Manifolds, tensor analysis, and applications

✍ Scribed by Marsden, Ratiu, Abraham.

• Publisher: Springer
• Year: 2002
• Tongue: English
• Leaves: 617
• Series: Global analysis, pure and applied
• Edition: 3ed, draft
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
......Page 1
Title......Page 2
Copyright
......Page 3
Contents
......Page 4
Preface......Page 6
Topological Spaces......Page 8
Metric Spaces......Page 15
Continuity......Page 19
Subspaces, Products, and Quotients......Page 22
Compactness......Page 27
Connectedness......Page 33
Baire Spaces......Page 38
Banach Spaces......Page 41
Linear and Multilinear Mappings......Page 56
The Derivative......Page 73
Properties of the Derivative......Page 79
The Inverse and Implicit Function Theorems......Page 108
Manifolds......Page 133
Submanifolds, Products, and Mappings......Page 141
The Tangent Bundle......Page 147
Vector Bundles......Page 156
Submersions, Immersions, and Transversality......Page 180
The Sard and Smale Theorems......Page 200
Vector Fields and Flows......Page 216
Vector Fields as Differential Operators......Page 237
An Introduction to Dynamical Systems......Page 264
Frobenius' Theorem and Foliations......Page 287
Basic Definitions and Properties
......Page 298
Some Classical Lie Groups
......Page 310
Actions of Lie Groups
......Page 329
Tensors on Linear Spaces......Page 343
Tensor Bundles and Tensor Fields......Page 352
The Lie Derivative: Algebraic Approach......Page 360
The Lie Derivative: Dynamic Approach......Page 369
Partitions of Unity......Page 375
Exterior Algebra......Page 388
Determinants, Volumes, and the Hodge Star Operator......Page 396
Differential Forms......Page 408
The Exterior Derivative, Interior Product, & Lie Derivative......Page 413
Orientation, Volume Elements and the Codifferential......Page 438
The Definition of the Integral......Page 452
Stokes' Theorem......Page 463
The Classical Theorems of Green, Gauss, and Stokes......Page 487
Induced Flows on Function Spaces and Ergodicity......Page 495
Introduction to Hodge--deRham Theory......Page 515
Hamiltonian Mechanics......Page 535
Fluid Mechanics......Page 555
Electromagnetism......Page 567
The Lie--Poisson Bracket in Continuum Mechanics and Plasmas......Page 575
Constraints and Control......Page 588
References
......Page 595