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 Frobenius' 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 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. - 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. Language: English