Mathematical Methods for Physics and Engineering - Matematica.NET

More documents

Recommendations

Info

31.7 HYPOTHESIS TESTINGP (t|H 0 )αt crittP (t|H 1 )βt crittFigure 31.10 The sampling distributions P (t|H 0 )andP (t|H 1 ) of a test statistict. The shaded areas indicate the (one-tailed) regions for which Pr(t >t crit |H 0 )=α and Pr(t t crit |H 0 )= P (t|H 0 ) dt = α; (31.106)t critthis is indicated by the shaded region in the upper half of figure 31.10. Equally,a (one-tailed) rejection region could consist of values of t less than some valuet crit . Alternatively, one could define a (two-tailed) rejection region by two valuest 1 and t 2 such that Pr(t 1
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:
Page 870:
Page 874:
LINE, SURFACE AND VOLUME INTEGRALSS
Page 878:
LINE, SURFACE AND VOLUME INTEGRALSi
Page 882:
Page 886:
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 922:
FOURIER SERIES12.21 c n =[(−1) n
Page 926:
INTEGRAL TRANSFORMSc(ω)expiωt−
Page 930:
INTEGRAL TRANSFORMS◮Find the Four
Page 934:
INTEGRAL TRANSFORMSYyk ′k0θx−Y
Page 938:
INTEGRAL TRANSFORMSequals zero. Thi
Page 942:
INTEGRAL TRANSFORMS◮Prove relatio
Page 946:
INTEGRAL TRANSFORMS(i) Differentiat
Page 950:
INTEGRAL TRANSFORMSg(y)(a)(b)(c)(d)
Page 954:
INTEGRAL TRANSFORMSgiven by∫1 ∞
Page 958:
INTEGRAL TRANSFORMS◮Prove the Wie
Page 962:
INTEGRAL TRANSFORMStwo- or three-di
Page 966:
INTEGRAL TRANSFORMS(iii) Once again
Page 970:
INTEGRAL TRANSFORMS◮Find the Lapl
Page 974:
INTEGRAL TRANSFORMSFigure 13.7text)
Page 978:
INTEGRAL TRANSFORMS13.4 Exercises13
Page 982:
INTEGRAL TRANSFORMS13.10 In many ap
Page 986:
INTEGRAL TRANSFORMS13.18 The equiva
Page 990:
INTEGRAL TRANSFORMS13.27 The functi
Page 994:
14First-order ordinary differential
Page 998:
FIRST-ORDER ORDINARY DIFFERENTIAL E
Page 1002:
Page 1006:
Page 1010:
Page 1014:
Page 1018:
Page 1022:
Page 1026:
Page 1030:
Page 1034:
Page 1038:
15Higher-order ordinary differentia
Page 1042:
HIGHER-ORDER ORDINARY DIFFERENTIAL
Page 1046:
Page 1050:
Page 1054:
Page 1058:
Page 1062:
Page 1066:
Page 1070:
Page 1074:
Page 1078:
Page 1082:
Page 1086:
Page 1090:
Page 1094:
Page 1098:
Page 1102:
Page 1106:
Page 1110:
Page 1114:
Page 1118:
Page 1122:
SERIES SOLUTIONS OF ORDINARY DIFFER
Page 1126:
Page 1130:
Page 1134:
Page 1138:
Page 1142:
Page 1146:
Page 1150:
Page 1154:
Page 1158:
Page 1162:
Page 1166:
17Eigenfunction methods fordifferen
Page 1170:
EIGENFUNCTION METHODS FOR DIFFERENT
Page 1174:
Page 1178:
Page 1182:
Page 1186:
Page 1190:
Page 1194:
Page 1198:
Page 1202:
Page 1206:
Page 1210:
Page 1214:
SPECIAL FUNCTIONSwhich on collectin
Page 1218:
SPECIAL FUNCTIONSwhere P l (x) is a
Page 1222:
SPECIAL FUNCTIONSwhich reduces to(x
Page 1226:
SPECIAL FUNCTIONS◮Prove the expre
Page 1230:
SPECIAL FUNCTIONSr and r ′ must b
Page 1234:
SPECIAL FUNCTIONSin (18.3) and (18.
Page 1238:
SPECIAL FUNCTIONSto be zero, since
Page 1242:
SPECIAL FUNCTIONSGenerating functio
Page 1246:
SPECIAL FUNCTIONSorthonormal set, i
Page 1250:
SPECIAL FUNCTIONSand has three regu
Page 1254:
SPECIAL FUNCTIONS4U 2U 32U 0U 1−1
Page 1258:
SPECIAL FUNCTIONSThe normalisation,
Page 1262:
SPECIAL FUNCTIONSUsing (18.65) and
Page 1266:
SPECIAL FUNCTIONSthe form of a Frob
Page 1270:
SPECIAL FUNCTIONS1.51J 0J 1J 20.52
Page 1274:
SPECIAL FUNCTIONS10.5Y 0Y 1 Y22 4 6
Page 1278:
SPECIAL FUNCTIONSevaluated using l
Page 1282:
SPECIAL FUNCTIONSFinally, subtracti
Page 1286:
SPECIAL FUNCTIONSUsing de Moivre’
Page 1290:
SPECIAL FUNCTIONS◮Show that the l
Page 1294:
SPECIAL FUNCTIONS105L 2L 3L 0L 11 2
Page 1298:
SPECIAL FUNCTIONSThe above orthogon
Page 1302:
SPECIAL FUNCTIONSIn particular, we
Page 1306:
SPECIAL FUNCTIONSwhere, in the seco
Page 1310:
SPECIAL FUNCTIONS18.9.1 Properties
Page 1314:
SPECIAL FUNCTIONSDifferentiating th
Page 1318:
SPECIAL FUNCTIONSgamma function. §
Page 1322:
SPECIAL FUNCTIONSwhere in the secon
Page 1326:
SPECIAL FUNCTIONSsecond solution to
Page 1330:
SPECIAL FUNCTIONSThe gamma function
Page 1334:
SPECIAL FUNCTIONSIf we let x = n +
Page 1338:
SPECIAL FUNCTIONSwhich is the requi
Page 1342:
SPECIAL FUNCTIONSand hence that the
Page 1346:
SPECIAL FUNCTIONSDeduce the value o
Page 1350:
SPECIAL FUNCTIONS18.24 The solution
Page 1354:
19Quantum operatorsAlthough the pre
Page 1358:
QUANTUM OPERATORSis to produce a sc
Page 1362:
QUANTUM OPERATORSIf A| a n 〉 = a|
Page 1366:
QUANTUM OPERATORSSimple identities
Page 1370:
QUANTUM OPERATORSlater algebraic co
Page 1374:
QUANTUM OPERATORSRHS gives(−i) 2
Page 1378:
QUANTUM OPERATORSwith[L 2 ,L z]=[L2
Page 1382:
QUANTUM OPERATORSoperate repeatedly
Page 1386:
QUANTUM OPERATORS19.2.2 Uncertainty
Page 1390:
QUANTUM OPERATORShence formally an
Page 1394:
QUANTUM OPERATORSan arbitrary compl
Page 1398:
QUANTUM OPERATORSThe proof, which m
Page 1402:
QUANTUM OPERATORSFor a particle of
Page 1406:
QUANTUM OPERATORS19.4 Hints and ans
Page 1410:
PDES: GENERAL AND PARTICULAR SOLUTI
Page 1414:
Page 1418:
Page 1422:
Page 1426:
Page 1430:
Page 1434:
Page 1438:
Page 1442:
Page 1446:
Page 1450:
Page 1454:
Page 1458:
Page 1462:
Page 1466:
Page 1470:
Page 1474:
Page 1478:
Page 1482:
Page 1486:
PDES: SEPARATION OF VARIABLES AND O
Page 1490:
Page 1494:
Page 1498:
Page 1502:
Page 1506:
Page 1510:
Page 1514:
Page 1518:
Page 1522:
Page 1526:
Page 1530:
Page 1534:
Page 1538:
Page 1542:
Page 1546:
Page 1550:
Page 1554:
Page 1558:
Page 1562:
Page 1566:
Page 1570:
Page 1574:
Page 1578:
Page 1582:
Page 1586:
Page 1590:
Page 1594:
Page 1598:
Page 1602:
Page 1606:
Page 1610:
CALCULUS OF VARIATIONSyabxFigure 22
Page 1614:
CALCULUS OF VARIATIONSB(b, y(b))dsd
Page 1618:
CALCULUS OF VARIATIONSwe can use (2
Page 1622:
CALCULUS OF VARIATIONS22.3.1 Severa
Page 1626:
CALCULUS OF VARIATIONSAx = x 0xyBFi
Page 1630:
CALCULUS OF VARIATIONS−ayOaxFigur
Page 1634:
CALCULUS OF VARIATIONSyBn 2xθ 1θ
Page 1638:
CALCULUS OF VARIATIONSUsing (22.13)
Page 1642:
CALCULUS OF VARIATIONS◮Show that
Page 1646:
CALCULUS OF VARIATIONSy(x)1(c)0.80.
Page 1650:
CALCULUS OF VARIATIONSoperator H is
Page 1654:
CALCULUS OF VARIATIONS22.8 Derive t
Page 1658:
CALCULUS OF VARIATIONS22.23 For the
Page 1662:
CALCULUS OF VARIATIONS22.5 (a) ∂x
Page 1666:
INTEGRAL EQUATIONSWe shall illustra
Page 1670:
INTEGRAL EQUATIONSinhomogeneous Fre
Page 1674:
INTEGRAL EQUATIONSThese two simulta
Page 1678:
INTEGRAL EQUATIONSThus, using (23.1
Page 1682:
INTEGRAL EQUATIONSSubstituting (23.
Page 1686:
INTEGRAL EQUATIONSwe may write the
Page 1690:
INTEGRAL EQUATIONSNeumann series, w
Page 1694:
INTEGRAL EQUATIONSsides of (23.51)
Page 1698:
INTEGRAL EQUATIONS23.5 Solve for φ
Page 1702:
INTEGRAL EQUATIONSBy examining the
Page 1706:
24Complex variablesThroughout this
Page 1710:
COMPLEX VARIABLESWe then find that[
Page 1714:
COMPLEX VARIABLESFor f to be differ
Page 1718:
COMPLEX VARIABLESwhere i and j are
Page 1722:
COMPLEX VARIABLESwhich is an altern
Page 1726:
COMPLEX VARIABLESderived from them
Page 1730:
COMPLEX VARIABLESy Cy yrθxrθxxC
Page 1734:
COMPLEX VARIABLESwhere a is a finit
Page 1738:
COMPLEX VARIABLESyz 1z 2sC ′ 1w 1
Page 1742:
COMPLEX VARIABLESysi Pw = g(z)Q R S
Page 1746:
COMPLEX VARIABLESysibw 3w = g(z)φ
Page 1750:
COMPLEX VARIABLESyBC 2C 1xC 3AFigur
Page 1754:
COMPLEX VARIABLESmust be made in te
Page 1758:
COMPLEX VARIABLESyBC 1RC 2AxFigure
Page 1762:
COMPLEX VARIABLEScontour C and z 0
Page 1766:
COMPLEX VARIABLESwhere a n is given
Page 1770:
COMPLEX VARIABLESyRz 0C 1C 2xFigure
Page 1774:
COMPLEX VARIABLESDifferentiating bo
Page 1778:
COMPLEX VARIABLESCC ′C(a)(b)Figur
Page 1782:
COMPLEX VARIABLESformula (24.56) wi
Page 1786:
COMPLEX VARIABLESyΓγ−RORxFigure
Page 1790:
COMPLEX VARIABLESyΓγABCDxFigure 2
Page 1794:
COMPLEX VARIABLES24.7 Find the real
Page 1798:
COMPLEX VARIABLES24.22 The equation
Page 1802:
APPLICATIONS OF COMPLEX VARIABLESyx
Page 1806:
APPLICATIONS OF COMPLEX VARIABLESis
Page 1810:
APPLICATIONS OF COMPLEX VARIABLES
Page 1814:
APPLICATIONS OF COMPLEX VARIABLESde
Page 1818:
APPLICATIONS OF COMPLEX VARIABLESwh
Page 1822:
APPLICATIONS OF COMPLEX VARIABLESyY
Page 1826:
APPLICATIONS OF COMPLEX VARIABLESIm
Page 1830:
APPLICATIONS OF COMPLEX VARIABLESId
Page 1834:
APPLICATIONS OF COMPLEX VARIABLES25
Page 1838:
APPLICATIONS OF COMPLEX VARIABLESfo
Page 1842:
APPLICATIONS OF COMPLEX VARIABLESIm
Page 1846:
Page 1850:
APPLICATIONS OF COMPLEX VARIABLESx-
Page 1854:
APPLICATIONS OF COMPLEX VARIABLESwh
Page 1858:
APPLICATIONS OF COMPLEX VARIABLES
Page 1862:
APPLICATIONS OF COMPLEX VARIABLESTh
Page 1866:
APPLICATIONS OF COMPLEX VARIABLESex
Page 1870:
APPLICATIONS OF COMPLEX VARIABLESFi
Page 1874:
APPLICATIONS OF COMPLEX VARIABLESan
Page 1878:
APPLICATIONS OF COMPLEX VARIABLESsi
Page 1882:
APPLICATIONS OF COMPLEX VARIABLESFi
Page 1886:
APPLICATIONS OF COMPLEX VARIABLESfr
Page 1890:
APPLICATIONS OF COMPLEX VARIABLESst
Page 1894:
APPLICATIONS OF COMPLEX VARIABLESvv
Page 1898:
APPLICATIONS OF COMPLEX VARIABLESAV
Page 1902:
APPLICATIONS OF COMPLEX VARIABLESTh
Page 1906:
Page 1910:
Page 1914:
TENSORS26.1 Some notationBefore pro
Page 1918:
TENSORSScalars behave differently u
Page 1922:
TENSORS26.4 First- and zero-order C
Page 1926:
TENSORSIn fact any scalar product o
Page 1930:
TENSORSanother, without reference t
Page 1934:
TENSORSPhysical examples involving
Page 1938:
TENSORSdoes this imply that the A p
Page 1942:
TENSORSLet us begin, however, by no
Page 1946:
TENSORSA useful application of (26.
Page 1950:
TENSORSRotate by π/2 about the Ox
Page 1954:
TENSORSbut since |L| = ±1 we may r
Page 1958:
TENSORS◮Using (26.40), show that
Page 1962:
TENSORS(iii) referred to these axes
Page 1966:
TENSORSFurther, Poisson’s ratio i
Page 1970:
TENSORScontrary is specifically sta
Page 1974:
TENSORS◮Calculate the elements g
Page 1978:
TENSORSThus, by inverting the matri
Page 1982:
TENSORSwhere the elements L ij are
Page 1986:
TENSORSu i to another u ′i , we m
Page 1990:
TENSORS◮Using (26.76), deduce the
Page 1994:
TENSORS26.19 Covariant differentiat
Page 1998:
TENSORSand sov i ; i = ∂vρ∂ρ
Page 2002:
TENSORSIn order to compare the resu
Page 2006:
TENSORSand so the covariant compone
Page 2010:
TENSORScomponents of a second-order
Page 2014:
TENSORS26.3 In section 26.3 the tra
Page 2018:
TENSORS(b) Find the principal axes
Page 2022:
TENSORS26.28 A curve r(t) is parame
Page 2026:
27Numerical methodsIt happens frequ
Page 2030:
NUMERICAL METHODSf(x)141210 f(x) =x
Page 2034:
NUMERICAL METHODSn x n f(x n )1 1.7
Page 2038:
NUMERICAL METHODSn A n f(A n ) B n
Page 2042:
NUMERICAL METHODSOf the four method
Page 2046:
NUMERICAL METHODSn x n+1 ɛ n1 8.5
Page 2050:
NUMERICAL METHODSappreciate how thi
Page 2054:
NUMERICAL METHODSn x 1 x 2 x 31 2 2
Page 2058:
NUMERICAL METHODS◮Solve the follo
Page 2062:
NUMERICAL METHODSother exact expres
Page 2066:
NUMERICAL METHODSThe difference bet
Page 2070:
NUMERICAL METHODSand orthogonal ove
Page 2074:
NUMERICAL METHODSGauss-Legendre int
Page 2078:
NUMERICAL METHODSGauss-Laguerre and
Page 2082:
NUMERICAL METHODSaveraged:t = 1 nn
Page 2086:
NUMERICAL METHODSsampling, in both
Page 2090:
NUMERICAL METHODSh(ξ 1 ) >ξ 2 . T
Page 2094:
NUMERICAL METHODSon (0, 1) and then
Page 2098:
NUMERICAL METHODSderivatives beyond
Page 2102:
NUMERICAL METHODSx y(estim.) y(exac
Page 2106:
NUMERICAL METHODSx y(estim.) y(exac
Page 2110:
NUMERICAL METHODSSteps (ii) and (ii
Page 2114:
NUMERICAL METHODS(ii) To order h 4
Page 2118:
NUMERICAL METHODSThese can then be
Page 2122:
NUMERICAL METHODSOur final example
Page 2126:
NUMERICAL METHODS27.8 A possible ru
Page 2130:
NUMERICAL METHODS(b) Substitute the
Page 2134:
NUMERICAL METHODS27.25 Laplace’s
Page 2138:
NUMERICAL METHODSV =80−∞40.541.
Page 2142:
GROUP THEORYNHMHHH(a)(b)HFigure 28.
Page 2146:
GROUP THEORYif matrices are involve
Page 2150:
GROUP THEORYand setting X = I ′ g
Page 2154:
GROUP THEORYIt is clear that cyclic
Page 2158:
GROUP THEORY1 3 5 71 1 3 5 73 3 1 7
Page 2162:
GROUP THEORY1 i −1 −i1 1 i −1
Page 2166:
GROUP THEORYI R R ′ K L MI I R R
Page 2170:
GROUP THEORYThe multiplication tabl
Page 2174:
GROUP THEORYeach number appears onc
Page 2178:
GROUP THEORYThree immediate consequ
Page 2182:
GROUP THEORY(a)I A BI I A BA A B IB
Page 2186:
GROUP THEORYthan they are like any
Page 2190:
GROUP THEORY(iii) Transitivity: X
Page 2194:
GROUP THEORYif H is a normal subgro
Page 2198:
GROUP THEORYmathematical details, a
Page 2202:
GROUP THEORY(a) Identify the distin
Page 2206:
GROUP THEORYm 1 (π)m 2 (π)m 3 (π
Page 2210:
29Representation theoryAs indicated
Page 2214:
REPRESENTATION THEORYFinally, for o
Page 2218:
REPRESENTATION THEORY◮For the gro
Page 2222:
REPRESENTATION THEORYcolumn matrice
Page 2226:
REPRESENTATION THEORYone comprises
Page 2230:
REPRESENTATION THEORYa representati
Page 2234:
REPRESENTATION THEORYof a particula
Page 2238:
REPRESENTATION THEORYof simple pair
Page 2242:
REPRESENTATION THEORY(d) No explici
Page 2246:
REPRESENTATION THEORY29.6.1 Orthogo
Page 2250:
REPRESENTATION THEORYconjugacy clas
Page 2254:
REPRESENTATION THEORY(a)I A BI I A
Page 2258:
REPRESENTATION THEORYwhere the λ i
Page 2262:
REPRESENTATION THEORY4mm I Q R, R
Page 2266:
REPRESENTATION THEORYbe non-zero th
Page 2270:
REPRESENTATION THEORY1y42x3Figure 2
Page 2274:
REPRESENTATION THEORY(i) Under I al
Page 2278:
REPRESENTATION THEORYy 3x 3y 1y2x 1
Page 2282:
REPRESENTATION THEORYMMMFigure 29.6
Page 2286:
REPRESENTATION THEORY(d) Complete t
Page 2290:
REPRESENTATION THEORYthe correspond
Page 2294:
REPRESENTATION THEORY(because of or
Page 2298:
PROBABILITYAiiiiiiBSivFigure 30.1A
Page 2302:
PROBABILITYA1B24756 3C8SFigure 30.4
Page 2306:
PROBABILITYSince A ∩ X = X we mus
Page 2310:
PROBABILITYThis is particularly use
Page 2314:
PROBABILITY◮Find the probability
Page 2318:
PROBABILITYTwo events A and B are s
Page 2322:
PROBABILITY30.2.3 Bayes’ theoremI
Page 2326:
PROBABILITYIn calculating the numbe
Page 2330:
PROBABILITYAnother useful result th
Page 2334:
PROBABILITYparticles can be distrib
Page 2338:
PROBABILITYf(x)2pF(x)1p12 p 12 3 4
Page 2342:
PROBABILITY◮A random variable X h
Page 2346:
PROBABILITYthe series is absolutely
Page 2350:
PROBABILITY30.5.3 Variance and stan
Page 2354:
PROBABILITY◮A biased die has prob
Page 2358:
PROBABILITYand differentiate it rep
Page 2362:
PROBABILITYθlighthouseLbeamOcoastl
Page 2366:
PROBABILITYIf X and Y are both disc
Page 2370:
PROBABILITYthe variables X and Y .
Page 2374:
PROBABILITYIf, as previously, the p
Page 2378:
PROBABILITYnr = nrFigure 30.10The p
Page 2382:
PROBABILITYan expression for the PG
Page 2386:
PROBABILITYScaling and shiftingIf Y
Page 2390:
PROBABILITYso that C X (t) =M X (it
Page 2394:
PROBABILITYDistribution Probability
Page 2398:
PROBABILITYFor evaluating binomial
Page 2402:
PROBABILITYi =1, 2,...,N, is distri
Page 2406:
PROBABILITYwhere in the last line p
Page 2410:
PROBABILITYt +∆t is given byP x (
Page 2414:
PROBABILITYfrom which we obtainM X(
Page 2418:
PROBABILITYDistribution Probability
Page 2422:
PROBABILITYIt is usual only to tabu
Page 2426:
PROBABILITY◮Sawmill A produces bo
Page 2430:
PROBABILITYx f(x) (binomial) f(x) (
Page 2434:
PROBABILITYStirling’s approximati
Page 2438:
PROBABILITYThe above results may be
Page 2442:
PROBABILITYf(x)10.80.6r =10.40.2r =
Page 2446:
PROBABILITYf(x)0.80.6x 0 =0,x 0 =2,
Page 2450:
PROBABILITYwhere M Xi (t) istheMGFo
Page 2454:
PROBABILITY30.11.2 Continuous bivar
Page 2458:
PROBABILITY◮Show that if X and Y
Page 2462:
PROBABILITY◮A biased die gives pr
Page 2466:
PROBABILITY◮A card is drawn at ra
Page 2470:
PROBABILITYFinally we note that, by
Page 2474:
PROBABILITY30.15.1 The multinomial
Page 2478:
PROBABILITYand thus, using (30.135)
Page 2482:
PROBABILITY30.6 X 1 ,X 2 ,...,X n a
Page 2486:
PROBABILITYfighting, kittens are re
Page 2490:
PROBABILITY30.23 A point P is chose
Page 2494:
PROBABILITY[ You will need the resu
Page 2498:
PROBABILITY30.23 Mean = 4/π. Varia
Page 2502:
STATISTICStherefore consider the x
Page 2506:
STATISTICSIt should be noted that,
Page 2510:
STATISTICS◮Calculate ∑ Ni=1 x i
Page 2514:
STATISTICSr xy =0.0 r xy =0.1 r xy
Page 2518:
STATISTICSSuppose, we wish to estim
Page 2522:
STATISTICSand describes the spread
Page 2526:
STATISTICSis trivially extended to
Page 2530:
STATISTICSP (â|a)αβâ α (a)â
Page 2534:
STATISTICS31.3.5 Confidence limits
Page 2538:
STATISTICSwhere in the last line we
Page 2542:
STATISTICSa region ˆR in â-space,
Page 2546:
STATISTICS% 99 95 10 5 2.5 1 0.5 0.
Page 2550:
STATISTICSwhere µ r is the rth pop
Page 2554:
STATISTICSwhere in the last line we
Page 2558:
STATISTICSHowever, since the sample
Page 2562:
STATISTICS31.4.6 Population covaria
Page 2566: STATISTICS31.4.7 A worked exampleTo
Page 2570: STATISTICSL(x; τ)1N =5L(x; τ)1N =
Page 2574: STATISTICSmaximum that occurs at a
Page 2578: STATISTICSDifferentiating with resp
Page 2582: STATISTICS31.5.3 Efficiency of ML e
Page 2586: STATISTICSP (ˆτ|τ)0.40.30.20.100
Page 2590: STATISTICSThus, a Bayesian statisti
Page 2594: STATISTICSBy comparing this result
Page 2598: STATISTICSand the matrix V −1 is
Page 2602: STATISTICSwhere the quantity denote
Page 2606: STATISTICSSetting the expression (3
Page 2610: STATISTICShence does not possess an
Page 2614: STATISTICShowever, one wishes to us
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?