Ivancevic_Applied-Diff-Geom

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Contentsxxi3.6.3.4 Stokes Theorem and de RhamCohomology . . . . . . . . . . . . . . . 1863.6.3.5 Euler–Poincaré Characteristics of M . . 1883.6.3.6 Duality of Chains and Forms on M . . 1883.6.3.7 Hodge Star Operator and HarmonicForms . . . . . . . . . . . . . . . . . . . 1903.7 Lie Derivatives on Smooth Manifolds . . . . . . . . . . . . 1923.7.1 Lie Derivative Operating on Functions . . . . . . 1923.7.2 Lie Derivative of Vector Fields . . . . . . . . . . . 1943.7.3 Time Derivative of the Evolution Operator . . . . 1973.7.4 Lie Derivative of Differential Forms . . . . . . . . 1973.7.5 Lie Derivative of Various Tensor Fields . . . . . . 1983.7.6 Application: Lie–Derivative Neurodynamics . . 2003.7.7 Lie Algebras . . . . . . . . . . . . . . . . . . . . . 2023.8 Lie Groups and Associated Lie Algebras . . . . . . . . . . 2023.8.1 Definition of a Lie Group . . . . . . . . . . . . . . 2033.8.2 Actions of Lie Groups on Smooth Manifolds . . . 2073.8.3 Basic Dynamical Lie Groups . . . . . . . . . . . . 2103.8.3.1 Galilei Group . . . . . . . . . . . . . . . 2103.8.3.2 General Linear Group . . . . . . . . . . 2113.8.4 Application: Lie Groups in Biodynamics . . . . 2123.8.4.1 Lie Groups of Joint Rotations . . . . . . 2123.8.4.2 Euclidean Groups of Total JointMotions . . . . . . . . . . . . . . . . . . 2163.8.4.3 Group Structure of BiodynamicalManifold . . . . . . . . . . . . . . . . . 2223.8.5 Application: Dynamical Games onSE(n)−Groups . . . . . . . . . . . . . . . . . . . 2273.8.5.1 Configuration Models for PlanarVehicles . . . . . . . . . . . . . . . . . . 2273.8.5.2 Two–Vehicles Conflict ResolutionManoeuvres . . . . . . . . . . . . . . . . 2283.8.5.3 Symplectic Reduction and DynamicalGames on SE(2) . . . . . . . . . . . . . 2303.8.5.4 Nash Solutions for Multi–VehicleManoeuvres . . . . . . . . . . . . . . . . 2333.8.6 Classical Lie Theory . . . . . . . . . . . . . . . . 2353.8.6.1 Basic Tables of Lie Groups and theirLie Algebras . . . . . . . . . . . . . . . 236
xxiiApplied Differential Geometry: A Modern Introduction3.8.6.2 Representations of Lie groups . . . . . . 2393.8.6.3 Root Systems and Dynkin Diagrams . . 2403.8.6.4 Simple and Semisimple Lie Groupsand Algebras . . . . . . . . . . . . . . . 2453.9 Lie Symmetries and Prolongations on Manifolds . . . . . 2473.9.1 Lie Symmetry Groups . . . . . . . . . . . . . . . . 2473.9.1.1 Exponentiation of Vector Fields on M . 2473.9.1.2 Lie Symmetry Groups and General DEs 2493.9.2 Prolongations . . . . . . . . . . . . . . . . . . . . 2503.9.2.1 Prolongations of Functions . . . . . . . 2503.9.2.2 Prolongations of Differential Equations 2513.9.2.3 Prolongations of Group Actions . . . . 2523.9.2.4 Prolongations of Vector Fields . . . . . 2533.9.2.5 General Prolongation Formula . . . . . 2543.9.3 Generalized Lie Symmetries . . . . . . . . . . . . 2563.9.3.1 Noether Symmetries . . . . . . . . . . . 2573.9.4 Application: Biophysical PDEs . . . . . . . . . 2613.9.4.1 The Heat Equation . . . . . . . . . . . . 2613.9.4.2 The Korteveg–De Vries Equation . . . . 2623.9.5 Lie–Invariant Geometric Objects . . . . . . . . . . 2623.9.5.1 Robot Kinematics . . . . . . . . . . . . 2623.9.5.2 Maurer–Cartan 1–Forms . . . . . . . . . 2643.9.5.3 General Structure of IntegrableOne–Forms . . . . . . . . . . . . . . . . 2653.9.5.4 Lax Integrable Dynamical Systems . . . 2673.9.5.5 Application: Burgers DynamicalSystem . . . . . . . . . . . . . . . . . . 2683.10 Riemannian Manifolds and Their Applications . . . . . . 2713.10.1 Local Riemannian Geometry . . . . . . . . . . . . 2713.10.1.1 Riemannian Metric on M . . . . . . . . 2723.10.1.2 Geodesics on M . . . . . . . . . . . . . 2773.10.1.3 Riemannian Curvature on M . . . . . . 2783.10.2 Global Riemannian Geometry . . . . . . . . . . . 2813.10.2.1 The Second Variation Formula . . . . . 2813.10.2.2 Gauss–Bonnet Formula . . . . . . . . . 2843.10.2.3 Ricci Flow on M . . . . . . . . . . . . . 2853.10.2.4 Structure Equations on M . . . . . . . 2873.10.3 Application: Autonomous LagrangianDynamics . . . . . . . . . . . . . . . . . . . . . . 289
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 18 and 19: ContentsPrefaceGlossary of Frequent
Page 20 and 21: Contentsxix2.1.5.2 Forces Acting on
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.
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Introduction 37theory. 50 Fig. 1.3
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Introduction 39to describe a univer
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Introduction 41which have two disti
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Introduction 43first theory can be
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Introduction 45the Casimir effect,
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Introduction 47• External coordin
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Introduction 49a connection-base de
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Chapter 2Technical Preliminaries: T
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Technical Preliminaries: Tensors, A
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Chapter 3Applied Manifold Geometry3
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Applied Manifold Geometry 139where
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Applied Manifold Geometry 141In loc
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Applied Manifold Geometry 143on the
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Applied Manifold Geometry 145exist
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Applied Manifold Geometry 147then E
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Applied Manifold Geometry 1490D man
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Applied Manifold Geometry 151Fig. 3
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Applied Manifold Geometry 153the fo
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Applied Manifold Geometry 155contin
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Applied Manifold Geometry 157In oth
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Applied Manifold Geometry 159scalar
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Applied Manifold Geometry 1633.6 Te
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Applied Manifold Geometry 165Let ϕ
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Applied Manifold Geometry 1673.6.1.
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Applied Manifold Geometry 169or, if
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Applied Manifold Geometry 171If X i
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Applied Manifold Geometry 173along
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Applied Manifold Geometry 175is dua
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Applied Manifold Geometry 179(1) α
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Applied Manifold Geometry 181diagra
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Applied Manifold Geometry 183Clearl
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Applied Manifold Geometry 187duces,
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Applied Manifold Geometry 189This i
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Applied Manifold Geometry 191Hodge
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Applied Manifold Geometry 193The in
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Applied Manifold Geometry 195since
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Applied Manifold Geometry 197If a l
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Applied Manifold Geometry 199asGive
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Applied Manifold Geometry 201The co
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Applied Manifold Geometry 203at the
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Applied Manifold Geometry 205For ex
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Applied Manifold Geometry 207axis,
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Applied Manifold Geometry 209For ex
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Applied Manifold Geometry 211The sp
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Applied Manifold Geometry 213corres
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Applied Manifold Geometry 215corres
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Applied Manifold Geometry 217Specia
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Applied Manifold Geometry 219the co
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Applied Manifold Geometry 221moment
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Applied Manifold Geometry 223produc
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Applied Manifold Geometry 225has a
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Applied Manifold Geometry 2273.8.5
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Applied Manifold Geometry 229in whi
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Applied Manifold Geometry 231with
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Applied Manifold Geometry 233itly:(
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Applied Manifold Geometry 235play t
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Applied Manifold Geometry 237Some r
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Applied Manifold Geometry 239Basic
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Applied Manifold Geometry 241orthog
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Applied Manifold Geometry 243is adm
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Applied Manifold Geometry 245Root S
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Applied Manifold Geometry 247groups
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Applied Manifold Geometry 251second
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Applied Manifold Geometry 255be a v
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Applied Manifold Geometry 257Now, a
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Applied Manifold Geometry 263other
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Applied Manifold Geometry 267result
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Applied Manifold Geometry 2713.10 R
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Applied Manifold Geometry 273More p
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Applied Manifold Geometry 275Y (t 0
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Applied Manifold Geometry 277which
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Applied Manifold Geometry 279for an
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Applied Manifold Geometry 283Thus w
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Applied Manifold Geometry 287evolut
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Applied Manifold Geometry 293‘ren
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Applied Manifold Geometry 295mute;
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Applied Manifold Geometry 297Rather
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Applied Manifold Geometry 299of the
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Applied Manifold Geometry 301clidea
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Applied Manifold Geometry 303Second
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Applied Manifold Geometry 305i.e.,
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Applied Manifold Geometry 307invari
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Applied Manifold Geometry 309The Ba
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Applied Manifold Geometry 311Let us
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Applied Manifold Geometry 313all cu
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Applied Manifold Geometry 315ders.
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Applied Manifold Geometry 317this c
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Applied Manifold Geometry 319In par
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Applied Manifold Geometry 321As I p
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Applied Manifold Geometry 323the fo
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Applied Manifold Geometry 325per (1
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Applied Manifold Geometry 327Genera
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Applied Manifold Geometry 329The ma
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Applied Manifold Geometry 331pretat
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Applied Manifold Geometry 333byMDL
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Applied Manifold Geometry 3353.12 S
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Applied Manifold Geometry 337H 2 (M
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Applied Manifold Geometry 339bracke
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Applied Manifold Geometry 341Any so
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Applied Manifold Geometry 343All on
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Applied Manifold Geometry 345The ac
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Applied Manifold Geometry 347linear
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Applied Manifold Geometry 349such t
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Applied Manifold Geometry 351Let γ
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Applied Manifold Geometry 355chart
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Applied Manifold Geometry 357phase-
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Applied Manifold Geometry 359( ) 0
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Applied Manifold Geometry 361hetero
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Applied Manifold Geometry 363thus c
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Applied Manifold Geometry 365Action
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Applied Manifold Geometry 367and we
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Applied Manifold Geometry 369action
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Applied Manifold Geometry 371and we
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Applied Manifold Geometry 373by (J(
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Applied Manifold Geometry 375fields
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Applied Manifold Geometry 377Let Tx
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Applied Manifold Geometry 381The ve
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Applied Manifold Geometry 383and gi
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Applied Manifold Geometry 385The co
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Applied Manifold Geometry 387tor Ca
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Applied Manifold Geometry 3912 −
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Applied Manifold Geometry 393comple
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Applied Manifold Geometry 395F[C]
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Applied Manifold Geometry 397become
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Applied Manifold Geometry 399of fuz
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Applied Manifold Geometry 401In our
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Applied Manifold Geometry 403Biodyn
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Applied Manifold Geometry 405for T
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Applied Manifold Geometry 407A vari
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Applied Manifold Geometry 409such t
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Applied Manifold Geometry 411of neg
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Applied Manifold Geometry 415Follow
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Applied Manifold Geometry 417and J
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Applied Manifold Geometry 419This t
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Applied Manifold Geometry 421above,
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Applied Manifold Geometry 423As a L
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Applied Manifold Geometry 425Theore
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Applied Manifold Geometry 427real v
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Applied Manifold Geometry 429indepe
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Applied Manifold Geometry 431In ter
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Applied Manifold Geometry 433M, we
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Applied Manifold Geometry 435The cu
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Applied Manifold Geometry 437(iv) (
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. ..Applied Manifold Geometry 441in
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Applied Manifold Geometry 443⌋ is
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Applied Manifold Geometry 445Now, a
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Applied Manifold Geometry 447symmet
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Applied Manifold Geometry 449study
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Applied Manifold Geometry 451togeth
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Applied Manifold Geometry 453result
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Applied Manifold Geometry 455will b
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Applied Manifold Geometry 457g 11 =
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Applied Manifold Geometry 459is def
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Applied Manifold Geometry 461ordere
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Applied Manifold Geometry 463Now, r
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Applied Manifold Geometry 465Simila
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Applied Manifold Geometry 467which
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Applied Manifold Geometry 469From t
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Applied Manifold Geometry 475On the
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Applied Manifold Geometry 477ing di
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Applied Manifold Geometry 479Note t
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Applied Manifold Geometry 481Then w
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Applied Manifold Geometry 483consis
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Chapter 4Applied Bundle Geometry4.1
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Applied Bundle Geometry 487is calle
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Applied Bundle Geometry 489V and an
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Applied Bundle Geometry 4914.3 Vect
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Applied Bundle Geometry 493dimensio
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Applied Bundle Geometry 495orthogon
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Applied Bundle Geometry 497t, is a
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Applied Bundle Geometry 499Let T Y
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Applied Bundle Geometry 501In other
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Applied Bundle Geometry 503we shall
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Applied Bundle Geometry 505Pendulum
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Applied Bundle Geometry 507Using L
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Applied Bundle Geometry 509the isom
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Applied Bundle Geometry 511theory f
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Applied Bundle Geometry 513of the c
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Applied Bundle Geometry 515where Ω
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Applied Bundle Geometry 517nals, th
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Applied Bundle Geometry 519(1985)])
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Applied Bundle Geometry 521finite-d
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Applied Bundle Geometry 523and thes
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Applied Bundle Geometry 525Now, let
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Applied Bundle Geometry 527gravity
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Applied Bundle Geometry 529a Dp−b
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Applied Bundle Geometry 531A princi
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Applied Bundle Geometry 533is the r
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Applied Bundle Geometry 535Now, let
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Applied Bundle Geometry 537By cycli
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Applied Bundle Geometry 5394.9.2 Fe
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Applied Bundle Geometry 541This lin
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Applied Bundle Geometry 543(1) L g
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Applied Bundle Geometry 545and ω 0
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Applied Bundle Geometry 547isfiesω
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Applied Bundle Geometry 549of the c
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Applied Bundle Geometry 551Controll
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Applied Bundle Geometry 553Foliatio
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Applied Bundle Geometry 555ψ : (x,
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Applied Bundle Geometry 557from res
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Applied Bundle Geometry 559Motion a
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Applied Bundle Geometry 561where Z
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Applied Bundle Geometry 563Systems
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Applied Bundle Geometry 5653. Recal
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Applied Bundle Geometry 567Therefor
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Applied Bundle Geometry 569smooth m
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Applied Bundle Geometry 571state va
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Applied Bundle Geometry 573The geom
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Applied Bundle Geometry 575Y ∋ y
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Applied Bundle Geometry 577autonomo
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Applied Bundle Geometry 579(see 3.1
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Applied Bundle Geometry 581and init
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Applied Bundle Geometry 583potentia
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Applied Bundle Geometry 585effector
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Applied Bundle Geometry 587Fig. 4.9
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Applied Bundle Geometry 589physical
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Applied Bundle Geometry 591open Lio
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Applied Bundle Geometry 593et al. (
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Applied Bundle Geometry 595(2) Acti
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Applied Bundle Geometry 597˙Φ ≤
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Applied Bundle Geometry 599number o
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Applied Bundle Geometry 601where Ψ
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Applied Bundle Geometry 603ing to c
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Applied Bundle Geometry 605where [
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Applied Bundle Geometry 607interval
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Applied Bundle Geometry 609The foll
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Applied Bundle Geometry 611In parti
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Applied Bundle Geometry 613The pull
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Applied Bundle Geometry 615(2) δ
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Applied Bundle Geometry 617of unita
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Applied Bundle Geometry 619system o
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Applied Bundle Geometry 621of H pre
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Applied Bundle Geometry 623It follo
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Applied Bundle Geometry 625a produc
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Applied Bundle Geometry 627B Ham(M)
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Applied Bundle Geometry 629b.Let us
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Applied Bundle Geometry 631If P is
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Applied Bundle Geometry 633question
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Applied Bundle Geometry 635spheres
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Applied Bundle Geometry 637be thoug
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Applied Bundle Geometry 639of the b
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Applied Bundle Geometry 641in C 1 (
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Applied Bundle Geometry 643from a t
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Applied Bundle Geometry 645Now let
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Applied Bundle Geometry 647for t <
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Applied Bundle Geometry 649is a Ham
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Applied Bundle Geometry 651The foll
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Applied Bundle Geometry 653is the u
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Applied Bundle Geometry 655every no
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Applied Bundle Geometry 657P × P w
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Applied Bundle Geometry 659that the
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Applied Bundle Geometry 661extend t
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Applied Bundle Geometry 663identity
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Applied Bundle Geometry 665The mapw
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Applied Bundle Geometry 667It is a
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Applied Bundle Geometry 669I Q is i
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Applied Bundle Geometry 671whereC i
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Applied Bundle Geometry 673form Q.
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Applied Bundle Geometry 675τ( exp(
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Applied Bundle Geometry 6774.13.2.2
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Applied Bundle Geometry 679Therefor
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Applied Bundle Geometry 683⎛⎞co
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Applied Bundle Geometry 685Usually,
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Applied Bundle Geometry 687on the s
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Applied Bundle Geometry 6914.13.3 P
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Applied Bundle Geometry 693Since T
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Applied Bundle Geometry 695On the o
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Applied Bundle Geometry 697may writ
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Applied Bundle Geometry 699determin
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Applied Bundle Geometry 701section,
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Applied Bundle Geometry 703for deri
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Applied Bundle Geometry 705dictions
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Applied Bundle Geometry 707too sing
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Applied Bundle Geometry 709In (4.18
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Applied Bundle Geometry 713sis repr
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Applied Bundle Geometry 717a (linea
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Applied Bundle Geometry 719past few
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Applied Bundle Geometry 721global q
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Applied Bundle Geometry 7239. It ha
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Applied Bundle Geometry 725needs to
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Applied Bundle Geometry 727pear at
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Applied Bundle Geometry 729X, F A i
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Applied Bundle Geometry 731ifolds,
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Applied Bundle Geometry 733and not
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Applied Bundle Geometry 735under U(
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Applied Bundle Geometry 737N = 2 br
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Applied Bundle Geometry 739thusa
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Applied Bundle Geometry 741antisymm
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Applied Bundle Geometry 743see this
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Applied Bundle Geometry 745Coupling
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Applied Bundle Geometry 747acts as
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Applied Bundle Geometry 749good coo
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Applied Bundle Geometry 751Φ. Brea
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Applied Bundle Geometry 753equating
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Applied Bundle Geometry 755differen
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Applied Bundle Geometry 7574.14.10
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Applied Bundle Geometry 759deformed
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Applied Bundle Geometry 761phenomen
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Applied Bundle Geometry 763appears
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Applied Bundle Geometry 7654.14.10.
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Applied Bundle Geometry 767For J >
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Applied Bundle Geometry 769K 1/2
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Applied Bundle Geometry 771(i) The
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Applied Bundle Geometry 773by α =
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Applied Bundle Geometry 775(4.279)
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Applied Bundle Geometry 777Λ ⊗
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Applied Bundle Geometry 779coupling
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Applied Bundle Geometry 781where we
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Applied Bundle Geometry 783The posi
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Applied Bundle Geometry 785Although
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Applied Bundle Geometry 787Let us a
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Applied Bundle Geometry 789of the t
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Applied Bundle Geometry 791The Riem
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Applied Bundle Geometry 793To compl
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Applied Bundle Geometry 795Ω 1 :Ω
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Chapter 5Applied Jet GeometryModern
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Applied Jet Geometry 799x, with coe
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Applied Jet Geometry 801s i (x), as
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Applied Jet Geometry 803It is conve
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Applied Jet Geometry 805has the 1
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Applied Jet Geometry 807which split
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Applied Jet Geometry 809gives the h
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Applied Jet Geometry 811Furthermore
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Applied Jet Geometry 813This is a c
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Applied Jet Geometry 815Due to this
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Applied Jet Geometry 817The affine
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Applied Jet Geometry 819In particul
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Applied Jet Geometry 821Building on
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Applied Jet Geometry 823on J ∞ (X
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Applied Jet Geometry 825its tangent
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Applied Jet Geometry 827Fig. 5.3 Hi
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Applied Jet Geometry 829It follows
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Applied Jet Geometry 831affine, whi
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Applied Jet Geometry 833where Γ i
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Applied Jet Geometry 8355.6.5 Jacob
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Applied Jet Geometry 837is also the
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Applied Jet Geometry 839A generic m
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Applied Jet Geometry 841where ϑ eH
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Applied Jet Geometry 843Since E L|
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Applied Jet Geometry 845Thus, every
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Applied Jet Geometry 847along u. Ev
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Applied Jet Geometry 849equations c
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Applied Jet Geometry 851A Hamiltoni
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Applied Jet Geometry 853Connections
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Applied Jet Geometry 855Lagrangian
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Applied Jet Geometry 857on V ∗ M.
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Applied Jet Geometry 859contain no
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Applied Jet Geometry 861(ii) For an
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Applied Jet Geometry 8635.6.16 Lyap
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Applied Jet Geometry 865for any t >
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Applied Jet Geometry 867vertical co
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Applied Jet Geometry 869difficulty,
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Applied Jet Geometry 871the energy
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Applied Jet Geometry 873by the tran
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Applied Jet Geometry 875The conditi
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Applied Jet Geometry 877Let Γ = (M
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Applied Jet Geometry 879geometrical
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Applied Jet Geometry 881Let (M, I)
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Applied Jet Geometry 883so referrin
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Applied Jet Geometry 885The Poincar
Page 922 and 923:
Applied Jet Geometry 887where N n
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Applied Jet Geometry 889be a projec
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Applied Jet Geometry 891to differen
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Applied Jet Geometry 893for every c
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Applied Jet Geometry 895vector-fiel
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Applied Jet Geometry 897This is phr
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Applied Jet Geometry 899where a, b
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Applied Jet Geometry 901represent t
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Applied Jet Geometry 903̂L : J 1 (
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Applied Jet Geometry 905space J 1 (
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Applied Jet Geometry 907First of al
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Applied Jet Geometry 909Therefore,
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Applied Jet Geometry 911Recall that
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Applied Jet Geometry 9135.11 Applic
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Applied Jet Geometry 915the vector
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Applied Jet Geometry 9175.11.4 Gaug
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Applied Jet Geometry 9195.11.5 Lagr
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Applied Jet Geometry 921are represe
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Applied Jet Geometry 923which are t
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Applied Jet Geometry 925of this fun
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Applied Jet Geometry 927a principal
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Applied Jet Geometry 929Recall that
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Applied Jet Geometry 931Moreover, t
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Applied Jet Geometry 933other only
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Applied Jet Geometry 935In particul
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Applied Jet Geometry 937Note that,
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Applied Jet Geometry 939This equali
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Applied Jet Geometry 941Then, the s
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Applied Jet Geometry 943In the Lagr
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Applied Jet Geometry 945Let τ be a
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Applied Jet Geometry 947Using (5.42
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Applied Jet Geometry 949associated
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Applied Jet Geometry 951parts,F α
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Applied Jet Geometry 953of the Hilb
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Applied Jet Geometry 955For example
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Applied Jet Geometry 957the convent
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Applied Jet Geometry 959left ideal
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Applied Jet Geometry 961LX → Σ i
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Applied Jet Geometry 963Levi-Civita
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Applied Jet Geometry 965see this fr
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Applied Jet Geometry 967curves. And
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Applied Jet Geometry 969One sees th
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Applied Jet Geometry 971sums over a
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Applied Jet Geometry 973all this to
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Applied Jet Geometry 975universe ev
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Applied Jet Geometry 977on Σ. If M
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Applied Jet Geometry 979(1) Lorentz
Page 1016 and 1017:
Applied Jet Geometry 981terpret thi
Page 1018 and 1019:
Chapter 6Geometrical Path Integrals
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Geometrical Path Integrals and Thei
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BibliographyAblowitz, M.J., Clarkso
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Bibliography 1255equations. Springe
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Bibliography 1257connections, Phys.
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Bibliography 1259JHEP 0003:007.Bran
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Bibliography 1261Clementi, C. Petti
Page 1298 and 1299:
Bibliography 1263Domany, E., Van He
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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
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Bibliography 1275Krener, A. (1984).
Page 1312 and 1313:
Bibliography 1277Li, M., Vitanyi, P
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Bibliography 1279Nature, 261(5560),
Page 1316 and 1317:
Bibliography 1281Lambert and R. Gol
Page 1318 and 1319:
Bibliography 1283dynamics of human
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Bibliography 1285Reshetikhin, N.Yu,
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Bibliography 1287Rev. E, 58, 6333-6
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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
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Index 1299Dirac-Born-Infeld theory,
Page 1336 and 1337:
Index 1301Gauss, 5Gauss map, 284Gau
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Index 1303isometric, 17isomorphic,
Page 1340 and 1341:
Index 1305Maxwell Lagrangian, 34Max
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Index 1307probability manifold, 333
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Index 1309super-space, 1215supercov
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Index 1311Yang-Mills theory, 39, 29
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