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Differential Geometry With Applications to Mechanics and Physics

Yves Talpaert
Format
Book
Published
New York : Marcel Dekker, c2001.
Language
English
Series
Monographs and Textbooks in Pure and Applied Mathematics
ISBN
0824703855 (alk. paper)
Contents
  • Lecture 0. Topology and Differential Calculus Requirements 1
  • 1. Topology 1
  • 1.1 Topological space 1
  • 1.2 Topological space basis 2
  • 1.3 Haussdorff space 4
  • 1.4 Homeomorphism 5
  • 1.5 Connected spaces 6
  • 1.6 Compact spaces 6
  • 1.7 Partition of unity 7
  • 2. Differential calculus in Banach spaces 8
  • 2.1 Banach space 8
  • 2.2 Differential calculus in Banach spaces 10
  • 2.3 Differentiation of R[superscript n] into Banach 17
  • 2.4 Differentiation of R[superscript n] into R[superscript n] 19
  • 2.5 Differentiation of R[superscript n] into R[superscript n] 22
  • Lecture 1. Manifolds 37
  • 1. Differentiable manifolds 40
  • 1.1 Chart and local coordinates 40
  • 1.2 Differentiable manifold structure 41
  • 1.3 Differentiable manifolds 43
  • 2. Differentiable mappings 50
  • 2.1 Generalities on differentiable mappings 50
  • 2.2 Particular differentiable mappings 55
  • 2.3 Pull-back of function 57
  • 3. Submanifolds 59
  • 3.1 Submanifolds of R[superscript n] 59
  • 3.2 Submanifold of manifold 64
  • Lecture 2. Tangent Vector Space 71
  • 1. Tangent vector 71
  • 1.1 Tangent curves 71
  • 1.2 Tangent vector 74
  • 2. Tangent space 80
  • 2.1 Definition of a tangent space 80
  • 2.2 Basis of tangent space 81
  • 2.3 Change of basis 82
  • 3. Differential at a point 83
  • 3.2 Image in local coordinates 85
  • 3.3 Differential of a function 86
  • Lecture 3. Tangent Bundle--Vector Field--One-Parameter Group Lie Algebra 91
  • 1. Tangent bundle 93
  • 1.1 Natural manifold TM 93
  • 1.2 Extension and commutative diagram 94
  • 2. Vector field on manifold 96
  • 2.2 Properties of vector fields 96
  • 3. Lie algebra structure 97
  • 3.1 Bracket 97
  • 3.2 Lie algebra 100
  • 3.3 Lie derivative 101
  • 4. One-parameter group of diffeomorphisms 102
  • 4.1 Differential equations in Banach 102
  • 4.2 One-parameter group of diffeomorphisms 104
  • Lecture 4. Cotangent Bundle--Vector Bundle of Tensors 125
  • 1. Cotangent bundle and covector field 125
  • 1.1 1-form 125
  • 1.2 Cotangent bundle 129
  • 1.3 Field of covectors 130
  • 2. Tensor algebra 130
  • 2.1 Tensor at a point and tensor algebra 130
  • 2.2 Tensor fields and tensor algebra 137
  • Lecture 5. Exterior Differential Forms 153
  • 1. Exterior form at a point 153
  • 1.1 Definition of a p-form 153
  • 1.2 Exterior product of 1-forms 155
  • 1.3 Expression of a p-form 156
  • 1.4 Exterior product of forms 158
  • 1.5 Exterior algebra 159
  • 2. Differential forms on a manifold 162
  • 2.1 Exterior algebra (Grassmann algebra) 162
  • 2.2 Change of basis 165
  • 3. Pull-back of a differential form 167
  • 3.1 Definition and representation 167
  • 3.2 Pull-back properties 168
  • 4. Exterior differentiation 170
  • 4.2 Exterior differential and pull-back 173
  • 5. Orientable manifolds 174
  • Lecture 6. Lie Derivative--Lie Group 185
  • 1. Lie derivative 186
  • 1.1 First presentation of Lie derivative 186
  • 1.2 Alternative interpretation of Lie derivative 195
  • 2. Inner product and Lie derivative 199
  • 2.1 Definition and properties 199
  • 2.2 Fundamental theorem 201
  • 3. Frobenius theorem 204
  • 4. Exterior differential systems 207
  • 4.1 Generalities 207
  • 4.2 Pfaff systems and Frobenius theorem 208
  • 5. Invariance of tensor fields 211
  • 5.2 Invariance of differential forms 212
  • 5.3 Lie algebra 214
  • 6. Lie group and algebra 214
  • 6.1 Lie group definition 215
  • 6.2 Lie algebra of Lie group 215
  • 6.3 Invariant differential forms on G 217
  • 6.4 One-parameter subgroup of a Lie group 218
  • Lecture 7. Integration of Forms: Stokes' Theorem, Cohomology and Integral Invariants 235
  • 1. n-form integration on n-manifold 235
  • 1.1 Integration definition 235
  • 1.2 Pull-back of a form and integral evaluation 237
  • 2.1 Integral over a chain 239
  • 2.1 Integral over a chain element 239
  • 2.2 Integral over a chain 239
  • 3. Stokes' theorem 240
  • 3.1 Stokes' formula for a closed p-interval 240
  • 3.2 Stokes' formula for a chain 242
  • 4. An introduction to cohomology theory 243
  • 4.1 Closed and exact forms--Cohomology 243
  • 4.2 Poincare lemma 244
  • 4.3 Cycle--Boundary--Homology 247
  • 5. Integral invariants 248
  • 5.1 Absolute integral invariant 248
  • 5.2 Relative integral invariant 252
  • Lecture 8. Riemannian Geometry 257
  • 1. Riemannian manifolds 257
  • 1.1 Metric tensor and manifolds 257
  • 1.2 Canonical isomorphism and conjugate tensor 262
  • 1.3 Orthonormal bases 266
  • 1.4 Hyperbolic manifold and special relativity 267
  • 1.5 Killing vector field 274
  • 1.6 Volume 275
  • 1.7 Hodge operator and adjoint 277
  • 1.8 Special relativity and Maxwell equations 280
  • 1.9 Induced metric and isometry 283
  • 2. Affine connection 285
  • 2.1 Affine connection definition 285
  • 2.2 Christoffel symbols 286
  • 2.3 Interpretation of the covariant derivative 288
  • 2.4 Torsion 291
  • 2.5 Levi-Civita (or Riemannian) connection 291
  • 2.6 Gradient--Divergence--Laplace operators 293
  • 3. Geodesic and Euler equation 300
  • 4. Curvatures--Ricci tensor--Bianchi identity--Einstein equations 302
  • 4.1 Curvature tensor 302
  • 4.2 Ricci tensor 305
  • 4.3 Bianchi identity 308
  • 4.4 Einstein equations 309
  • Lecture 9. Lagrange and Hamilton Mechanics 325
  • 1. Classical mechanics spaces and metric 325
  • 1.1 Generalized coordinates and spaces 325
  • 1.2 Kinetic energy and Riemannian manifold 327
  • 2. Hamilton principle, Motion equations, Phase space 329
  • 2.1 Lagrangian 329
  • 2.2 Principle of least action 329
  • 2.3 Lagrange equations 331
  • 2.4 Canonical equations of Hamilton 332
  • 2.5 Phase space 337
  • 3. D'Alembert-Lagrange principle--Lagrange equations 338
  • 3.1 D'Alembert-Lagrange principle 338
  • 3.2 Lagrange equations 340
  • 3.3 Euler-Noether theorem 341
  • 3.4 Motion equations on Riemannian manifolds 343
  • 4. Canonical transformations and integral invariants 344
  • 4.1 Diffeomorphisms on phase spacetime 344
  • 4.2 Integral invariants 346
  • 4.3 Integral invariants and canonical transformations 348
  • 4.4 Liouville theorem 352
  • 5. N-body problem and a problem of statistical mechanics 352
  • 5.1 N-body problem and fundamental equations 353
  • 5.2 A problem of statistical mechanics 358
  • 6. Isolating integrals 369
  • 6.1 Definition and examples 369
  • 6.2 Jeans theorem 372
  • 6.3 Stellar trajectories in the galaxy 373
  • 6.4 Third integral 375
  • 6.5 Invariant curve and third integral existence 379
  • Lecture 10. Symplectic Geometry--Hamilton--Jacobi Mechanics 385
  • 1. Symplectic geometry 388
  • 1.1 Darboux theorem and symplectic matrix 388
  • 1.2 Canonical isomorphism 391
  • 1.3 Poisson bracket of one-forms 393
  • 1.4 Poisson bracket of functions 396
  • 1.5 Symplectic mapping and canonical transformation 399
  • 2. Canonical transformations in mechanics 404
  • 2.1 Hamilton vector field 404
  • 2.2 Canonical transformations--Lagrange brackets 408
  • 2.3 Generating functions 412
  • 3. Hamilton-Jacobi equation 415
  • 3.1 Hamilton-Jacobi equation and Jacobi theorem 415
  • 3.2 Separability 419
  • 4. A variational principle of analytical mechanics 422
  • 4.1 Variational principle (with one degree of freedom) 423
  • 4.2 Variational principle (with n degrees of freedom) 427.
Description
xiii, 454 p. : ill. ; 26 cm.
Notes
Includes bibliographical references (p. 443-444) and index.
Series Statement
Monographs and textbooks in pure and applied mathematics ; 237
Technical Details
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  • Staff View

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    g| Lecture 0. t| Topology and Differential Calculus Requirements g| 1 -- g| 1. t| Topology g| 1 -- g| 1.1 t| Topological space g| 1 -- g| 1.2 t| Topological space basis g| 2 -- g| 1.3 t| Haussdorff space g| 4 -- g| 1.4 t| Homeomorphism g| 5 -- g| 1.5 t| Connected spaces g| 6 -- g| 1.6 t| Compact spaces g| 6 -- g| 1.7 t| Partition of unity g| 7 -- g| 2. t| Differential calculus in Banach spaces g| 8 -- g| 2.1 t| Banach space g| 8 -- g| 2.2 t| Differential calculus in Banach spaces g| 10 -- g| 2.3 t| Differentiation of R[superscript n] into Banach g| 17 -- g| 2.4 t| Differentiation of R[superscript n] into R[superscript n] g| 19 -- g| 2.5 t| Differentiation of R[superscript n] into R[superscript n] g| 22 -- g| Lecture 1. t| Manifolds g| 37 -- g| 1. t| Differentiable manifolds g| 40 -- g| 1.1 t| Chart and local coordinates g| 40 -- g| 1.2 t| Differentiable manifold structure g| 41 -- g| 1.3 t| Differentiable manifolds g| 43 -- g| 2. t| Differentiable mappings g| 50 -- g| 2.1 t| Generalities on differentiable mappings g| 50 -- g| 2.2 t| Particular differentiable mappings g| 55 -- g| 2.3 t| Pull-back of function g| 57 -- g| 3. t| Submanifolds g| 59 -- g| 3.1 t| Submanifolds of R[superscript n] g| 59 -- g| 3.2 t| Submanifold of manifold g| 64 -- g| Lecture 2. t| Tangent Vector Space g| 71 -- g| 1. t| Tangent vector g| 71 -- g| 1.1 t| Tangent curves g| 71 -- g| 1.2 t| Tangent vector g| 74 -- g| 2. t| Tangent space g| 80 -- g| 2.1 t| Definition of a tangent space g| 80 -- g| 2.2 t| Basis of tangent space g| 81 -- g| 2.3 t| Change of basis g| 82 -- g| 3. t| Differential at a point g| 83 -- g| 3.2 t| Image in local coordinates g| 85 -- g| 3.3 t| Differential of a function g| 86 -- g| Lecture 3. t| Tangent Bundle--Vector Field--One-Parameter Group Lie Algebra g| 91 -- g| 1. t| Tangent bundle g| 93 -- g| 1.1 t| Natural manifold TM g| 93 -- g| 1.2 t| Extension and commutative diagram g| 94 -- g| 2. t| Vector field on manifold g| 96 -- g| 2.2 t| Properties of vector fields g| 96 -- g| 3. t| Lie algebra structure g| 97 -- g| 3.1 t| Bracket g| 97 -- g| 3.2 t| Lie algebra g| 100 -- g| 3.3 t| Lie derivative g| 101 -- g| 4. t| One-parameter group of diffeomorphisms g| 102 -- g| 4.1 t| Differential equations in Banach g| 102 -- g| 4.2 t| One-parameter group of diffeomorphisms g| 104 -- g| Lecture 4. t| Cotangent Bundle--Vector Bundle of Tensors g| 125 -- g| 1. t| Cotangent bundle and covector field g| 125 -- g| 1.1 t| 1-form g| 125 -- g| 1.2 t| Cotangent bundle g| 129 -- g| 1.3 t| Field of covectors g| 130 -- g| 2. t| Tensor algebra g| 130 -- g| 2.1 t| Tensor at a point and tensor algebra g| 130 -- g| 2.2 t| Tensor fields and tensor algebra g| 137 -- g| Lecture 5. t| Exterior Differential Forms g| 153 -- g| 1. t| Exterior form at a point g| 153 -- g| 1.1 t| Definition of a p-form g| 153 -- g| 1.2 t| Exterior product of 1-forms g| 155 -- g| 1.3 t| Expression of a p-form g| 156 -- g| 1.4 t| Exterior product of forms g| 158 -- g| 1.5 t| Exterior algebra g| 159 -- g| 2. t| Differential forms on a manifold g| 162 -- g| 2.1 t| Exterior algebra (Grassmann algebra) g| 162 -- g| 2.2 t| Change of basis g| 165 -- g| 3. t| Pull-back of a differential form g| 167 -- g| 3.1 t| Definition and representation g| 167 -- g| 3.2 t| Pull-back properties g| 168 -- g| 4. t| Exterior differentiation g| 170 -- g| 4.2 t| Exterior differential and pull-back g| 173 -- g| 5. t| Orientable manifolds g| 174 -- g| Lecture 6. t| Lie Derivative--Lie Group g| 185 -- g| 1. t| Lie derivative g| 186 -- g| 1.1 t| First presentation of Lie derivative g| 186 -- g| 1.2 t| Alternative interpretation of Lie derivative g| 195 -- g| 2. t| Inner product and Lie derivative g| 199 -- g| 2.1 t| Definition and properties g| 199 -- g| 2.2 t| Fundamental theorem g| 201 -- g| 3. t| Frobenius theorem g| 204 -- g| 4. t| Exterior differential systems g| 207 -- g| 4.1 t| Generalities g| 207 -- g| 4.2 t| Pfaff systems and Frobenius theorem g| 208 -- g| 5. t| Invariance of tensor fields g| 211 -- g| 5.2 t| Invariance of differential forms g| 212 -- g| 5.3 t| Lie algebra g| 214 -- g| 6. t| Lie group and algebra g| 214 -- g| 6.1 t| Lie group definition g| 215 -- g| 6.2 t| Lie algebra of Lie group g| 215 -- g| 6.3 t| Invariant differential forms on G g| 217 -- g| 6.4 t| One-parameter subgroup of a Lie group g| 218 -- g| Lecture 7. t| Integration of Forms: Stokes' Theorem, Cohomology and Integral Invariants g| 235 -- g| 1. t| n-form integration on n-manifold g| 235 -- g| 1.1 t| Integration definition g| 235 -- g| 1.2 t| Pull-back of a form and integral evaluation g| 237 -- g| 2.1 t| Integral over a chain g| 239 -- g| 2.1 t| Integral over a chain element g| 239 -- g| 2.2 t| Integral over a chain g| 239 -- g| 3. t| Stokes' theorem g| 240 -- g| 3.1 t| Stokes' formula for a closed p-interval g| 240 -- g| 3.2 t| Stokes' formula for a chain g| 242 -- g| 4. t| An introduction to cohomology theory g| 243 -- g| 4.1 t| Closed and exact forms--Cohomology g| 243 -- g| 4.2 t| Poincare lemma g| 244 -- g| 4.3 t| Cycle--Boundary--Homology g| 247 -- g| 5. t| Integral invariants g| 248 -- g| 5.1 t| Absolute integral invariant g| 248 -- g| 5.2 t| Relative integral invariant g| 252 -- g| Lecture 8. t| Riemannian Geometry g| 257 -- g| 1. t| Riemannian manifolds g| 257 -- g| 1.1 t| Metric tensor and manifolds g| 257 -- g| 1.2 t| Canonical isomorphism and conjugate tensor g| 262 -- g| 1.3 t| Orthonormal bases g| 266 -- g| 1.4 t| Hyperbolic manifold and special relativity g| 267 -- g| 1.5 t| Killing vector field g| 274 -- g| 1.6 t| Volume g| 275 -- g| 1.7 t| Hodge operator and adjoint g| 277 -- g| 1.8 t| Special relativity and Maxwell equations g| 280 -- g| 1.9 t| Induced metric and isometry g| 283 -- g| 2. t| Affine connection g| 285 -- g| 2.1 t| Affine connection definition g| 285 -- g| 2.2 t| Christoffel symbols g| 286 -- g| 2.3 t| Interpretation of the covariant derivative g| 288 -- g| 2.4 t| Torsion g| 291 -- g| 2.5 t| Levi-Civita (or Riemannian) connection g| 291 -- g| 2.6 t| Gradient--Divergence--Laplace operators g| 293 -- g| 3. t| Geodesic and Euler equation g| 300 -- g| 4. t| Curvatures--Ricci tensor--Bianchi identity--Einstein equations g| 302 -- g| 4.1 t| Curvature tensor g| 302 -- g| 4.2 t| Ricci tensor g| 305 -- g| 4.3 t| Bianchi identity g| 308 -- g| 4.4 t| Einstein equations g| 309 -- g| Lecture 9. t| Lagrange and Hamilton Mechanics g| 325 -- g| 1. t| Classical mechanics spaces and metric g| 325 -- g| 1.1 t| Generalized coordinates and spaces g| 325 -- g| 1.2 t| Kinetic energy and Riemannian manifold g| 327 -- g| 2. t| Hamilton principle, Motion equations, Phase space g| 329 -- g| 2.1 t| Lagrangian g| 329 -- g| 2.2 t| Principle of least action g| 329 -- g| 2.3 t| Lagrange equations g| 331 -- g| 2.4 t| Canonical equations of Hamilton g| 332 -- g| 2.5 t| Phase space g| 337 -- g| 3. t| D'Alembert-Lagrange principle--Lagrange equations g| 338 -- g| 3.1 t| D'Alembert-Lagrange principle g| 338 -- g| 3.2 t| Lagrange equations g| 340 -- g| 3.3 t| Euler-Noether theorem g| 341 -- g| 3.4 t| Motion equations on Riemannian manifolds g| 343 -- g| 4. t| Canonical transformations and integral invariants g| 344 -- g| 4.1 t| Diffeomorphisms on phase spacetime g| 344 -- g| 4.2 t| Integral invariants g| 346 -- g| 4.3 t| Integral invariants and canonical transformations g| 348 -- g| 4.4 t| Liouville theorem g| 352 -- g| 5. t| N-body problem and a problem of statistical mechanics g| 352 -- g| 5.1 t| N-body problem and fundamental equations g| 353 -- g| 5.2 t| A problem of statistical mechanics g| 358 -- g| 6. t| Isolating integrals g| 369 -- g| 6.1 t| Definition and examples g| 369 -- g| 6.2 t| Jeans theorem g| 372 -- g| 6.3 t| Stellar trajectories in the galaxy g| 373 -- g| 6.4 t| Third integral g| 375 -- g| 6.5 t| Invariant curve and third integral existence g| 379 -- g| Lecture 10. t| Symplectic Geometry--Hamilton--Jacobi Mechanics g| 385 -- g| 1. t| Symplectic geometry g| 388 -- g| 1.1 t| Darboux theorem and symplectic matrix g| 388 -- g| 1.2 t| Canonical isomorphism g| 391 -- g| 1.3 t| Poisson bracket of one-forms g| 393 -- g| 1.4 t| Poisson bracket of functions g| 396 -- g| 1.5 t| Symplectic mapping and canonical transformation g| 399 -- g| 2. t| Canonical transformations in mechanics g| 404 -- g| 2.1 t| Hamilton vector field g| 404 -- g| 2.2 t| Canonical transformations--Lagrange brackets g| 408 -- g| 2.3 t| Generating functions g| 412 -- g| 3. t| Hamilton-Jacobi equation g| 415 -- g| 3.1 t| Hamilton-Jacobi equation and Jacobi theorem g| 415 -- g| 3.2 t| Separability g| 419 -- g| 4. t| A variational principle of analytical mechanics g| 422 -- g| 4.1 t| Variational principle (with one degree of freedom) g| 423 -- g| 4.2 t| Variational principle (with n degrees of freedom) g| 427.
    596
      
      
    a| 8
    650
      
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    a| Geometry, Differential.
    999
      
      
    a| QA641 .T215 2001 w| LC i| X004478926 l| STACKS m| MATH t| BOOK
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