Author | Abraham, Ralph. author |
---|---|
Title | Manifolds, Tensor Analysis, and Applications [electronic resource] / by Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1988 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1029-0 |
Descript | XI, 656 p. online resource |
1 Topology -- 1.1 Topological Spaces -- 1.2 Metric Spaces -- 1.3 Continuity -- 1.4 Subspaces, Products, and Quotients -- 1.5 Compactness -- 1.6 Connectedness -- 1.7 Baire Spaces -- 2 Banach Spaces and Differential Calculus -- 2.1 Banach Spaces -- 2.2 Linear and Multilinear Mappings -- 2.3 The Derivative -- 2.4 Properties of the Derivative -- 2.5 The Inverse and Implicit Function Theorems -- 3 Manifolds and Vector Bundles -- 3.1 Manifolds -- 3.2 Submanifolds, Products, and Mappings -- 3.3 The Tangent Bundle -- 3.4 Vector Bundles -- 3.5 Submersions, Immersions and Transversality -- 4 Vector Fields and Dynamical Systems -- 4.1 Vector Fields and Flows -- 4.2 Vector Fields as Differential Operators -- 4.3 An Introduction to Dynamical Systems -- 4.4 Frobeniusโ Theorem and Foliations -- 5 Tensors -- 5.1 Tensors in Linear Spaces -- 5.2 Tensor Bundles and Tensor Fields -- 5.3 The Lie Derivative: Algebraic Approach -- 5.4 The Lie Derivative: Dynamic Approach -- 5.5 Partitions of Unity -- 6 Differential Forms -- 6. I Exterior Algebra -- 6.2 Determinants, Volumes, and the Hodge Star Operator -- 6.3 Differential Forms -- 6.4 The Exterior Derivative, Interior Product, and Lie Derivative -- 6.5 Orientation, Volume Elements, and the Codifferential -- 7 Integration on Manifolds -- 7.1 The Definition of the Integral -- 7.2 Stokesโ Theorem -- 7.3 The Classical Theorems of Green, Gauss, and Stokes -- 7.4 Induced Flows on Function Spaces and Ergodicity -- 7.5 Introduction to Hodge-deRham Theory and Topological Applications of Differential Forms -- 8 Applications -- 8.1 Hamiltonian Mechanics -- 8.2 Fluid Mechanics -- 8.3 Electromagnetism -- 8.3 The Lie-Poisson Bracket in Continuum Mechanics and Plasma Physics -- 8.4 Constraints and Control -- References