动态系统与流体流的几何理论GEOMETRICAL THEORY OF DYNAMICAL SYSTEMS AND FLUID FLOWS

This is an introductory textbook on the geometrical theory of dynamical systems, fluid flows, and certain integrable systems. The subjects are interdisciplinary and extend from mathematics, mechanics and physics to mechanical engineering, and the approach is very fundamental. The underlying concepts are based on differential geometry and theory of Lie groups in the mathematical aspect, and on transformation symmetries and gauge theory in the physical aspect. A great deal of effort has been directed toward making the description elementary, clear and concise, so that beginners will have an access to the topics.

Preface
Ⅰ Mathematical Bases
　1. Manifolds, Flows, Lie Groups and Lie Algebras
　　1.1 Dynamical Systems
1.2 Manifolds and Diffeomorphisms
1.3 Flows and Vector Fields
　1.3.1 A steady flow and its velocity field
1.3.2 Tangent vector and differential operator
1.3.3 Tangent space
1.3.4 Time-dependent (unsteady) velocity field
　　1.4 Dynamical Trajectory
1.4.1 Fiber bundle (tangent bundle)
1.4.2 Lagrangian and Halniltonian
1.4.3 Legendre transformation
1.5 Differential and Inner Product
1.5.1 Covector (1-form)
1.5.2 Inner (scalar) product
1.6 Mapping of Vectors and Covectors
1.6.1 Push-forward transformation
1.6.2 Pull-back transformation
1.6.3 Coordinate transformation
1.7 Lie Group and Invariant Vector Fields
1.8 Lie Algebra and Lie Derivative
1.8.1 Lie algebra, adjoint operator and Lie bracket
1.8.2 An example of the rotation group SO(3)
1.8.3 Lie derivative and Lagrange derivative
1.9 Diffeomorphisms of a Circle S1
1.10 Transformation of Tensors and Invariance
1.10.1 Transformations of vectors and metric tensors . . .
1.10.2 Covariant tensors
1.10.3 Mixed tensors
1.10.4 Contravariant tensors
　2. Geometry of Surfaces in R3
2.1 First Fundamental Form
2.2 Second Fundamental Form
2.3 Gauss's Surface Equation and an Induced Connection
2.4 Gauss Mainardi Codazzi Equation and Integrability . .
2.5 Gaussian Curvature of a Surface
2.5.1 Riemann tensors
2.5.2 Gaussian curvature
2.5.3 Geodesic curvature and normal curvature
2.5.4 Principal curvatures
2.6 Geodesic Equation
2.7 Structure Equations in Differential Forms
2.7.1 Smooth surfaces in IRa and integrability
2.7.2 Structure equations
2.7.3 Geodesic equation
2.8 Gauss Spherical Map
2.9 Gauss Bonnet Theorem I
2.10 Gauss Bonnet Theorem II
2.11 Uniqueness: First and Second Fundamental Tensors
3. Riemannian Geometry
3.1 Tangent Space
3.1.1 Tangent vectors and inner product
3.1.2 Riemannian metric
3.1.3 Examples of metric tensor
　……
Ⅱ Dynamical Systems
Ⅲ　Flows of Ideal Fluids
Ⅳ　Geometry of Integrable Systems
Appendix A Topological Space and Mappings
Appendix B Exterior Forms, Products and Differentials
Appendix C Lie Groups and Rotation Groups
Appendix D A Curve and a Surface in R3
Appendix E Curvature Transformation
Appendix F Function Spaces Lp, Hs and Orthogonal Decomposition
Appendix G Derivation of KdV Equation of a Shallow Water Wave
Appendix H Two-Cocycle, Central Extension and Bott Cocycle
Appendix I Additional Comment on the Gauge THeory of 7.3
Appendix J Frobenius Integration Theorem and Pfaffian System
Appendix K Orthogonal Coordinate Net and Lines of Curvature
References
Index