Ivancevic_Applied-Diff-Geom

More documents

Recommendations

Info

Applied Bundle Geometry 673form Q. Then any vector v ∈ V may be written as v = v k e k and theClifford algebra isC(R 4 ) = span{ c n11 cn2 2 cn3 3 cn4 4 | n i ∈ {0, 1} }, where c i = q( e i ).For convenience, we may denote c i by e i without ambiguity. Especially,C 2 (R 4 ) = span{ e 1 e 2 , e 1 e 3 , e 1 e 4 , e 2 e 3 , e 2 e 4 , e 3 e 4 }Now consider the isomorphism τ : C 2 (R 4 ) −→ so(4) = so(R 4 ). And thecorresponding τ : Spin(4) −→ so(4). We haveτ(e i e j ) · v = [ e i e j ,==4∑v k e k ] =k=14∑v k [ e i e j , e k ] =k=14∑v k ( e i e j e k − e k e i e j )k=14∑v k ( e i e j e k + e i e k e j − e i e k e j − e k e i e j )k=14∑v k ( −2 Q( e j , e k ) e i + 2 Q( e k , e i ) e j ) = −2 v j e i + 2 v i e j .k=1Hence τ ( e i e j ) corresponds to 2 · m(i, j) ∈ so(4) where m(i, j) =(m(i, j) αβ ) is a matrix with entriesExplicitly, we have⎧⎪⎨ 1, if α = j and β = i,m(i, j) αβ = −1, if α = i and β = j,⎪⎩ 0, otherwise.⎛ ⎞⎛ ⎞0 −1 0 00 0 −1 0τ( e 1 e 2 ) = 2 ⎜1 0 0 0⎟⎝0 0 0 0⎠ τ( e 1 e 3 ) = 2 ⎜0 0 0 0⎟⎝1 0 0 0⎠0 0 0 00 0 0 0⎛ ⎞⎛ ⎞0 0 0 −10 0 0 0τ( e 1 e 4 ) = 2 ⎜0 0 0 0⎟⎝0 0 0 0 ⎠ τ( e 2 e 3 ) = 2 ⎜0 0 −1 0⎟⎝0 1 0 0⎠1 0 0 00 0 0 0
674 Applied Differential Geometry: A Modern Introduction⎛ ⎞⎛ ⎞0 0 0 00 0 0 0τ( e 2 e 4 ) = 2 ⎜0 0 0 −1⎟⎝0 0 0 0 ⎠ τ( e 3 e 4 ) = 2 ⎜0 0 0 0⎟⎝0 0 0 −1⎠ .0 1 0 00 0 1 0Notice that for t ∈ [0, 2π), τ( t 2 e ie j ) corresponds to the matrix t · m(i, j)and(t · m(i, j) αβ ) 2 = −t 2 · ∆(i, j),where ∆(i, j) is a matrix with(t · m(i, j) αβ ) 4 = t 4 · ∆(i, j),∆(i, j) αβ ={1, if α = β = i or α = β = j,0, otherwise.Therefore, we haveexp τ ( t 2 e ie j ) = (cos t − 1) · ∆(i, j) + 1 SO(4)+ (sin t) m(i, j).Since we have the commutative relation τ · exp = exp · τ, it followsthatτ exp ( t e i e j ) = exp τ ( te i e j ) = (cos 2t−1)·∆(i, j) + 1 SO(4)+ (sin 2t) m(i, j).Explicitly, we haveτ( exp( t e 1 e 2 ) ) =τ( exp( t e 1 e 3 ) ) =⎛⎞cos 2t − sin 2t 0 0⎜sin 2t cos 2t 0 0⎟⎝ 0 0 1 0⎠0 0 0 1⎛⎞cos 2t 0 − sin 2t 0⎜ 0 1 0 0⎟⎝sin 2t 0 cos 2t 0⎠0 0 0 1τ( exp( t e 1 e 4 ) ) =⎛⎞cos 2t 0 0 − sin 2t⎜ 0 1 0 0⎟⎝ 0 0 1 0 ⎠sin 2t 0 0 cos 2t
Page 2 and 3:
APPLIEDDIFFERENTIALGEOMETRYA Modern
Page 4 and 5:
APPLIEDDIFFERENTIALGEOMETRYA Modern
Page 6 and 7:
Dedicated to:Nitya, Atma and Kali
Page 8 and 9:
PrefaceApplied Differential Geometr
Page 10 and 11:
ixAcknowledgmentsThe authors wish t
Page 12 and 13:
Glossary of Frequently Used Symbols
Page 14 and 15:
Page 16 and 17:
Page 18 and 19:
ContentsPrefaceGlossary of Frequent
Page 20 and 21:
Contentsxix2.1.5.2 Forces Acting on
Page 22 and 23:
Contentsxxi3.6.3.4 Stokes Theorem a
Page 24 and 25:
Contentsxxiii3.10.3.1 Basis of Lagr
Page 26 and 27:
Contentsxxv3.17 Applied Unorthodox
Page 28 and 29:
Contentsxxvii4.9.8.2 Geometrodynami
Page 30 and 31:
Contentsxxix4.14.7.5 Monopole Conde
Page 32 and 33:
Contentsxxxi5.12.3 Hawking-Penrose
Page 34 and 35:
Contentsxxxiii6.5.2.4 Dimensional R
Page 36 and 37:
Chapter 1IntroductionIn this introd
Page 38 and 39:
Introduction 3Fig. 1.1 The four cha
Page 40 and 41:
Introduction 5• Riemannian manifo
Page 42 and 43:
Introduction 7charts.The atlas cont
Page 44 and 45:
Introduction 9every point has a nei
Page 46 and 47:
Introduction 11of curves include ci
Page 48 and 49:
Introduction 13the variables define
Page 50 and 51:
Introduction 15U i denotes one of t
Page 52 and 53:
Introduction 17which varies smoothl
Page 54 and 55:
Introduction 19interactions.1.1.5.2
Page 56 and 57:
Introduction 21real number. This co
Page 58 and 59:
Introduction 23Such a force is inde
Page 60 and 61:
Introduction 25tons. In this formul
Page 62 and 63:
Introduction 27vector-field are sol
Page 64 and 65:
Introduction 29or simply connected)
Page 66 and 67:
Introduction 311.1.8 Application: A
Page 68 and 69:
Introduction 33theory itself existe
Page 70 and 71:
Introduction 35mediate the forces.
Page 72 and 73:
Introduction 37theory. 50 Fig. 1.3
Page 74 and 75:
Introduction 39to describe a univer
Page 76 and 77:
Introduction 41which have two disti
Page 78 and 79:
Introduction 43first theory can be
Page 80 and 81:
Introduction 45the Casimir effect,
Page 82 and 83:
Introduction 47• External coordin
Page 84 and 85:
Introduction 49a connection-base de
Page 86 and 87:
Chapter 2Technical Preliminaries: T
Page 88 and 89:
Technical Preliminaries: Tensors, A
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Chapter 3Applied Manifold Geometry3
Page 174 and 175:
Applied Manifold Geometry 139where
Page 176 and 177:
Applied Manifold Geometry 141In loc
Page 178 and 179:
Applied Manifold Geometry 143on the
Page 180 and 181:
Applied Manifold Geometry 145exist
Page 182 and 183:
Applied Manifold Geometry 147then E
Page 184 and 185:
Applied Manifold Geometry 1490D man
Page 186 and 187:
Applied Manifold Geometry 151Fig. 3
Page 188 and 189:
Applied Manifold Geometry 153the fo
Page 190 and 191:
Applied Manifold Geometry 155contin
Page 192 and 193:
Applied Manifold Geometry 157In oth
Page 194 and 195:
Applied Manifold Geometry 159scalar
Page 196 and 197:
Page 198 and 199:
Applied Manifold Geometry 1633.6 Te
Page 200 and 201:
Applied Manifold Geometry 165Let ϕ
Page 202 and 203:
Applied Manifold Geometry 1673.6.1.
Page 204 and 205:
Applied Manifold Geometry 169or, if
Page 206 and 207:
Applied Manifold Geometry 171If X i
Page 208 and 209:
Applied Manifold Geometry 173along
Page 210 and 211:
Applied Manifold Geometry 175is dua
Page 212 and 213:
Page 214 and 215:
Applied Manifold Geometry 179(1) α
Page 216 and 217:
Applied Manifold Geometry 181diagra
Page 218 and 219:
Applied Manifold Geometry 183Clearl
Page 220 and 221:
Page 222 and 223:
Applied Manifold Geometry 187duces,
Page 224 and 225:
Applied Manifold Geometry 189This i
Page 226 and 227:
Applied Manifold Geometry 191Hodge
Page 228 and 229:
Applied Manifold Geometry 193The in
Page 230 and 231:
Applied Manifold Geometry 195since
Page 232 and 233:
Applied Manifold Geometry 197If a l
Page 234 and 235:
Applied Manifold Geometry 199asGive
Page 236 and 237:
Applied Manifold Geometry 201The co
Page 238 and 239:
Applied Manifold Geometry 203at the
Page 240 and 241:
Applied Manifold Geometry 205For ex
Page 242 and 243:
Applied Manifold Geometry 207axis,
Page 244 and 245:
Applied Manifold Geometry 209For ex
Page 246 and 247:
Applied Manifold Geometry 211The sp
Page 248 and 249:
Applied Manifold Geometry 213corres
Page 250 and 251:
Applied Manifold Geometry 215corres
Page 252 and 253:
Applied Manifold Geometry 217Specia
Page 254 and 255:
Applied Manifold Geometry 219the co
Page 256 and 257:
Applied Manifold Geometry 221moment
Page 258 and 259:
Applied Manifold Geometry 223produc
Page 260 and 261:
Applied Manifold Geometry 225has a
Page 262 and 263:
Applied Manifold Geometry 2273.8.5
Page 264 and 265:
Applied Manifold Geometry 229in whi
Page 266 and 267:
Applied Manifold Geometry 231with
Page 268 and 269:
Applied Manifold Geometry 233itly:(
Page 270 and 271:
Applied Manifold Geometry 235play t
Page 272 and 273:
Applied Manifold Geometry 237Some r
Page 274 and 275:
Applied Manifold Geometry 239Basic
Page 276 and 277:
Applied Manifold Geometry 241orthog
Page 278 and 279:
Applied Manifold Geometry 243is adm
Page 280 and 281:
Applied Manifold Geometry 245Root S
Page 282 and 283:
Applied Manifold Geometry 247groups
Page 284 and 285:
Page 286 and 287:
Applied Manifold Geometry 251second
Page 288 and 289:
Page 290 and 291:
Applied Manifold Geometry 255be a v
Page 292 and 293:
Applied Manifold Geometry 257Now, a
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
Applied Manifold Geometry 263other
Page 300 and 301:
Page 302 and 303:
Applied Manifold Geometry 267result
Page 304 and 305:
Page 306 and 307:
Applied Manifold Geometry 2713.10 R
Page 308 and 309:
Applied Manifold Geometry 273More p
Page 310 and 311:
Applied Manifold Geometry 275Y (t 0
Page 312 and 313:
Applied Manifold Geometry 277which
Page 314 and 315:
Applied Manifold Geometry 279for an
Page 316 and 317:
Page 318 and 319:
Applied Manifold Geometry 283Thus w
Page 320 and 321:
Page 322 and 323:
Applied Manifold Geometry 287evolut
Page 324 and 325:
Page 326 and 327:
Page 328 and 329:
Applied Manifold Geometry 293‘ren
Page 330 and 331:
Applied Manifold Geometry 295mute;
Page 332 and 333:
Applied Manifold Geometry 297Rather
Page 334 and 335:
Applied Manifold Geometry 299of the
Page 336 and 337:
Applied Manifold Geometry 301clidea
Page 338 and 339:
Applied Manifold Geometry 303Second
Page 340 and 341:
Applied Manifold Geometry 305i.e.,
Page 342 and 343:
Applied Manifold Geometry 307invari
Page 344 and 345:
Applied Manifold Geometry 309The Ba
Page 346 and 347:
Applied Manifold Geometry 311Let us
Page 348 and 349:
Applied Manifold Geometry 313all cu
Page 350 and 351:
Applied Manifold Geometry 315ders.
Page 352 and 353:
Applied Manifold Geometry 317this c
Page 354 and 355:
Applied Manifold Geometry 319In par
Page 356 and 357:
Applied Manifold Geometry 321As I p
Page 358 and 359:
Applied Manifold Geometry 323the fo
Page 360 and 361:
Applied Manifold Geometry 325per (1
Page 362 and 363:
Applied Manifold Geometry 327Genera
Page 364 and 365:
Applied Manifold Geometry 329The ma
Page 366 and 367:
Applied Manifold Geometry 331pretat
Page 368 and 369:
Applied Manifold Geometry 333byMDL
Page 370 and 371:
Applied Manifold Geometry 3353.12 S
Page 372 and 373:
Applied Manifold Geometry 337H 2 (M
Page 374 and 375:
Applied Manifold Geometry 339bracke
Page 376 and 377:
Applied Manifold Geometry 341Any so
Page 378 and 379:
Applied Manifold Geometry 343All on
Page 380 and 381:
Applied Manifold Geometry 345The ac
Page 382 and 383:
Applied Manifold Geometry 347linear
Page 384 and 385:
Applied Manifold Geometry 349such t
Page 386 and 387:
Applied Manifold Geometry 351Let γ
Page 388 and 389:
Page 390 and 391:
Applied Manifold Geometry 355chart
Page 392 and 393:
Applied Manifold Geometry 357phase-
Page 394 and 395:
Applied Manifold Geometry 359( ) 0
Page 396 and 397:
Applied Manifold Geometry 361hetero
Page 398 and 399:
Applied Manifold Geometry 363thus c
Page 400 and 401:
Applied Manifold Geometry 365Action
Page 402 and 403:
Applied Manifold Geometry 367and we
Page 404 and 405:
Applied Manifold Geometry 369action
Page 406 and 407:
Applied Manifold Geometry 371and we
Page 408 and 409:
Applied Manifold Geometry 373by (J(
Page 410 and 411:
Applied Manifold Geometry 375fields
Page 412 and 413:
Applied Manifold Geometry 377Let Tx
Page 414 and 415:
Page 416 and 417:
Applied Manifold Geometry 381The ve
Page 418 and 419:
Applied Manifold Geometry 383and gi
Page 420 and 421:
Applied Manifold Geometry 385The co
Page 422 and 423:
Applied Manifold Geometry 387tor Ca
Page 424 and 425:
Page 426 and 427:
Applied Manifold Geometry 3912 −
Page 428 and 429:
Applied Manifold Geometry 393comple
Page 430 and 431:
Applied Manifold Geometry 395F[C]
Page 432 and 433:
Applied Manifold Geometry 397become
Page 434 and 435:
Applied Manifold Geometry 399of fuz
Page 436 and 437:
Applied Manifold Geometry 401In our
Page 438 and 439:
Applied Manifold Geometry 403Biodyn
Page 440 and 441:
Applied Manifold Geometry 405for T
Page 442 and 443:
Applied Manifold Geometry 407A vari
Page 444 and 445:
Applied Manifold Geometry 409such t
Page 446 and 447:
Applied Manifold Geometry 411of neg
Page 448 and 449:
Page 450 and 451:
Applied Manifold Geometry 415Follow
Page 452 and 453:
Applied Manifold Geometry 417and J
Page 454 and 455:
Applied Manifold Geometry 419This t
Page 456 and 457:
Applied Manifold Geometry 421above,
Page 458 and 459:
Applied Manifold Geometry 423As a L
Page 460 and 461:
Applied Manifold Geometry 425Theore
Page 462 and 463:
Applied Manifold Geometry 427real v
Page 464 and 465:
Applied Manifold Geometry 429indepe
Page 466 and 467:
Applied Manifold Geometry 431In ter
Page 468 and 469:
Applied Manifold Geometry 433M, we
Page 470 and 471:
Applied Manifold Geometry 435The cu
Page 472 and 473:
Applied Manifold Geometry 437(iv) (
Page 474 and 475:
Page 476 and 477:
. ..Applied Manifold Geometry 441in
Page 478 and 479:
Applied Manifold Geometry 443⌋ is
Page 480 and 481:
Applied Manifold Geometry 445Now, a
Page 482 and 483:
Applied Manifold Geometry 447symmet
Page 484 and 485:
Applied Manifold Geometry 449study
Page 486 and 487:
Applied Manifold Geometry 451togeth
Page 488 and 489:
Applied Manifold Geometry 453result
Page 490 and 491:
Applied Manifold Geometry 455will b
Page 492 and 493:
Applied Manifold Geometry 457g 11 =
Page 494 and 495:
Applied Manifold Geometry 459is def
Page 496 and 497:
Applied Manifold Geometry 461ordere
Page 498 and 499:
Applied Manifold Geometry 463Now, r
Page 500 and 501:
Applied Manifold Geometry 465Simila
Page 502 and 503:
Applied Manifold Geometry 467which
Page 504 and 505:
Applied Manifold Geometry 469From t
Page 506 and 507:
Page 508 and 509:
Page 510 and 511:
Applied Manifold Geometry 475On the
Page 512 and 513:
Applied Manifold Geometry 477ing di
Page 514 and 515:
Applied Manifold Geometry 479Note t
Page 516 and 517:
Applied Manifold Geometry 481Then w
Page 518 and 519:
Applied Manifold Geometry 483consis
Page 520 and 521:
Chapter 4Applied Bundle Geometry4.1
Page 522 and 523:
Applied Bundle Geometry 487is calle
Page 524 and 525:
Applied Bundle Geometry 489V and an
Page 526 and 527:
Applied Bundle Geometry 4914.3 Vect
Page 528 and 529:
Applied Bundle Geometry 493dimensio
Page 530 and 531:
Applied Bundle Geometry 495orthogon
Page 532 and 533:
Applied Bundle Geometry 497t, is a
Page 534 and 535:
Applied Bundle Geometry 499Let T Y
Page 536 and 537:
Applied Bundle Geometry 501In other
Page 538 and 539:
Applied Bundle Geometry 503we shall
Page 540 and 541:
Applied Bundle Geometry 505Pendulum
Page 542 and 543:
Applied Bundle Geometry 507Using L
Page 544 and 545:
Applied Bundle Geometry 509the isom
Page 546 and 547:
Applied Bundle Geometry 511theory f
Page 548 and 549:
Applied Bundle Geometry 513of the c
Page 550 and 551:
Applied Bundle Geometry 515where Ω
Page 552 and 553:
Applied Bundle Geometry 517nals, th
Page 554 and 555:
Applied Bundle Geometry 519(1985)])
Page 556 and 557:
Applied Bundle Geometry 521finite-d
Page 558 and 559:
Applied Bundle Geometry 523and thes
Page 560 and 561:
Applied Bundle Geometry 525Now, let
Page 562 and 563:
Applied Bundle Geometry 527gravity
Page 564 and 565:
Applied Bundle Geometry 529a Dp−b
Page 566 and 567:
Applied Bundle Geometry 531A princi
Page 568 and 569:
Applied Bundle Geometry 533is the r
Page 570 and 571:
Applied Bundle Geometry 535Now, let
Page 572 and 573:
Applied Bundle Geometry 537By cycli
Page 574 and 575:
Applied Bundle Geometry 5394.9.2 Fe
Page 576 and 577:
Applied Bundle Geometry 541This lin
Page 578 and 579:
Applied Bundle Geometry 543(1) L g
Page 580 and 581:
Applied Bundle Geometry 545and ω 0
Page 582 and 583:
Applied Bundle Geometry 547isfiesω
Page 584 and 585:
Applied Bundle Geometry 549of the c
Page 586 and 587:
Applied Bundle Geometry 551Controll
Page 588 and 589:
Applied Bundle Geometry 553Foliatio
Page 590 and 591:
Applied Bundle Geometry 555ψ : (x,
Page 592 and 593:
Applied Bundle Geometry 557from res
Page 594 and 595:
Applied Bundle Geometry 559Motion a
Page 596 and 597:
Applied Bundle Geometry 561where Z
Page 598 and 599:
Applied Bundle Geometry 563Systems
Page 600 and 601:
Applied Bundle Geometry 5653. Recal
Page 602 and 603:
Applied Bundle Geometry 567Therefor
Page 604 and 605:
Applied Bundle Geometry 569smooth m
Page 606 and 607:
Applied Bundle Geometry 571state va
Page 608 and 609:
Applied Bundle Geometry 573The geom
Page 610 and 611:
Applied Bundle Geometry 575Y ∋ y
Page 612 and 613:
Applied Bundle Geometry 577autonomo
Page 614 and 615:
Applied Bundle Geometry 579(see 3.1
Page 616 and 617:
Applied Bundle Geometry 581and init
Page 618 and 619:
Applied Bundle Geometry 583potentia
Page 620 and 621:
Applied Bundle Geometry 585effector
Page 622 and 623:
Applied Bundle Geometry 587Fig. 4.9
Page 624 and 625:
Applied Bundle Geometry 589physical
Page 626 and 627:
Applied Bundle Geometry 591open Lio
Page 628 and 629:
Applied Bundle Geometry 593et al. (
Page 630 and 631:
Applied Bundle Geometry 595(2) Acti
Page 632 and 633:
Applied Bundle Geometry 597˙Φ ≤
Page 634 and 635:
Applied Bundle Geometry 599number o
Page 636 and 637:
Applied Bundle Geometry 601where Ψ
Page 638 and 639:
Applied Bundle Geometry 603ing to c
Page 640 and 641:
Applied Bundle Geometry 605where [
Page 642 and 643:
Applied Bundle Geometry 607interval
Page 644 and 645:
Applied Bundle Geometry 609The foll
Page 646 and 647:
Applied Bundle Geometry 611In parti
Page 648 and 649:
Applied Bundle Geometry 613The pull
Page 650 and 651:
Applied Bundle Geometry 615(2) δ
Page 652 and 653:
Applied Bundle Geometry 617of unita
Page 654 and 655:
Applied Bundle Geometry 619system o
Page 656 and 657:
Applied Bundle Geometry 621of H pre
Page 658 and 659: Applied Bundle Geometry 623It follo
Page 660 and 661: Applied Bundle Geometry 625a produc
Page 662 and 663: Applied Bundle Geometry 627B Ham(M)
Page 664 and 665: Applied Bundle Geometry 629b.Let us
Page 666 and 667: Applied Bundle Geometry 631If P is
Page 668 and 669: Applied Bundle Geometry 633question
Page 670 and 671: Applied Bundle Geometry 635spheres
Page 672 and 673: Applied Bundle Geometry 637be thoug
Page 674 and 675: Applied Bundle Geometry 639of the b
Page 676 and 677: Applied Bundle Geometry 641in C 1 (
Page 678 and 679: Applied Bundle Geometry 643from a t
Page 680 and 681: Applied Bundle Geometry 645Now let
Page 682 and 683: Applied Bundle Geometry 647for t <
Page 684 and 685: Applied Bundle Geometry 649is a Ham
Page 686 and 687: Applied Bundle Geometry 651The foll
Page 688 and 689: Applied Bundle Geometry 653is the u
Page 690 and 691: Applied Bundle Geometry 655every no
Page 692 and 693: Applied Bundle Geometry 657P × P w
Page 694 and 695: Applied Bundle Geometry 659that the
Page 696 and 697: Applied Bundle Geometry 661extend t
Page 698 and 699: Applied Bundle Geometry 663identity
Page 700 and 701: Applied Bundle Geometry 665The mapw
Page 702 and 703: Applied Bundle Geometry 667It is a
Page 704 and 705: Applied Bundle Geometry 669I Q is i
Page 706 and 707: Applied Bundle Geometry 671whereC i
Page 710 and 711: Applied Bundle Geometry 675τ( exp(
Page 712 and 713: Applied Bundle Geometry 6774.13.2.2
Page 714 and 715: Applied Bundle Geometry 679Therefor
Page 718 and 719: Applied Bundle Geometry 683⎛⎞co
Page 720 and 721: Applied Bundle Geometry 685Usually,
Page 722 and 723: Applied Bundle Geometry 687on the s
Page 726 and 727: Applied Bundle Geometry 6914.13.3 P
Page 728 and 729: Applied Bundle Geometry 693Since T
Page 730 and 731: Applied Bundle Geometry 695On the o
Page 732 and 733: Applied Bundle Geometry 697may writ
Page 734 and 735: Applied Bundle Geometry 699determin
Page 736 and 737: Applied Bundle Geometry 701section,
Page 738 and 739: Applied Bundle Geometry 703for deri
Page 740 and 741: Applied Bundle Geometry 705dictions
Page 742 and 743: Applied Bundle Geometry 707too sing
Page 744 and 745: Applied Bundle Geometry 709In (4.18
Page 748 and 749: Applied Bundle Geometry 713sis repr
Page 752 and 753: Applied Bundle Geometry 717a (linea
Page 754 and 755: Applied Bundle Geometry 719past few
Page 756 and 757: Applied Bundle Geometry 721global q
Page 758 and 759:
Applied Bundle Geometry 7239. It ha
Page 760 and 761:
Applied Bundle Geometry 725needs to
Page 762 and 763:
Applied Bundle Geometry 727pear at
Page 764 and 765:
Applied Bundle Geometry 729X, F A i
Page 766 and 767:
Applied Bundle Geometry 731ifolds,
Page 768 and 769:
Applied Bundle Geometry 733and not
Page 770 and 771:
Applied Bundle Geometry 735under U(
Page 772 and 773:
Applied Bundle Geometry 737N = 2 br
Page 774 and 775:
Applied Bundle Geometry 739thusa
Page 776 and 777:
Applied Bundle Geometry 741antisymm
Page 778 and 779:
Applied Bundle Geometry 743see this
Page 780 and 781:
Applied Bundle Geometry 745Coupling
Page 782 and 783:
Applied Bundle Geometry 747acts as
Page 784 and 785:
Applied Bundle Geometry 749good coo
Page 786 and 787:
Applied Bundle Geometry 751Φ. Brea
Page 788 and 789:
Applied Bundle Geometry 753equating
Page 790 and 791:
Applied Bundle Geometry 755differen
Page 792 and 793:
Applied Bundle Geometry 7574.14.10
Page 794 and 795:
Applied Bundle Geometry 759deformed
Page 796 and 797:
Applied Bundle Geometry 761phenomen
Page 798 and 799:
Applied Bundle Geometry 763appears
Page 800 and 801:
Applied Bundle Geometry 7654.14.10.
Page 802 and 803:
Applied Bundle Geometry 767For J >
Page 804 and 805:
Applied Bundle Geometry 769K 1/2
Page 806 and 807:
Applied Bundle Geometry 771(i) The
Page 808 and 809:
Applied Bundle Geometry 773by α =
Page 810 and 811:
Applied Bundle Geometry 775(4.279)
Page 812 and 813:
Applied Bundle Geometry 777Λ ⊗
Page 814 and 815:
Applied Bundle Geometry 779coupling
Page 816 and 817:
Applied Bundle Geometry 781where we
Page 818 and 819:
Applied Bundle Geometry 783The posi
Page 820 and 821:
Applied Bundle Geometry 785Although
Page 822 and 823:
Applied Bundle Geometry 787Let us a
Page 824 and 825:
Applied Bundle Geometry 789of the t
Page 826 and 827:
Applied Bundle Geometry 791The Riem
Page 828 and 829:
Applied Bundle Geometry 793To compl
Page 830 and 831:
Applied Bundle Geometry 795Ω 1 :Ω
Page 832 and 833:
Chapter 5Applied Jet GeometryModern
Page 834 and 835:
Applied Jet Geometry 799x, with coe
Page 836 and 837:
Applied Jet Geometry 801s i (x), as
Page 838 and 839:
Applied Jet Geometry 803It is conve
Page 840 and 841:
Applied Jet Geometry 805has the 1
Page 842 and 843:
Applied Jet Geometry 807which split
Page 844 and 845:
Applied Jet Geometry 809gives the h
Page 846 and 847:
Applied Jet Geometry 811Furthermore
Page 848 and 849:
Applied Jet Geometry 813This is a c
Page 850 and 851:
Applied Jet Geometry 815Due to this
Page 852 and 853:
Applied Jet Geometry 817The affine
Page 854 and 855:
Applied Jet Geometry 819In particul
Page 856 and 857:
Applied Jet Geometry 821Building on
Page 858 and 859:
Applied Jet Geometry 823on J ∞ (X
Page 860 and 861:
Applied Jet Geometry 825its tangent
Page 862 and 863:
Applied Jet Geometry 827Fig. 5.3 Hi
Page 864 and 865:
Applied Jet Geometry 829It follows
Page 866 and 867:
Applied Jet Geometry 831affine, whi
Page 868 and 869:
Applied Jet Geometry 833where Γ i
Page 870 and 871:
Applied Jet Geometry 8355.6.5 Jacob
Page 872 and 873:
Applied Jet Geometry 837is also the
Page 874 and 875:
Applied Jet Geometry 839A generic m
Page 876 and 877:
Applied Jet Geometry 841where ϑ eH
Page 878 and 879:
Applied Jet Geometry 843Since E L|
Page 880 and 881:
Applied Jet Geometry 845Thus, every
Page 882 and 883:
Applied Jet Geometry 847along u. Ev
Page 884 and 885:
Applied Jet Geometry 849equations c
Page 886 and 887:
Applied Jet Geometry 851A Hamiltoni
Page 888 and 889:
Applied Jet Geometry 853Connections
Page 890 and 891:
Applied Jet Geometry 855Lagrangian
Page 892 and 893:
Applied Jet Geometry 857on V ∗ M.
Page 894 and 895:
Applied Jet Geometry 859contain no
Page 896 and 897:
Applied Jet Geometry 861(ii) For an
Page 898 and 899:
Applied Jet Geometry 8635.6.16 Lyap
Page 900 and 901:
Applied Jet Geometry 865for any t >
Page 902 and 903:
Applied Jet Geometry 867vertical co
Page 904 and 905:
Applied Jet Geometry 869difficulty,
Page 906 and 907:
Applied Jet Geometry 871the energy
Page 908 and 909:
Applied Jet Geometry 873by the tran
Page 910 and 911:
Applied Jet Geometry 875The conditi
Page 912 and 913:
Applied Jet Geometry 877Let Γ = (M
Page 914 and 915:
Applied Jet Geometry 879geometrical
Page 916 and 917:
Applied Jet Geometry 881Let (M, I)
Page 918 and 919:
Applied Jet Geometry 883so referrin
Page 920 and 921:
Applied Jet Geometry 885The Poincar
Page 922 and 923:
Applied Jet Geometry 887where N n
Page 924 and 925:
Applied Jet Geometry 889be a projec
Page 926 and 927:
Applied Jet Geometry 891to differen
Page 928 and 929:
Applied Jet Geometry 893for every c
Page 930 and 931:
Applied Jet Geometry 895vector-fiel
Page 932 and 933:
Applied Jet Geometry 897This is phr
Page 934 and 935:
Applied Jet Geometry 899where a, b
Page 936 and 937:
Applied Jet Geometry 901represent t
Page 938 and 939:
Applied Jet Geometry 903̂L : J 1 (
Page 940 and 941:
Applied Jet Geometry 905space J 1 (
Page 942 and 943:
Applied Jet Geometry 907First of al
Page 944 and 945:
Applied Jet Geometry 909Therefore,
Page 946 and 947:
Applied Jet Geometry 911Recall that
Page 948 and 949:
Applied Jet Geometry 9135.11 Applic
Page 950 and 951:
Applied Jet Geometry 915the vector
Page 952 and 953:
Applied Jet Geometry 9175.11.4 Gaug
Page 954 and 955:
Applied Jet Geometry 9195.11.5 Lagr
Page 956 and 957:
Applied Jet Geometry 921are represe
Page 958 and 959:
Applied Jet Geometry 923which are t
Page 960 and 961:
Applied Jet Geometry 925of this fun
Page 962 and 963:
Applied Jet Geometry 927a principal
Page 964 and 965:
Applied Jet Geometry 929Recall that
Page 966 and 967:
Applied Jet Geometry 931Moreover, t
Page 968 and 969:
Applied Jet Geometry 933other only
Page 970 and 971:
Applied Jet Geometry 935In particul
Page 972 and 973:
Applied Jet Geometry 937Note that,
Page 974 and 975:
Applied Jet Geometry 939This equali
Page 976 and 977:
Applied Jet Geometry 941Then, the s
Page 978 and 979:
Applied Jet Geometry 943In the Lagr
Page 980 and 981:
Applied Jet Geometry 945Let τ be a
Page 982 and 983:
Applied Jet Geometry 947Using (5.42
Page 984 and 985:
Applied Jet Geometry 949associated
Page 986 and 987:
Applied Jet Geometry 951parts,F α
Page 988 and 989:
Applied Jet Geometry 953of the Hilb
Page 990 and 991:
Applied Jet Geometry 955For example
Page 992 and 993:
Applied Jet Geometry 957the convent
Page 994 and 995:
Applied Jet Geometry 959left ideal
Page 996 and 997:
Applied Jet Geometry 961LX → Σ i
Page 998 and 999:
Applied Jet Geometry 963Levi-Civita
Page 1000 and 1001:
Applied Jet Geometry 965see this fr
Page 1002 and 1003:
Applied Jet Geometry 967curves. And
Page 1004 and 1005:
Applied Jet Geometry 969One sees th
Page 1006 and 1007:
Applied Jet Geometry 971sums over a
Page 1008 and 1009:
Applied Jet Geometry 973all this to
Page 1010 and 1011:
Applied Jet Geometry 975universe ev
Page 1012 and 1013:
Applied Jet Geometry 977on Σ. If M
Page 1014 and 1015:
Applied Jet Geometry 979(1) Lorentz
Page 1016 and 1017:
Applied Jet Geometry 981terpret thi
Page 1018 and 1019:
Chapter 6Geometrical Path Integrals
Page 1020 and 1021:
Geometrical Path Integrals and Thei
Page 1022 and 1023:
Page 1024 and 1025:
Page 1026 and 1027:
Page 1028 and 1029:
Page 1030 and 1031:
Page 1032 and 1033:
Page 1034 and 1035:
Page 1036 and 1037:
Page 1038 and 1039:
Page 1040 and 1041:
Page 1042 and 1043:
Page 1044 and 1045:
Page 1046 and 1047:
Page 1048 and 1049:
Page 1050 and 1051:
Page 1052 and 1053:
Page 1054 and 1055:
Page 1056 and 1057:
Page 1058 and 1059:
Page 1060 and 1061:
Page 1062 and 1063:
Page 1064 and 1065:
Page 1066 and 1067:
Page 1068 and 1069:
Page 1070 and 1071:
Page 1072 and 1073:
Page 1074 and 1075:
Page 1076 and 1077:
Page 1078 and 1079:
Page 1080 and 1081:
Page 1082 and 1083:
Page 1084 and 1085:
Page 1086 and 1087:
Page 1088 and 1089:
Page 1090 and 1091:
Page 1092 and 1093:
Page 1094 and 1095:
Page 1096 and 1097:
Page 1098 and 1099:
Page 1100 and 1101:
Page 1102 and 1103:
Page 1104 and 1105:
Page 1106 and 1107:
Page 1108 and 1109:
Page 1110 and 1111:
Page 1112 and 1113:
Page 1114 and 1115:
Page 1116 and 1117:
Page 1118 and 1119:
Page 1120 and 1121:
Page 1122 and 1123:
Page 1124 and 1125:
Page 1126 and 1127:
Page 1128 and 1129:
Page 1130 and 1131:
Page 1132 and 1133:
Page 1134 and 1135:
Page 1136 and 1137:
Page 1138 and 1139:
Page 1140 and 1141:
Page 1142 and 1143:
Page 1144 and 1145:
Page 1146 and 1147:
Page 1148 and 1149:
Page 1150 and 1151:
Page 1152 and 1153:
Page 1154 and 1155:
Page 1156 and 1157:
Page 1158 and 1159:
Page 1160 and 1161:
Page 1162 and 1163:
Page 1164 and 1165:
Page 1166 and 1167:
Page 1168 and 1169:
Page 1170 and 1171:
Page 1172 and 1173:
Page 1174 and 1175:
Page 1176 and 1177:
Page 1178 and 1179:
Page 1180 and 1181:
Page 1182 and 1183:
Page 1184 and 1185:
Page 1186 and 1187:
Page 1188 and 1189:
Page 1190 and 1191:
Page 1192 and 1193:
Page 1194 and 1195:
Page 1196 and 1197:
Page 1198 and 1199:
Page 1200 and 1201:
Page 1202 and 1203:
Page 1204 and 1205:
Page 1206 and 1207:
Page 1208 and 1209:
Page 1210 and 1211:
Page 1212 and 1213:
Page 1214 and 1215:
Page 1216 and 1217:
Page 1218 and 1219:
Page 1220 and 1221:
Page 1222 and 1223:
Page 1224 and 1225:
Page 1226 and 1227:
Page 1228 and 1229:
Page 1230 and 1231:
Page 1232 and 1233:
Page 1234 and 1235:
Page 1236 and 1237:
Page 1238 and 1239:
Page 1240 and 1241:
Page 1242 and 1243:
Page 1244 and 1245:
Page 1246 and 1247:
Page 1248 and 1249:
Page 1250 and 1251:
Page 1252 and 1253:
Page 1254 and 1255:
Page 1256 and 1257:
Page 1258 and 1259:
Page 1260 and 1261:
Page 1262 and 1263:
Page 1264 and 1265:
Page 1266 and 1267:
Page 1268 and 1269:
Page 1270 and 1271:
Page 1272 and 1273:
Page 1274 and 1275:
Page 1276 and 1277:
Page 1278 and 1279:
Page 1280 and 1281:
Page 1282 and 1283:
Page 1284 and 1285:
Page 1286 and 1287:
Page 1288 and 1289:
BibliographyAblowitz, M.J., Clarkso
Page 1290 and 1291:
Bibliography 1255equations. Springe
Page 1292 and 1293:
Bibliography 1257connections, Phys.
Page 1294 and 1295:
Bibliography 1259JHEP 0003:007.Bran
Page 1296 and 1297:
Bibliography 1261Clementi, C. Petti
Page 1298 and 1299:
Bibliography 1263Domany, E., Van He
Page 1300 and 1301:
Bibliography 1265geometrical signat
Page 1302 and 1303:
Bibliography 1267Bergmann theory of
Page 1304 and 1305:
Bibliography 1269to Complex Systems
Page 1306 and 1307:
Bibliography 1271Action for String
Page 1308 and 1309:
Bibliography 1273Ivancevic, V., Bea
Page 1310 and 1311:
Bibliography 1275Krener, A. (1984).
Page 1312 and 1313:
Bibliography 1277Li, M., Vitanyi, P
Page 1314 and 1315:
Bibliography 1279Nature, 261(5560),
Page 1316 and 1317:
Bibliography 1281Lambert and R. Gol
Page 1318 and 1319:
Bibliography 1283dynamics of human
Page 1320 and 1321:
Bibliography 1285Reshetikhin, N.Yu,
Page 1322 and 1323:
Bibliography 1287Rev. E, 58, 6333-6
Page 1324 and 1325:
Bibliography 1289Stasheff, J.D. (19
Page 1326 and 1327:
Bibliography 1291matical Physics. S
Page 1328 and 1329:
Bibliography 1293Witten, E. (1991).
Page 1330 and 1331:
Index1−jet bundle, 8271−jet lif
Page 1332 and 1333:
Index 1297Chan-Paton factor, 1238Ch
Page 1334 and 1335:
Index 1299Dirac-Born-Infeld theory,
Page 1336 and 1337:
Index 1301Gauss, 5Gauss map, 284Gau
Page 1338 and 1339:
Index 1303isometric, 17isomorphic,
Page 1340 and 1341:
Index 1305Maxwell Lagrangian, 34Max
Page 1342 and 1343:
Index 1307probability manifold, 333
Page 1344 and 1345:
Index 1309super-space, 1215supercov
Page 1346:
Index 1311Yang-Mills theory, 39, 29
show all

Ivancevic_Applied-Diff-Geom

Create successful ePaper yourself

Delete template?

Save as template?