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Manifolds, tensor analysis, and applications

Author: Ralph Abraham; Jerrold E Marsden; Tudor S Rațiu
Publisher: New York : Springer-Verlag, ©1988.
Series: Applied mathematical sciences (Springer-Verlag New York Inc.), v. 75.
Edition/Format:   Book : English : 2nd edView all editions and formats
Database:WorldCat
Summary:

This book presents core material in nonlinear analysis, offering working knowledge of manifolds, dynamical systems, tensors and differential forms. Coverage includes Hamiltonian mechanics, fluid  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Ralph Abraham; Jerrold E Marsden; Tudor S Rațiu
ISBN: 0387967907 9780387967905 3540967907 9783540967903 9781461210290 1461210291 9781461269908 1461269903
OCLC Number: 18562688
Description: 654 pages : illustrations ; 25 cm.
Contents: 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.
Series Title: Applied mathematical sciences (Springer-Verlag New York Inc.), v. 75.
Responsibility: R. Abraham, J.E. Marsden, T. Ratiu.
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