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The Geometry of Physics: An Introduction
 
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The Geometry of Physics: An Introduction (Paperback)
by Theodore Frankel (Author)
5.0 out of 5 stars 1 customer review (1 customer review)

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Book Description
This book is intended to provide a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism, thermodynamics, the deformation tensors of elasticity, soap films, special and general relativity, the Dirac operator and spinors, and gauge fields, including Yang-Mills, the Aharonov-Bohm effect, Berry phase, and instanton winding numbers. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space; consequently, the book should also be of interest to mathematics students. This book will be useful to graduate and advanced undergraduate students of physics, engineering and mathematics. It can be used as a course text or for self study.

Synopsis
This book is intended to provide a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism, thermodynamics, the deformation tensors of elasticity, soap films, special and general relativity, the Dirac operator and spinors, and gauge fields, including Yang-Mills, the Aharonov-Bohm effect, Berry phase, and instanton winding numbers. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space; consequently, the book should also be of interest to mathematics students. This book will be useful to graduate and advanced undergraduate students of physics, engineering and mathematics. It can be used as a course text or for self study.
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45 of 46 people found the following review helpful:
5.0 out of 5 stars This book could well be on its way to becoming a classic., 17 April 2001
By A Customer
The machinery of Differential Geometry is important because it allows us to rewrite equations in a more general form. The aim is understand what the equations are saying, rather than how they are saying it. The Maxwell equations describing Electromagnetism, for example, have a particular form. How much of the content of these equations relates to the electromagnetic fields, how much to the background space in which those fields live and how much to the arbitrary way we choose to label points in that space? The answer to this question is not obvious from the form of the equations we are taught at university, but becomes manifest when expressed in the language of differential geometry.

Seeing your favourite equations expressed in this strange language can take a little getting used to. Frankel's book gently takes the reader through the basics of reinterpreting physics through geometry. Having set down firm foundations, the book then explains the more esoteric aspects of differential geometry, but again showing how these apparent complications further allow one to abstract the meaning of the physical theory from the mathematical grit.

There are lots of exercises that provide you with the chance to try out the new techniques for yourself. I would strongly encourage people to at least attempt the questions. Physics is an active pastime, not a passive one. The book doesn't include answers to the questions, but it is usually obvious whether you have managed to get the correct answer, or not. There are also plenty of worked examples in the text (although they aren't flagged as such) to help you on your way. I also found that some of the main questions were referenced in later chapter in a way that provided the answer (e.g. In Question 2.3.2 we saw that it was possible to write this in the form ...).

The book covers the use of differential geometry in a wide range of physical theories: Electromagnetism; Special & General Relativity; Field Theory; Thermodynamics; Classical Dynamics. The last sections of the book deal with expressing Yang-Mills theories using Fibre Bundles, so making explicit the relationship between the General Relativity and the other three forces of Nature.

More than once while reading this book I have had a real 'Oh Wow!' moment (I have never really stopped reading the book). In one sense the extra machinery seems to say no more than the old mathematics (and Frankel shows the reader how to recover the familiar from the new more elegant formulation), and to only add complexity. But the new language makes clear the deep connections between the different branches of physics and mathematics. It really is beautiful.

Reading this book is just a beginning. But it's a fine beginning. This book will prepare you to explore the more advanced works, some of which are also written up in text books; some of which form the cutting edge of understanding theories such as String Theory (and are available on Preprint Servers online).

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