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Theodore Frankel

The geometry of physics : an introduction

Modern physics is taking more and more of a geometrical viewpoint - particle physics is full of terms like SU(2) and SO(3). Unfortunately, when students get to the point of needing to study such things in detail they are often 'thrown in at the deep end' - many books devote just a short space to the mathematics, so that they can get on to the physics more quickly. This means that students may struggle, or worse, end up with just a superficial idea of the subject. In 'The geometry of physics' Theodore Frankel goes for a more gentle approach. Rather than writing for graduate students, the book is aimed at undergraduates. It is steadily paced, and has plenty of diagrams, so that it can be worked through by the student, including those studying on their own.

The book is in three sections. The first part gives an introduction to differential geometry, looking at the properties of surfaces in three dimenstions. It then reformulates some of classical physics, such as electromagnetism, in the language of exterior differential forms. The second part goes further into tensor calculus, and demonstrates the application of this to General Relativity. The third part looks at quantum theory and particle physics, in particular the use of Lie groups in describing symmetries. It then shows how this has been used in the development of Yang-Mills theory.

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Paperback 720 pages
ISBN: 0521539277
Salesrank: 326030
Weight:2.65 lbs

Published: 2003 Cambridge University Press

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Amazon.co.uk info

Paperback 720 pages
ISBN: 0521539277
Salesrank: 374547
Weight:2.65 lbs

Published: 2003 Cambridge University Press

Amazon price £35.15

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Paperback 720 pages
ISBN: 0521539277
Salesrank: 100584
Weight:2.65 lbs

Published: 2003 Cambridge University Press

Amazon price CDN$ 36.51

Marketplace:New from CDN$ 36.51:Used from CDN$ 77.85

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Product Description
Theodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms essential to a better understanding of classical and modern physics and engineering. Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact Lie groups. Geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. First Edition Hb (1997): 0-521-38334-X First Edition Pb (1999): 0-521-38753-1

Product Description

Theodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms essential to a better understanding of classical and modern physics and engineering. Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact Lie groups. Geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. First Edition Hb (1997): 0-521-38334-X First Edition Pb (1999): 0-521-38753-1

Fantastic - for the scientist *****
A very good book: buy it. But only if you are a scientist or student of physics/mathematics. This is not popular-science-common-public level.

Fantastic - for the scientist *****

A very good book: buy it. But only if you are a scientist or student of physics/mathematics. This is not popular-science-common-public level.

a book worth keeping *****
This book can be quite confusing if you start without any background on the idea of manifold or knows nothing about general relativity. However, it does have strong points: 1. The notation is very up-to-date, and is entirely coordinate-independant approach. 2. The author explains in great details of formulation of modern differential geometry, and the details are comparatively lacking in other reference books. 3. The author never hesitate to use graphs and diagrams to illustrate points, and stroke nice balance in between mathematics rigor and physical insight. Although it appears quite verbose at some point, it is mainly because differential geometry is such a heavy subject. Another book nice to have as companion reading is Goldburg's "Tensor analysis on Manifold", a terse, well-written text book.

a book worth keeping *****

This book can be quite confusing if you start without any background on the idea of manifold or knows nothing about general relativity. However, it does have strong points:

1. The notation is very up-to-date, and is entirely coordinate-independant approach.

2. The author explains in great details of formulation of modern differential geometry, and the details are comparatively lacking in other reference books.

3. The author never hesitate to use graphs and diagrams to illustrate points, and stroke nice balance in between mathematics rigor and physical insight.

Although it appears quite verbose at some point, it is mainly because differential geometry is such a heavy subject. Another book nice to have as companion reading is Goldburg's "Tensor analysis on Manifold", a terse, well-written text book.

Phenomenal *****
I just finished reading this book and I found it phenomenal. The physical ideas are made very clear in a natural mathematical framework.

Phenomenal *****

I just finished reading this book and I found it phenomenal. The physical ideas are made very clear in a natural mathematical framework.

You should buy this, despite its flaws *****
The other reviews on this page give this book anywhere from 1 to 5 stars, and they are all correct in their own way. The book is inspired, deep and full of physics applications and insights. On the other hand, it skims over mathematical rigor to a large degree and focuses more on defining things, getting a feel for them and moving on to application. My advice: buy the book for its strengths, and read other books in parallel if you need more rigor. But still, buy it. Also, things can be confusing on the first two or three reads, but keep at it and you will be glad you did.

You should buy this, despite its flaws *****

The other reviews on this page give this book anywhere from 1 to 5 stars, and they are all correct in their own way. The book is inspired, deep and full of physics applications and insights. On the other hand, it skims over mathematical rigor to a large degree and focuses more on defining things, getting a feel for them and moving on to application.

My advice: buy the book for its strengths, and read other books in parallel if you need more rigor. But still, buy it.

Also, things can be confusing on the first two or three reads, but keep at it and you will be glad you did.

The perfect first book in differential geometry *****
Differential geometry can be a very intimidating subject due to its heavy formalism. There are complete books (such as Kobayashi& Nomizu) very good as reference books, and there very few books that show the reader the picture behind the formulas. This is one such book. It tells you the intuition behind each construction and from this point of view it has many things in common with Arnold's famous book on Math. Methods in Classical Mechanics. But where as Arnold does not pay too much attention to formalism, this book achieves this task as well. It shows the reader how to do those impossible computations as well. This is definitely the first place to look at if you want to really learn differential geometry. If it seems difficult it is only because the subject is so.

The perfect first book in differential geometry *****

Differential geometry can be a very intimidating subject due to its heavy formalism. There are complete books (such as Kobayashi& Nomizu) very good as reference books, and there very few books that show the reader the picture behind the formulas.

This is one such book. It tells you the intuition behind each construction and from this point of view it has many things in common with Arnold's famous book on Math. Methods in Classical Mechanics. But where as Arnold does not pay too much attention to formalism, this book achieves this task as well. It shows the reader how to do those impossible computations as well.

This is definitely the first place to look at if you want to really learn differential geometry. If it seems difficult it is only because the subject is so.

This book could well be on its way to becoming a classic. *****
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).

This book could well be on its way to becoming a classic. *****

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).

Bad book. **
Frankel's book is provbably the most confusing book I have ever looked into. As other readers noted, it is probably because of his approach not to define things properly. The book's style is extremely wordy, unnecessary wordy that is. The result - total confusion. Mr. Frankel probably thinks the readers are nearly morons, so he tries to re-express some (really simple) notions with words that supposedly will make things lucid. Well, he fails. Alternative book by Nakahara is way better.I also recommend "Analysis, Manifolds and Physics" by Yvonne Cgiqyet-Bruhat, et al 2 stars for effort.

Bad book. **

Frankel's book is provbably the most confusing book I have ever looked into. As other readers noted, it is probably because of his approach not to define things properly. The book's style is extremely wordy, unnecessary wordy that is. The result - total confusion. Mr. Frankel probably thinks the readers are nearly morons, so he tries to re-express some (really simple) notions with words that supposedly will make things lucid. Well, he fails.
Alternative book by Nakahara is way better.I also recommend "Analysis, Manifolds and Physics" by Yvonne Cgiqyet-Bruhat, et al
2 stars for effort.

Good one, even if not the best, probably ****
This is a valuable reference for students pursuing a support or who want to get themselves deeper in the mathemathical part connected with QFT and GR. I particularly appreciated the first chapter about Manifolds and vector fields, the part about algebraic topology (chapter 13: chains, homology groups and De Rahm's theorem, Betti numbers) and the part about homotopy groups. On the other hand the first part about tensors, exterior forms, integration of differential forms and the Lie derivative seems to me a bit uneven compared to the one I've mentioned above. For this section I'd recommend: Aldrovandi - Pereira, "Introduction to geometrical Physics", or V.I. Arnold, "Classical Mechanics" (first part) which is not complete if compared to the other two books (this is a book about the symplectic formulation of CM and not strictly a matemathical book) but things that are contained are exposed in a beautiful way. Another valuable book is Nakahara (a classic one), but I still have to finish reading it so I'll leave a comment about it in the next. The level of T. Frankel is at last yr undergrad - 1st yr graduate.

Good one, even if not the best, probably ****

This is a valuable reference for students pursuing a support or who want to get themselves deeper in the mathemathical part connected with QFT and GR. I particularly appreciated the first chapter about Manifolds and vector fields, the part about algebraic topology (chapter 13: chains, homology groups and De Rahm's theorem, Betti numbers) and the part about homotopy groups. On the other hand the first part about tensors, exterior forms, integration of differential forms and the Lie derivative seems to me a bit uneven compared to the one I've mentioned above. For this section I'd recommend: Aldrovandi - Pereira, "Introduction to geometrical Physics", or V.I. Arnold, "Classical Mechanics" (first part) which is not complete if compared to the other two books (this is a book about the symplectic formulation of CM and not strictly a matemathical book) but things that are contained are exposed in a beautiful way. Another valuable book is Nakahara (a classic one), but I still have to finish reading it so I'll leave a comment about it in the next. The level of T. Frankel is at last yr undergrad - 1st yr graduate.

There are better... ***
I have used this book in an independent study in Geometry of Differential Forms. It did not take me too long to start looking for other references. There is something about its content that makes it diffucult to follow. May be it's too wordy. There are several misprints in notation. After I few weeks of study, I turned to Morita's Geometry of Differential Forms. The mathematical presentation is much clear and it's only 300 pages. I really like Frankel's book mainly for its application to physics. But with respect to the math, I recommend Morita's and Thirring monographs.

There are better... ***

I have used this book in an independent study in Geometry of Differential Forms. It did not take me too long to start looking for other references. There is something about its content that makes it diffucult to follow. May be it's too wordy. There are several misprints in notation. After I few weeks of study, I turned to Morita's Geometry of Differential Forms. The mathematical presentation is much clear and it's only 300 pages. I really like Frankel's book mainly for its application to physics. But with respect to the math, I recommend Morita's and Thirring monographs.

Dissapointing! *
Having gone through the first 3 chapters of this book, I must say I am really dissapointed. The author is supposedly trying to avoid the mathematical rigor to the account of explaining things in a physical way. Well, he almost completely fails in that and, the worst, confuses the reader. For example, he devotes 15 pages to smoothly introduce the reader to the concept of a manifold, promising a more rigorous definition to be given later on. When the time comes, he uses two "brand new" concepts, namely Haussdorf space and countable base, for the meaning of which, the reader has to look up in other books!(however cited in bibliography) If one wants to understand what all these things (manifolds, diff. forms, Lie deriv, etc.) are about, the best thing to do is buy a mathematical book. After all, it is hardly possible to satisfactorily describe abstract mathematical concepts if you avoid using mathematical language! For the interested, I am NOT a mathematician.

Dissapointing! *

Having gone through the first 3 chapters of this book, I must say I am really dissapointed. The author is supposedly trying to avoid the mathematical rigor to the account of explaining things in a physical way. Well, he almost completely fails in that and, the worst, confuses the reader. For example, he devotes 15 pages to smoothly introduce the reader to the concept of a manifold, promising a more rigorous definition to be given later on. When the time comes, he uses two "brand new" concepts, namely Haussdorf space and countable base, for the meaning of which, the reader has to look up in other books!(however cited in bibliography) If one wants to understand what all these things (manifolds, diff. forms, Lie deriv, etc.) are about, the best thing to do is buy a mathematical book. After all, it is hardly possible to satisfactorily describe abstract mathematical concepts if you avoid using mathematical language! For the interested, I am NOT a mathematician.

over and over and over again *****
Having taken a course out of Frankel (over the first 7 chapters) and now having used it in my senior project (topology of circuit analysis) I have to say that I love this book more by the day. Beforewarned it is not an easy text and you may have to read a section or a chapter over a hundred times. I have found that the material is dense and deep but in a way that welcomes effort. It is weak as far as rigor goes, but rigor can sometimes get in the way of understanding. Use this book alongside mathematics texts in topology, differential geometry and linear algebra and there is much to gain. For an undergraduate in mathematical physics (which I am) I have come to love this book I highly recommend it to a serious student.

over and over and over again *****

Having taken a course out of Frankel (over the first 7 chapters) and now having used it in my senior project (topology of circuit analysis) I have to say that I love this book more by the day.

Beforewarned it is not an easy text and you may have to read a section or a chapter over a hundred times. I have found that the material is dense and deep but in a way that welcomes effort. It is weak as far as rigor goes, but rigor can sometimes get in the way of understanding. Use this book alongside mathematics texts in topology, differential geometry and linear algebra and there is much to gain.

For an undergraduate in mathematical physics (which I am) I have come to love this book I highly recommend it to a serious student.