPACESIn a simi
Page 562:
MATRICES AND VECTOR SPACES◮The ma
Page 566:
MATRICES AND VECTOR SPACESThese are
Page 570:
MATRICES AND VECTOR SPACES◮Find t
Page 574:
MATRICES AND VECTOR SPACESthe right
Page 578:
MATRICES AND VECTOR SPACESdetermina
Page 582:
MATRICES AND VECTOR SPACESIt follow
Page 586:
MATRICES AND VECTOR SPACESequivalen
Page 590:
MATRICES AND VECTOR SPACESand may b
Page 594:
MATRICES AND VECTOR SPACESmay be sh
Page 598:
MATRICES AND VECTOR SPACESClearly r
Page 602:
MATRICES AND VECTOR SPACESHence 〈
Page 606:
MATRICES AND VECTOR SPACESWe also s
Page 610:
MATRICES AND VECTOR SPACESa result
Page 614:
MATRICES AND VECTOR SPACESHence λ
Page 618:
MATRICES AND VECTOR SPACES8.14 Dete
Page 622:
MATRICES AND VECTOR SPACES◮Constr
Page 626:
MATRICES AND VECTOR SPACESComparing
Page 630:
MATRICES AND VECTOR SPACESthat is,
Page 634:
MATRICES AND VECTOR SPACES| exp A|.
Page 638:
MATRICES AND VECTOR SPACESalso. Ano
Page 642:
MATRICES AND VECTOR SPACES8.17.2 Qu
Page 646:
MATRICES AND VECTOR SPACESIf a vect
Page 650:
MATRICES AND VECTOR SPACES◮Show t
Page 654:
MATRICES AND VECTOR SPACESThis set
Page 658:
MATRICES AND VECTOR SPACESthe uniqu
Page 662:
MATRICES AND VECTOR SPACESthe numbe
Page 666:
MATRICES AND VECTOR SPACESnon-zero
Page 670:
MATRICES AND VECTOR SPACESUsing the
Page 674:
MATRICES AND VECTOR SPACES8.3 Using
Page 678:
MATRICES AND VECTOR SPACES(b) find
Page 682:
MATRICES AND VECTOR SPACES8.26 Show
Page 686:
MATRICES AND VECTOR SPACES8.40 Find
Page 690:
9Normal modesAny student of the phy
Page 694:
NORMAL MODESP P Pθ 1θ 2θ 2lθ 1
Page 698:
NORMAL MODESfrequency corresponds t
Page 702:
NORMAL MODESThe final and most comp
Page 706:
NORMAL MODESThe potential matrix is
Page 710:
NORMAL MODESneous equations for α
Page 714:
NORMAL MODESbe shown that they do p
Page 718:
NORMAL MODESunder gravity. At time
Page 722:
NORMAL MODES9.8 (It is recommended
Page 726:
10Vector calculusIn chapter 7 we di
Page 730:
VECTOR CALCULUSyê φjê ρρiφxFi
Page 734:
VECTOR CALCULUSThe order of the fac
Page 738:
VECTOR CALCULUSzCˆnPˆtˆbr(u)OyxF
Page 742:
VECTOR CALCULUSTherefore, rememberi
Page 746:
VECTOR CALCULUSFinally, we note tha
Page 750:
VECTOR CALCULUStotal derivative, th
Page 754:
VECTOR CALCULUSmathematical point o
Page 758:
VECTOR CALCULUS◮For the function
Page 762:
VECTOR CALCULUSIn addition to these
Page 766:
VECTOR CALCULUS∇(φ + ψ) =∇φ
Page 770:
VECTOR CALCULUSa is a vector field,
Page 774:
VECTOR CALCULUSρ, φ, z, wherex =
Page 778:
VECTOR CALCULUS∇Φ = ∂Φ∂ρ
Page 782:
VECTOR CALCULUSand r ≥ 0, 0 ≤
Page 786:
VECTOR CALCULUS10.10 General curvil
Page 790:
VECTOR CALCULUSFor orthogonal coord
Page 794:
VECTOR CALCULUS◮Prove the express
Page 798:
VECTOR CALCULUS10.3 The general equ
Page 802:
VECTOR CALCULUSUse this formula to
Page 806:
VECTOR CALCULUS10.21 Paraboloidal c
Page 810:
VECTOR CALCULUS10.23 The tangent ve
Page 814:
LINE, SURFACE AND VOLUME INTEGRALSE
Page 818:
LINE, SURFACE AND VOLUME INTEGRALSy
Page 822:
LINE, SURFACE AND VOLUME INTEGRALSi
Page 826:
Page 830:
Page 834:
LINE, SURFACE AND VOLUME INTEGRALSw
Page 838:
LINE, SURFACE AND VOLUME INTEGRALSS
Page 842:
LINE, SURFACE AND VOLUME INTEGRALSw
Page 846:
LINE, SURFACE AND VOLUME INTEGRALSd
Page 850:
LINE, SURFACE AND VOLUME INTEGRALSI
Page 854:
LINE, SURFACE AND VOLUME INTEGRALS1
Page 858:
LINE, SURFACE AND VOLUME INTEGRALSz
Page 862:
Page 866:
LINE, SURFACE AND VOLUME INTEGRALS1
Page 870: LINE, SURFACE AND VOLUME INTEGRALS1
Page 874: LINE, SURFACE AND VOLUME INTEGRALSS
Page 878: LINE, SURFACE AND VOLUME INTEGRALSi
Page 890: FOURIER SERIESf(x)xLLFigure 12.1 An
Page 894: FOURIER SERIESapply for r = 0 as we
Page 898: FOURIER SERIESare not used as often
Page 902: FOURIER SERIES(a)0L(b)0L2L(c)0L2L(d
Page 906: FOURIER SERIESconverge to the corre
Page 910: FOURIER SERIES12.8 Parseval’s the
Page 914: FOURIER SERIESbe better for numeric
Page 918: FOURIER SERIES12.21 Find the comple
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:
27.3 SIMULTANEOUS LINEAR EQUATIONSt
Page 2056:
. . .. . .27.3 SIMULTANEOUS LINEAR
Page 2060:
27.4 NUMERICAL INTEGRATION(a) (b) (
Page 2064:
27.4 NUMERICAL INTEGRATIONThis prov
Page 2068:
27.4 NUMERICAL INTEGRATION27.4.3 Ga
Page 2072:
27.4 NUMERICAL INTEGRATIONso, provi
Page 2076:
27.4 NUMERICAL INTEGRATIONfactor is
Page 2080:
27.4 NUMERICAL INTEGRATIONhas becom
Page 2084:
27.4 NUMERICAL INTEGRATIONwill have
Page 2088:
27.4 NUMERICAL INTEGRATIONy = f(x)y
Page 2092:
27.4 NUMERICAL INTEGRATIONIt will b
Page 2096:
27.5 FINITE DIFFERENCESmany values
Page 2100:
27.6 DIFFERENTIAL EQUATIONSx h y(ex
Page 2104:
27.6 DIFFERENTIAL EQUATIONSbut they
Page 2108:
27.6 DIFFERENTIAL EQUATIONSThe forw
Page 2112:
27.6 DIFFERENTIAL EQUATIONSWe assum
Page 2116:
27.7 HIGHER-ORDER EQUATIONSy1.00.80
Page 2120:
27.8 PARTIAL DIFFERENTIAL EQUATIONS
Page 2124:
27.9 EXERCISES27.9 Exercises27.1 Us
Page 2128:
27.9 EXERCISES(b) Try to repeat the
Page 2132:
27.9 EXERCISES27.21 Write a compute
Page 2136:
27.10 HINTS AND ANSWERS27.27 The Sc
Page 2140:
28Group theoryFor systems that have
Page 2144:
28.1 GROUPS28.1.1 Definition of a g
Page 2148:
28.1 GROUPS◮Using only the first
Page 2152:
28.1 GROUPSLMKFigure 28.2 Reflectio
Page 2156:
28.2 FINITE GROUPS28.2 Finite group
Page 2160:
28.2 FINITE GROUPS(a)1 5 7 111 1 5
Page 2164:
28.3 NON-ABELIAN GROUPSAs a first e
Page 2168:
28.3 NON-ABELIAN GROUPSI A B C D EI
Page 2172:
28.4 PERMUTATION GROUPSSuppose that
Page 2176:
28.5 MAPPINGS BETWEEN GROUPS28.5 Ma
Page 2180:
28.6 SUBGROUPS(a)I A B C D EI I A B
Page 2184:
28.7 SUBDIVIDING A GROUP(i) the set
Page 2188:
28.7 SUBDIVIDING A GROUPthis implie
Page 2192:
28.7 SUBDIVIDING A GROUP• Two cos
Page 2196:
28.7 SUBDIVIDING A GROUP(iii) In an
Page 2200:
28.8 EXERCISES28.4 Prove that the r
Page 2204:
28.8 EXERCISESSimilarly compute C 2
Page 2208:
28.9 HINTS AND ANSWERS≠For Φ 4 ,
Page 2212:
29.1 DIPOLE MOMENTS OF MOLECULESABA
Page 2216:
29.2 CHOOSING AN APPROPRIATE FORMAL
Page 2220:
Page 2224:
Page 2228:
29.3 EQUIVALENT REPRESENTATIONSresp
Page 2232:
29.4 REDUCIBILITY OF A REPRESENTATI
Page 2236:
29.4 REDUCIBILITY OF A REPRESENTATI
Page 2240:
29.5 THE ORTHOGONALITY THEOREM FOR
Page 2244:
29.6 CHARACTERS3m I A, B C, D, EA 1
Page 2248:
29.7 COUNTING IRREPS USING CHARACTE
Page 2252:
Page 2256:
Page 2260:
29.8 CONSTRUCTION OF A CHARACTER TA
Page 2264:
29.10 PRODUCT REPRESENTATIONSgive a
Page 2268:
29.11 PHYSICAL APPLICATIONS OF GROU
Page 2272:
Page 2276:
Page 2280:
Page 2284:
29.12 EXERCISESas the sum of two on
Page 2288:
29.12 EXERCISESUse this to show tha
Page 2292:
29.13 HINTS AND ANSWERS(a) Make an
Page 2296:
30ProbabilityAll scientists will kn
Page 2300:
30.1 VENN DIAGRAMSA42 6 3BS15Figure
Page 2304:
30.1 VENN DIAGRAMSgets beyond three
Page 2308:
30.2 PROBABILITYtimes then we expec
Page 2312:
30.2 PROBABILITYHowever, we may wri
Page 2316:
30.2 PROBABILITYace from a pack of
Page 2320:
30.2 PROBABILITYA 4A 3OA 1A 2BFigur
Page 2324:
30.3 PERMUTATIONS AND COMBINATIONSW
Page 2328:
30.3 PERMUTATIONS AND COMBINATIONSt
Page 2332:
30.3 PERMUTATIONS AND COMBINATIONSm
Page 2336:
30.4 RANDOM VARIABLES AND DISTRIBUT
Page 2340:
30.4 RANDOM VARIABLES AND DISTRIBUT
Page 2344:
30.5 PROPERTIES OF DISTRIBUTIONSIn
Page 2348:
30.5 PROPERTIES OF DISTRIBUTIONSInt
Page 2352:
30.5 PROPERTIES OF DISTRIBUTIONS|x
Page 2356:
30.5 PROPERTIES OF DISTRIBUTIONSWe
Page 2360:
30.6 FUNCTIONS OF RANDOM VARIABLESf
Page 2364:
30.6 FUNCTIONS OF RANDOM VARIABLESY
Page 2368:
30.6 FUNCTIONS OF RANDOM VARIABLESw
Page 2372:
30.7 GENERATING FUNCTIONSvariance o
Page 2376:
30.7 GENERATING FUNCTIONSand differ
Page 2380:
30.7 GENERATING FUNCTIONSi.e. the P
Page 2384:
30.7 GENERATING FUNCTIONSThe MGF wi
Page 2388:
30.7 GENERATING FUNCTIONSprobabilit
Page 2392:
30.7 GENERATING FUNCTIONSComparing
Page 2396:
30.8 IMPORTANT DISCRETE DISTRIBUTIO
Page 2400:
Page 2404:
Page 2408:
Page 2412:
Page 2416:
30.9 IMPORTANT CONTINUOUS DISTRIBUT
Page 2420:
Page 2424:
Page 2428:
Page 2432:
Page 2436:
Page 2440:
Page 2444:
Page 2448:
30.10 THE CENTRAL LIMIT THEOREMand
Page 2452:
30.11 JOINT DISTRIBUTIONSconsult on
Page 2456:
30.12 PROPERTIES OF JOINT DISTRIBUT
Page 2460:
Page 2464:
Page 2468:
30.13 GENERATING FUNCTIONS FOR JOIN
Page 2472:
30.15 IMPORTANT JOINT DISTRIBUTIONS
Page 2476:
30.15 IMPORTANT JOINT DISTRIBUTIONS
Page 2480:
30.16 EXERCISEStivariate Gaussian.
Page 2484:
30.16 EXERCISES30.11 A boy is selec
Page 2488:
30.16 EXERCISES30.18 A particle is
Page 2492:
30.16 EXERCISESaccording to one of
Page 2496:
30.17 HINTS AND ANSWERSconstraint
Page 2500:
31StatisticsIn this chapter, we tur
Page 2504:
31.2 SAMPLE STATISTICS188.7 204.7 1
Page 2508:
31.2 SAMPLE STATISTICSand the sampl
Page 2512:
31.2 SAMPLE STATISTICSmoments of th
Page 2516:
31.3 ESTIMATORS AND SAMPLING DISTRI
Page 2520:
Page 2524:
Page 2528:
Page 2532:
Page 2536:
Page 2540:
Page 2544:
31.4 SOME BASIC ESTIMATORSâ 2a 2(a
Page 2548:
31.4 SOME BASIC ESTIMATORSexact exp
Page 2552:
31.4 SOME BASIC ESTIMATORSwhere s 4
Page 2556:
31.4 SOME BASIC ESTIMATORSthe form(
Page 2560:
31.4 SOME BASIC ESTIMATORS(known) c
Page 2564:
31.4 SOME BASIC ESTIMATORSSince the
Page 2568:
31.5 MAXIMUM-LIKELIHOOD METHODSubst
Page 2572:
31.5 MAXIMUM-LIKELIHOOD METHODL(x;
Page 2576:
31.5 MAXIMUM-LIKELIHOOD METHOD◮In
Page 2580:
31.5 MAXIMUM-LIKELIHOOD METHOD31.5.
Page 2584:
31.5 MAXIMUM-LIKELIHOOD METHOD31.5.
Page 2588:
31.5 MAXIMUM-LIKELIHOOD METHODwhere
Page 2592:
31.5 MAXIMUM-LIKELIHOOD METHODL(x;
Page 2596:
31.5 MAXIMUM-LIKELIHOOD METHODBy su
Page 2600:
31.6 THE METHOD OF LEAST SQUARESThe
Page 2604:
31.6 THE METHOD OF LEAST SQUARESwhe
Page 2608:
31.6 THE METHOD OF LEAST SQUARESy76
Page 2612:
31.7 HYPOTHESIS TESTINGhowever, suc
Page 2616:
31.7 HYPOTHESIS TESTINGP (t|H 0 )α
Page 2620:
31.7 HYPOTHESIS TESTING◮Ten indep
Page 2624:
31.7 HYPOTHESIS TESTING◮Ten indep
Page 2628:
31.7 HYPOTHESIS TESTINGThe sum of s
Page 2632:
31.7 HYPOTHESIS TESTINGP (t|H 0 )0.
Page 2636:
31.7 HYPOTHESIS TESTINGdistribution
Page 2640:
31.7 HYPOTHESIS TESTINGλ(u)0.100.0
Page 2644:
31.7 HYPOTHESIS TESTINGWe now turn
Page 2648:
31.7 HYPOTHESIS TESTINGC n1 ,n 2(F)
Page 2652:
31.7 HYPOTHESIS TESTINGIn the last
Page 2656:
31.8 EXERCISES31.6 Prove that the s
Page 2660:
31.8 EXERCISES31.13 A similar techn
Page 2664:
31.9 HINTS AND ANSWERS31.9 Hints an
Page 2668:
IndexWhere the discussion of a topi
Page 2672:
INDEXrecurrence relations, 611-612s
Page 2676:
INDEXcomplement, 1121probability fo
Page 2680:
INDEXin spherical polars, 362Stoke
Page 2684:
INDEXin cylindrical polars, 360in s
Page 2688:
INDEXdiscontinuous functions, 420-4
Page 2692:
INDEXnomenclature, 1102non-Abelian,
Page 2696:
INDEXtriple, see multiple integrals
Page 2700:
INDEXlevel lines, 905, 906Levi-Civi
Page 2704:
INDEXMonte Carlo methods, of integr
Page 2708:
INDEXorthogonal transformations, 93
Page 2712:
INDEXstandard deviation σ, 1146var
Page 2716:
INDEXwave equation, 714-716, 737, 7
Page 2720:
INDEXsymmetric tensors, 938symmetry
Page 2724:
INDEXvolume integrals, 396and diver
show all

Mathematical Methods for Physics and Engineering - Matematica.NET

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?