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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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Topology

Abstract
The purpose of this chapter is to introduce just enough topology for later requirements. It is assumed that the reader has had a course in advanced calculus and so is acquainted with open, closed, compact, and connected sets in Euclidean space (see for example Marsden [1974a] and Rudin [1976]). If this background is weak, the reader may find the pace of this chapter too fast. If the background is under control, the chapter should serve to collect, review, and solidify concepts in a more general context. Readers already familiar with point set topology can safely skip this chapter.
Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

Chapter 2. Banach Spaces and Differential Calculus

Abstract
Manifolds have enough structure to allow differentiation of maps between them. To set the stage for these concepts requires a development of differential calculus in linear spaces from a geometric point of view. The goal of this chapter is to provide this perspective.
Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

Chapter 3. Manifolds and Vector Bundles

Abstract
We are now ready to study manifolds and the differential calculus of maps between manifolds. Manifolds are an abstraction of the idea of a smooth surface in Euclidean space. This abstraction has proved useful because many sets that are smooth in some sense are not presented to us as subsets of Euclidean space. The abstraction strips away the containing space and makes constructions intrinsic to the manifold itself. This point of view is well worth the geometric insight it provides.
Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

Chapter 4. Vector Fields and Dynamical Systems

Abstract
This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters will study the related topics of tensors and differential forms. A basic operation introduced in this chapter is the Lie derivative of a function or a vector field. It is introduced in two different ways, algebraically as a type of directional derivative and dynamically as a rate of change along a flow. The Lie derivative formula asserts the equivalence of these two definitions. The Lie derivative is a basic operation used extensively in differential geometry, general relativity, Hamiltonian mechanics, and continuum mechanics.
Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

Chapter 5. Tensors

Abstract
In the previous chapter we studied vector fields and functions on manifolds. Now these objects are generalized to tensor fields, which are sections of vector bundles built out of the tangent bundle. This study is continued in the next chapter when we discuss differential forms, which are tensors with special symmetry properties. One of the objectives of this chapter is to extend the pull-back and Lie derivative operations from functions and vector fields to tensor fields.
Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

Chapter 6. Differential Forms

Abstract
Differential k-forms are tensor fields of type (0, k) that are completely antisymmetric. Such tensor fields arise in many applications in physics, engineering, and mathematics. A hint at why this is so is the fact that the classical operations of grad, div, and curl and the theorems of Green, Gauss, and Stokes can all be expressed concisely in terms of differential forms. However, the examples of Hamiltonian mechanics and Maxwell’s equations (see Chapter 8) show that their applicability goes well beyond this.
Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

Chapter 7. Integration on Manifolds

Abstract
The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ℝn using a partition of unity subordinate to an atlas. The change-of-variables theorem guarantees that the integral is well defined, independent of the choice of atlas and partition of unity. Two basic theorems of integral calculus, the change-of-variables theorem and Stokes’ theorem, are discussed in detail along with some applications.
Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

Chapter 8. Applications

Abstract
This chapter presents some applications of manifold theory and tensor analysis to physics and engineering. Our selection is a of limited scope and depth, with the intention of providing an introduction to the techniques. There are many other applications of the ideas of this book as well. We list below a few selected references for further reading in the same spirt.
Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

Backmatter

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