Manifolds, Tensor Analysis, and Applications

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 me­ chanics, electromagnetism, plasma dynamics and control thcory arc given in Chapter 8, using both invariant and index notation. The current edition of the book does not deal with Riemannian geometry in much detail, and it does not treat Lie groups, principal bundles, or Morse theory. Some of this is planned for a subsequent edition. Meanwhile, the authors will make available to interested readers supplementary chapters on Lie Groups and Differential Topology and invite comments on the book's contents and development. Throughout the text supplementary topics are given, marked with the symbols ~ and {l:;J. This device enables the reader to skip various topics without disturbing the main flow of the text. Some of these provide additional background material intended for completeness, to minimize the necessity of consulting too many outside references. We treat finite and infinite-dimensional manifolds simultaneously. This is partly for efficiency of exposition. Without advanced applications, using manifolds of mappings, the study of infinite-dimensional manifolds can be hard to motivate.

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.