数学物理中的整体分析（英文版）
出版时间:
2003-06
版次:
1
ISBN:
9787506259484
定价:
38.00
装帧:
平装
开本:
其他
纸张:
胶版纸
页数:
213页
10人买过
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The first edition of this book entitled Analysis on Riemannian Manifolds and Some Problems of Mathematical Physics was published by Voronezh University Press in 1989. For its English edition, the book has been substantially revised and expanded. In particular, new material has been added to Sections 19 and 20. I am grateful to Viktor L. Ginzburg for his hard work on the translation and for writing Appendix F, and to Tomasz Zastawniak for his numerous suggestions. My special thanks go to the referee for his valuable remarks on the theory of stochastic processes. Finally, I would like to acknowledge the support of the AMS fSU Aid Fund and the International Science Foundation (Grant NZB000), which made possible my work on some of the new results included in the English edition of the book. Preface to the English Edition
Preface to the Russian Edition
Part I. Finite-Dimensional Differential Geometry and Mechanics
Chapter 1 Some Geometric Constructions in Calculus on Manifolds
1. Complete Riemannian Metrics and the Completeness of Vector Fields
2. Riemannian Manifolds
3. Integral Operators with Parallel Translation
Chapter 2 Geometric Formalism of Newtonian Mechanics
4. Geometric Mechanics: Introduction and Review of Standard Examples
5. Geometric Mechanics with Linear Constraints
6. Mechanical Systems with Discontinuous Forces cand Systems with Control Differential Inclusions
7. Integral Equations of Geometric Mechanics: The Velocity Hodograph
8. Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Force
Chapter 3 Accessible Points of Mechanical Systems
9. Examples of Points that Cannot Be Connected by a Trajectory
10. The Main Result on Accessible Points
11. Generalizations to Systems with Constraints
Part II. Stochastic Differential Geometry and its Applications to Physics
Chapter 4 Stochastic Differential Equations on Riemannian Manifolds
12. Review of the Theory of Stochastic Equations and Integrals on Finite-Dimensional Linear Spaces
13. Stochastic Differential Equations on Manifolds
14. Stochastic Parallel Translation and the Integral Formalism for the It5 Equations
15. Wiener Processes on Riemannian Manifolds and Related Stochastic Differential Equations
16. Stochastic Differential Equations with Constraints
Chapter 5 The Langevin Equation
17. The Langevin Equation of Geometric Mechanics
18. Strong Solutions of the Langevin Equation, Ornstein-Uhlenbeck Processes
Chapter 6 Mean Derivatives, Nelson's Stochastic Mechanics, and Quantization
19. More on Stochastic Equations and Stochastic Mechanics in IRn
20. Mean Derivatives and Stochastic Mechanics on Riemannian Manifolds
21. Relativistic Stochastic Mechanics
Part III. Infinite-Dimensional Differential Geometry and Hydrodynamics
Chapter 7 Geometry of Manifolds of Diffeomorphisms
22. Manifolds of Mappings and Groups of Diffeomorphisms
23. Weak Riemannian Metrics and Connections on Manifolds of Diffeomorphisms
24. Lagrangian Formalism of Hydrodynamics of an Ideal Barotropic Fluid
Chapter 8 Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid
25. Geometry of the Manifold of Volume-Preserving Diffeomorphisms and LHSs of an Ideal Incompressible Fluid
26. The Flow of an Ideal Incompressible Fluid on a Manifold with Boundary as an LHS with an Infinite-Dimensional Constraint on the Group of Diffeomorphisms of a Closed Manifold
27. The Regularity Theorem and a Review of Results on the Existen e of Solutions
Chapter 9 Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms
28. Stochastic Differential Geometry on the Groups of Diffeomorphisms of the n-Dimensional Torus
29. A Viscous Incompressible Fluid
References
Index
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内容简介:
The first edition of this book entitled Analysis on Riemannian Manifolds and Some Problems of Mathematical Physics was published by Voronezh University Press in 1989. For its English edition, the book has been substantially revised and expanded. In particular, new material has been added to Sections 19 and 20. I am grateful to Viktor L. Ginzburg for his hard work on the translation and for writing Appendix F, and to Tomasz Zastawniak for his numerous suggestions. My special thanks go to the referee for his valuable remarks on the theory of stochastic processes. Finally, I would like to acknowledge the support of the AMS fSU Aid Fund and the International Science Foundation (Grant NZB000), which made possible my work on some of the new results included in the English edition of the book.
-
目录:
Preface to the English Edition
Preface to the Russian Edition
Part I. Finite-Dimensional Differential Geometry and Mechanics
Chapter 1 Some Geometric Constructions in Calculus on Manifolds
1. Complete Riemannian Metrics and the Completeness of Vector Fields
2. Riemannian Manifolds
3. Integral Operators with Parallel Translation
Chapter 2 Geometric Formalism of Newtonian Mechanics
4. Geometric Mechanics: Introduction and Review of Standard Examples
5. Geometric Mechanics with Linear Constraints
6. Mechanical Systems with Discontinuous Forces cand Systems with Control Differential Inclusions
7. Integral Equations of Geometric Mechanics: The Velocity Hodograph
8. Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Force
Chapter 3 Accessible Points of Mechanical Systems
9. Examples of Points that Cannot Be Connected by a Trajectory
10. The Main Result on Accessible Points
11. Generalizations to Systems with Constraints
Part II. Stochastic Differential Geometry and its Applications to Physics
Chapter 4 Stochastic Differential Equations on Riemannian Manifolds
12. Review of the Theory of Stochastic Equations and Integrals on Finite-Dimensional Linear Spaces
13. Stochastic Differential Equations on Manifolds
14. Stochastic Parallel Translation and the Integral Formalism for the It5 Equations
15. Wiener Processes on Riemannian Manifolds and Related Stochastic Differential Equations
16. Stochastic Differential Equations with Constraints
Chapter 5 The Langevin Equation
17. The Langevin Equation of Geometric Mechanics
18. Strong Solutions of the Langevin Equation, Ornstein-Uhlenbeck Processes
Chapter 6 Mean Derivatives, Nelson's Stochastic Mechanics, and Quantization
19. More on Stochastic Equations and Stochastic Mechanics in IRn
20. Mean Derivatives and Stochastic Mechanics on Riemannian Manifolds
21. Relativistic Stochastic Mechanics
Part III. Infinite-Dimensional Differential Geometry and Hydrodynamics
Chapter 7 Geometry of Manifolds of Diffeomorphisms
22. Manifolds of Mappings and Groups of Diffeomorphisms
23. Weak Riemannian Metrics and Connections on Manifolds of Diffeomorphisms
24. Lagrangian Formalism of Hydrodynamics of an Ideal Barotropic Fluid
Chapter 8 Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid
25. Geometry of the Manifold of Volume-Preserving Diffeomorphisms and LHSs of an Ideal Incompressible Fluid
26. The Flow of an Ideal Incompressible Fluid on a Manifold with Boundary as an LHS with an Infinite-Dimensional Constraint on the Group of Diffeomorphisms of a Closed Manifold
27. The Regularity Theorem and a Review of Results on the Existen e of Solutions
Chapter 9 Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms
28. Stochastic Differential Geometry on the Groups of Diffeomorphisms of the n-Dimensional Torus
29. A Viscous Incompressible Fluid
References
Index
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