Manifolds, Tensor Analysis, and Applications, Volume 75

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Springer Science & Business Media, 1988 - Language Arts & Disciplines - 654 pages
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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.
  

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Contents

Topology
1
11 Topological Spaces
2
12 Metric Spaces
9
13 Continuity
14
14 Subspaces Products and Quotients
18
15 Compactness
24
16 Connectedness
31
17 Baire Spaces
37
52 Tensor Bundles and Tensor Fields
349
Algebraic Approach
359
Dynamic Approach
370
55 Partitions of Unity
377
Differential Forms
392
62 Determinants Volumes and the Hodge Star Operator
402
63 Differential Forms
417
64 The Exterior Derivative Interior Product and Lie Derivative
423

BANACH SPACES AND DIFFERENTIAL CALCULUS
40
22 Linear and Multilinear Mappings
56
23 The Derivative
75
24 Properties of the Derivative
83
25 The Inverse and Implicit Function Theorems
116
MANIFOLDS AND VECTOR BUNDLES
141
32 Submanifolds Products and Mappings
150
33 The Tangent Bundle
157
34 Vector Bundles
167
35 Submersions Immersions and Transversality
196
Vector Fields and Dynamical Systems
238
42 Vector Fields as Differential Operators
265
43 An Introduction to Dynamical Systems
298
44 Frobenius Theorem and Foliations
326
Tensors
338
65 Orientation Volume Elements and the Codifferential
449
Integration on Manifolds
464
72 StokesTheorem
476
73 The Classical Theorems of Green Gauss and Stokes
504
74 Induced Flows on Function Spaces and Ergodicity
513
75 Introduction to HodgeDeRham Theory and Topological Applications of Differential Forms
538
Applications
560
82 Fluid Mechanics
584
83 Electromagnetism
599
84 The LiePoisson Bracket in Continuum Mechanics and Plasma Physics
609
85 Constraints and Control
624
References
631
Index
643
Copyright

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About the author (1988)

Ralph Abraham is the author of Dynamics: The Geometry of Behavior. He lives in California.

Marsden, California Institute of Technology, Pasadena.

Ratiu, University of California, Santa Cruz.