Differential Geometry with Applications to Mechanics and Physics

by ;
Edition: 1st
Format: Hardcover
Pub. Date: 2000-09-12
Publisher(s): CRC Press
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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

Table of Contents

Prefacep. v
Topology and Differential Calculus Requirementsp. 1
Topologyp. 1
Topological spacep. 1
Topological space basisp. 2
Haussdorff spacep. 4
Homeomorphismp. 5
Connected spacesp. 6
Compact spacesp. 6
Partition of unityp. 7
Differential calculus in Banach spacesp. 8
Banach spacep. 8
Differential calculus in Banach spacesp. 10
Differentiation of R[superscript n] into Banachp. 17
Differentiation of R[superscript n] into R[superscript n]p. 19
Differentiation of R[superscript n] into R[superscript n]p. 22
Exercisesp. 30
Manifoldsp. 37
Introductionp. 37
Differentiable manifoldsp. 40
Chart and local coordinatesp. 40
Differentiable manifold structurep. 41
Differentiable manifoldsp. 43
Differentiable mappingsp. 50
Generalities on differentiable mappingsp. 50
Particular differentiable mappingsp. 55
Pull-back of functionp. 57
Submanifoldsp. 59
Submanifolds of R[superscript n]p. 59
Submanifold of manifoldp. 64
Exercisesp. 65
Tangent Vector Spacep. 71
Tangent vectorp. 71
Tangent curvesp. 71
Tangent vectorp. 74
Tangent spacep. 80
Definition of a tangent spacep. 80
Basis of tangent spacep. 81
Change of basisp. 82
Differential at a pointp. 83
Definitionsp. 84
The image in local coordinatesp. 85
Differential of a functionp. 86
Exercisesp. 87
Tangent Bundle--Vector Field--One-Parameter Group Lie Algebrap. 91
Introductionp. 91
Tangent bundlep. 93
Natural manifold TMp. 93
Extension and commutative diagramp. 94
Vector field on manifoldp. 96
Definitionsp. 96
Properties of vector fieldsp. 96
Lie algebra structurep. 97
Bracketp. 97
Lie algebrap. 100
Lie derivativep. 101
One-parameter group of diffeomorphismsp. 102
Differential equations in Banachp. 102
One-parameter group of diffeomorphismsp. 104
Exercisesp. 111
Cotangent Bundle--Vector Bundle of Tensorsp. 125
Cotangent bundle and covector fieldp. 125
1-formp. 125
Cotangent bundlep. 129
Field of covectorsp. 130
Tensor algebrap. 130
Tensor at a point and tensor algebrap. 130
Tensor fields and tensor algebrap. 137
Exercisesp. 144
Exterior Differential Formsp. 153
Exterior form at a pointp. 153
Definition of a p-formp. 153
Exterior product of 1-formsp. 155
Expression of a p-formp. 156
Exterior product of formsp. 158
Exterior algebrap. 159
Differential forms on a manifoldp. 162
Exterior algebra (Grassmann algebra)p. 162
Change of basisp. 165
Pull-back of a differential formp. 167
Definition and representationp. 167
Pull-back propertiesp. 168
Exterior differentiationp. 170
Definitionp. 170
Exterior differential and pull-backp. 173
Orientable manifoldsp. 174
Exercisesp. 178
Lie Derivative--Lie Groupp. 185
Lie derivativep. 186
First presentation of Lie derivativep. 186
Alternative interpretation of Lie derivativep. 195
Inner product and Lie derivativep. 199
Definition and propertiesp. 199
Fundamental theoremp. 201
Frobenius theoremp. 204
Exterior differential systemsp. 207
Generalitiesp. 207
Pfaff systems and Frobenius theoremp. 208
Invariance of tensor fieldsp. 211
Definitionsp. 211
Invariance of differential formsp. 212
Lie algebrap. 214
Lie group and algebrap. 214
Lie group definitionp. 215
Lie algebra of Lie groupp. 215
Invariant differential forms on Gp. 217
One-parameter subgroup of a Lie groupp. 218
Exercisesp. 224
Integration of Forms: Stokes' Theorem, Cohomology and Integral Invariantsp. 235
n-form integration on n-manifoldp. 235
Integration definitionp. 235
Pull-back of a form and integral evaluationp. 237
Integral over a chainp. 239
Integral over a chain elementp. 239
Integral over a chainp. 239
Stokes' theoremp. 240
Stokes' formula for a closed p-intervalp. 240
Stokes' formula for a chainp. 242
An introduction to cohomology theoryp. 243
Closed and exact forms--Cohomologyp. 243
Poincare lemmap. 244
Cycle--Boundary--Homologyp. 247
Integral invariantsp. 248
Absolute integral invariantp. 248
Relative integral invariantp. 252
Exercisesp. 253
Riemannian Geometryp. 257
Riemannian manifoldsp. 257
Metric tensor and manifoldsp. 257
Canonical isomorphism and conjugate tensorp. 262
Orthonormal basesp. 266
Hyperbolic manifold and special relativityp. 267
Killing vector fieldp. 274
Volumep. 275
The Hodge operator and adjointp. 277
Special relativity and Maxwell equationsp. 280
Induced metric and isometryp. 283
Affine connectionp. 285
Affine connection definitionp. 285
Christoffel symbolsp. 286
Interpretation of the covariant derivativep. 288
Torsionp. 291
Levi-Civita (or Riemannian) connectionp. 291
Gradient--Divergence--Laplace operatorsp. 293
Geodesic and Euler equationp. 300
Curvatures--Ricci tensor--Bianchi identity--Einstein equationsp. 302
Curvature tensorp. 302
Ricci tensorp. 305
Bianchi identityp. 308
Einstein equationsp. 309
Exercisesp. 310
Lagrange and Hamilton Mechanicsp. 325
Classical mechanics spaces and metricp. 325
Generalized coordinates and spacesp. 325
Kinetic energy and Riemannian manifoldp. 327
Hamilton principle, Motion equations, Phase spacep. 329
Lagrangianp. 329
Principle of least actionp. 329
Lagrange equationsp. 331
Canonical equations of Hamiltonp. 332
Phase spacep. 337
D'Alembert-Lagrange principle--Lagrange equationsp. 338
D'Alembert-Lagrange principlep. 338
Lagrange equationsp. 340
Euler-Noether theoremp. 341
Motion equations on Riemannian manifoldsp. 343
Canonical transformations and integral invariantsp. 344
Diffeomorphisms on phase spacetimep. 344
Integral invariantsp. 346
Integral invariants and canonical transformationsp. 348
Liouville theoremp. 352
The N-body problem and a problem of statistical mechanicsp. 352
N-body problem and fundamental equationsp. 353
A problem of statistical mechanicsp. 358
Isolating integralsp. 369
Definition and examplesp. 369
Jeans theoremp. 372
Stellar trajectories in the galaxyp. 373
The third integralp. 375
Invariant curve and third integral existencep. 379
Exercisesp. 381
Symplectic Geometry--Hamilton--Jacobi Mechanicsp. 385
Preliminariesp. 385
Symplectic geometryp. 388
Darboux theorem and symplectic matrixp. 388
Canonical isomorphismp. 391
Poisson bracket of one-formsp. 393
Poisson bracket of functionsp. 396
Symplectic mapping and canonical transformationp. 399
Canonical transformations in mechanicsp. 404
Hamilton vector fieldp. 404
Canonical transformations--Lagrange bracketsp. 408
Generating functionsp. 412
Hamilton-Jacobi equationp. 415
Hamilton-Jacobi equation and Jacobi theoremp. 415
Separabilityp. 419
A variational principle of analytical mechanicsp. 422
Variational principle (with one degree of freedom)p. 423
Variational principle (with n degrees of freedom)p. 427
Exercisesp. 429
Bibliographyp. 443
Glossaryp. 445
Table of Contents provided by Syndetics. All Rights Reserved.

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