The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid mechanics, electromagnetism, plasma dynamics and control theory are given using both invariant and index notation. The prerequisites required are solid undergraduate courses in linear algebra and advanced calculus.

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.