Differential Geometry with Applications to Mechanics and Physics

by Talpaert; Yves

Edition: 1st

ISBN13: 9780824703851

ISBN10: 0824703855

Format: Hardcover

Pub. Date: 2000-09-12

Publisher(s): CRC Press

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Summary

Compiling data on submanifolds, tangent bundles and spaces, integral invariants, tensor fields, and enterior differential forms, this text illustrates the fundamental concepts, definitions and properties of mechanical and analytical calculus. Also offers some topology and differential calculus. DLC: Geometry--Differential

Preface	p. v
Topology and Differential Calculus Requirements	p. 1
Topology	p. 1
Topological space	p. 1
Topological space basis	p. 2
Haussdorff space	p. 4
Homeomorphism	p. 5
Connected spaces	p. 6
Compact spaces	p. 6
Partition of unity	p. 7
Differential calculus in Banach spaces	p. 8
Banach space	p. 8
Differential calculus in Banach spaces	p. 10
Differentiation of R[superscript n] into Banach	p. 17
Differentiation of R[superscript n] into R[superscript n]	p. 19
Differentiation of R[superscript n] into R[superscript n]	p. 22
Exercises	p. 30
Manifolds	p. 37
Introduction	p. 37
Differentiable manifolds	p. 40
Chart and local coordinates	p. 40
Differentiable manifold structure	p. 41
Differentiable manifolds	p. 43
Differentiable mappings	p. 50
Generalities on differentiable mappings	p. 50
Particular differentiable mappings	p. 55
Pull-back of function	p. 57
Submanifolds	p. 59
Submanifolds of R[superscript n]	p. 59
Submanifold of manifold	p. 64
Exercises	p. 65
Tangent Vector Space	p. 71
Tangent vector	p. 71
Tangent curves	p. 71
Tangent vector	p. 74
Tangent space	p. 80
Definition of a tangent space	p. 80
Basis of tangent space	p. 81
Change of basis	p. 82
Differential at a point	p. 83
Definitions	p. 84
The image in local coordinates	p. 85
Differential of a function	p. 86
Exercises	p. 87
Tangent Bundle--Vector Field--One-Parameter Group Lie Algebra	p. 91
Introduction	p. 91
Tangent bundle	p. 93
Natural manifold TM	p. 93
Extension and commutative diagram	p. 94
Vector field on manifold	p. 96
Definitions	p. 96
Properties of vector fields	p. 96
Lie algebra structure	p. 97
Bracket	p. 97
Lie algebra	p. 100
Lie derivative	p. 101
One-parameter group of diffeomorphisms	p. 102
Differential equations in Banach	p. 102
One-parameter group of diffeomorphisms	p. 104
Exercises	p. 111
Cotangent Bundle--Vector Bundle of Tensors	p. 125
Cotangent bundle and covector field	p. 125
1-form	p. 125
Cotangent bundle	p. 129
Field of covectors	p. 130
Tensor algebra	p. 130
Tensor at a point and tensor algebra	p. 130
Tensor fields and tensor algebra	p. 137
Exercises	p. 144
Exterior Differential Forms	p. 153
Exterior form at a point	p. 153
Definition of a p-form	p. 153
Exterior product of 1-forms	p. 155
Expression of a p-form	p. 156
Exterior product of forms	p. 158
Exterior algebra	p. 159
Differential forms on a manifold	p. 162
Exterior algebra (Grassmann algebra)	p. 162
Change of basis	p. 165
Pull-back of a differential form	p. 167
Definition and representation	p. 167
Pull-back properties	p. 168
Exterior differentiation	p. 170
Definition	p. 170
Exterior differential and pull-back	p. 173
Orientable manifolds	p. 174
Exercises	p. 178
Lie Derivative--Lie Group	p. 185
Lie derivative	p. 186
First presentation of Lie derivative	p. 186
Alternative interpretation of Lie derivative	p. 195
Inner product and Lie derivative	p. 199
Definition and properties	p. 199
Fundamental theorem	p. 201
Frobenius theorem	p. 204
Exterior differential systems	p. 207
Generalities	p. 207
Pfaff systems and Frobenius theorem	p. 208
Invariance of tensor fields	p. 211
Definitions	p. 211
Invariance of differential forms	p. 212
Lie algebra	p. 214
Lie group and algebra	p. 214
Lie group definition	p. 215
Lie algebra of Lie group	p. 215
Invariant differential forms on G	p. 217
One-parameter subgroup of a Lie group	p. 218
Exercises	p. 224
Integration of Forms: Stokes' Theorem, Cohomology and Integral Invariants	p. 235
n-form integration on n-manifold	p. 235
Integration definition	p. 235
Pull-back of a form and integral evaluation	p. 237
Integral over a chain	p. 239
Integral over a chain element	p. 239
Integral over a chain	p. 239
Stokes' theorem	p. 240
Stokes' formula for a closed p-interval	p. 240
Stokes' formula for a chain	p. 242
An introduction to cohomology theory	p. 243
Closed and exact forms--Cohomology	p. 243
Poincare lemma	p. 244
Cycle--Boundary--Homology	p. 247
Integral invariants	p. 248
Absolute integral invariant	p. 248
Relative integral invariant	p. 252
Exercises	p. 253
Riemannian Geometry	p. 257
Riemannian manifolds	p. 257
Metric tensor and manifolds	p. 257
Canonical isomorphism and conjugate tensor	p. 262
Orthonormal bases	p. 266
Hyperbolic manifold and special relativity	p. 267
Killing vector field	p. 274
Volume	p. 275
The Hodge operator and adjoint	p. 277
Special relativity and Maxwell equations	p. 280
Induced metric and isometry	p. 283
Affine connection	p. 285
Affine connection definition	p. 285
Christoffel symbols	p. 286
Interpretation of the covariant derivative	p. 288
Torsion	p. 291
Levi-Civita (or Riemannian) connection	p. 291
Gradient--Divergence--Laplace operators	p. 293
Geodesic and Euler equation	p. 300
Curvatures--Ricci tensor--Bianchi identity--Einstein equations	p. 302
Curvature tensor	p. 302
Ricci tensor	p. 305
Bianchi identity	p. 308
Einstein equations	p. 309
Exercises	p. 310
Lagrange and Hamilton Mechanics	p. 325
Classical mechanics spaces and metric	p. 325
Generalized coordinates and spaces	p. 325
Kinetic energy and Riemannian manifold	p. 327
Hamilton principle, Motion equations, Phase space	p. 329
Lagrangian	p. 329
Principle of least action	p. 329
Lagrange equations	p. 331
Canonical equations of Hamilton	p. 332
Phase space	p. 337
D'Alembert-Lagrange principle--Lagrange equations	p. 338
D'Alembert-Lagrange principle	p. 338
Lagrange equations	p. 340
Euler-Noether theorem	p. 341
Motion equations on Riemannian manifolds	p. 343
Canonical transformations and integral invariants	p. 344
Diffeomorphisms on phase spacetime	p. 344
Integral invariants	p. 346
Integral invariants and canonical transformations	p. 348
Liouville theorem	p. 352
The N-body problem and a problem of statistical mechanics	p. 352
N-body problem and fundamental equations	p. 353
A problem of statistical mechanics	p. 358
Isolating integrals	p. 369
Definition and examples	p. 369
Jeans theorem	p. 372
Stellar trajectories in the galaxy	p. 373
The third integral	p. 375
Invariant curve and third integral existence	p. 379
Exercises	p. 381
Symplectic Geometry--Hamilton--Jacobi Mechanics	p. 385
Preliminaries	p. 385
Symplectic geometry	p. 388
Darboux theorem and symplectic matrix	p. 388
Canonical isomorphism	p. 391
Poisson bracket of one-forms	p. 393
Poisson bracket of functions	p. 396
Symplectic mapping and canonical transformation	p. 399
Canonical transformations in mechanics	p. 404
Hamilton vector field	p. 404
Canonical transformations--Lagrange brackets	p. 408
Generating functions	p. 412
Hamilton-Jacobi equation	p. 415
Hamilton-Jacobi equation and Jacobi theorem	p. 415
Separability	p. 419
A variational principle of analytical mechanics	p. 422
Variational principle (with one degree of freedom)	p. 423
Variational principle (with n degrees of freedom)	p. 427
Exercises	p. 429
Bibliography	p. 443
Glossary	p. 445
Table of Contents provided by Syndetics. All Rights Reserved.

Differential Geometry with Applications to Mechanics and Physics

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