Preface
Background Notation
CHAPTER 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
CHAPTER 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
CHAPTER 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
CHAPTER 4 Vector Fields and Dynamical Systems
4.1 Vector Fields and Rows
4.2 Vector Fields as Differential Operators
4.3 An Introduction to Dynamical Systems
4.4 Frobenius' Theorem and Foliations
CHAPTER 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
CHAPTER 6 Differential Forms
6.1 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
CHAPTER 7 Integration on Manifolds
7.1 The Definition of the Integral
7.2 Stokes 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
CHAPTER 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
Index
Supplementary Chapters--Available from the authors as they are produced
S-1 Lie Groups
S-2 Introduction to Differential Topology
S-3 Topics in Riemannian Geometry
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