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Differential Geometry, Analysis and Physics

By Lee, Jeffrey M.

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Book Id: WPLBN0000659790
Format Type: PDF eBook
File Size: 9.43 MB
Reproduction Date: 2005
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Title: Differential Geometry, Analysis and Physics  
Author: Lee, Jeffrey M.
Volume:
Language: English
Subject: Science., Mathematics, Logic
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Lee, J. M. (n.d.). Differential Geometry, Analysis and Physics. Retrieved from http://www.netlibrary.net/


Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: In this book I present differential geometry and related mathematical topics with the help of examples from physics. It is well known that there is something strikingly mathematical about the physical universe as it is conceived of in the physical sciences. The convergence of physics with mathematics, especially differential geometry, topology and global analysis is even more pronounced in the newer quantum theories such as gauge field theory and string theory. The amount of mathematical sophistication required for a good understanding of modern physics is astounding. On the other hand, the philosophy of this book is that mathematics itself is illuminated by physics and physical thinking.

Table of Contents
Contents 0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Preliminaries and Local Theory 1 1.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Chain Rule, Product rule and Taylor?s Theorem . . . . . . . . . 11 1.3 Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Differentiable Manifolds 15 2.1 Rough Ideas I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Differentiable Manifolds and Differentiable Maps . . . . . . . . . 17 2.4 Pseudo-Groups and Models Spaces . . . . . . . . . . . . . . . . . 22 2.5 Smooth Maps and Diffeomorphisms . . . . . . . . . . . . . . . . 27 2.6 Coverings and Discrete groups . . . . . . . . . . . . . . . . . . . 30 2.6.1 Covering spaces and the fundamental group . . . . . . . . 30 2.6.2 Discrete Group Actions . . . . . . . . . . . . . . . . . . . 36 2.7 Grassmannian manifolds . . . . . . . . . . . . . . . . . . . . . . . 39 2.8 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.9 Manifolds with boundary. . . . . . . . . . . . . . . . . . . . . . . 43 3 The Tangent Structure 47 3.1 Rough Ideas II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 The Tangent Map . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 The Tangent and Cotangent Bundles . . . . . . . . . . . . . . . . 55 3.5.1 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5.2 The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . 57 3.6 Important Special Situations. . . . . . . . . . . . . . . . . . . . . 59 4 Submanifold, Immersion and Submersion. 63 4.1 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Submanifolds of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Regular and Critical Points and Values . . . . . . . . . . . . . . . 66 4.4 Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

 

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