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## Summary

## Table of Contents

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 |

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