The Math of Physics type books? (Not a methods book)

[DrummingAtom]

Lately, I realized that I've grown tired of calculations and want to venture into more theory type books although I haven't taken any proof based math classes yet. I feel confident that I can do basic proofs, i.e. the "show that" type problems but nothing too crazy after that. I've read through a chunk of Boas' book and it's great to show how to calculate but I definitely don't understand how things are working. I've taken linear algebra, Diffy Q, Calc 1-3, and I'm just finishing up a class on special functions and advanced DE techniques. Also, I've just taken intro physics 1 and 2 but have learned a tiny bit of upper level physics on my own.

Anyway, I want a book that shows the math of physics without getting too proofy or require high level physics knowledge. The closest book I've found is "The Geometry of Physics" by Frankel but reading through the table of contents is very intimidating because I literally don't know anything of those topics before: manifolds, bundles, forms, etc. Although, according to the preface, my background does meet the requirements for the book. Anyone else use this book?

Ultimately, I want a book that shows the math for itself but then applies it later on to physics. Also, I'm very visual and need pictures to understand almost anything. I will be self studying from the book as well. Any suggestions?

Thank you.

 

[espen180]

Not knowing the topics in the book beforehand is sort of the point, no? ;)

I read a bit in the book in question, and it's not so bad. The exposition is fluent and it's not proofy. However, notions like manifolds, bundles and differential forms take a little time to get used to. It might not be immediately clear how it fits in with the physics or why it is useful. However, if you are willing to spend some time thinking about these things, I don't think it will be a problem. As for pictures, there are a lot of them, don't worry.

However, it certainly wouldn't hurt reading some proof based math. Some abstract algebra would be a good starting point.

 

[kloptok]

Frankel is in my opinion a nice book, sometimes there is a bit too much text but in general I think it is written in an understandable and pedagogic fashion. Don't be scared by the ToC - the book starts basic and you will soon understand the basic concepts of manifolds, forms and bundles (I do agree with espen180 though, it does take a while to get used to all this stuff). A similar book, with similar prerequisites is Schutz' "Geometrical methods in Theoretical Physics". If you know Schutz from his GR book the style is similar. That is, a non-intimidating, beginner-friendly style which still (in my mind) contains the necessary rigour (or suitable reference) for a book for physicists. Schutz contains almost no topology though.

If you're into topology and geometry a standard is also Nakahara's "Geometry, Topology and Physics" which deals with roughly the same subjects as Frankel (some more focus on topology) but perhaps with a bit more rigour and depth, and the style is a bit more mathematical in my opinion. Personally I would say that Frankel is more beginner-friendly but Nakahara is nice because it is quite dense. Regarding prerequisites it is of course always better the more you know, and you need some level of that thing called "mathematical maturity". It is probably good if you have some experience with for example differential geometry and Lie groups which you will probably have if you've taken courses in for example general relativity and quantum field theory.

I have only browsed through the contents, but another book which I think looked promising is Sadri Hassani's "Foundations of Mathematical Physics". There is also a book by Peter Szekeres called "A Course in Modern Mathematical Physics" which might actually suit your needs very well since it covers more than just geometry and topology.

Good luck in finding a book! And don't be scared just because it looks difficult - keep reading and sooner or later you'll get it. As one of my lecturers said: "you don't learn math, you get used to it".

 

[George Jones]

I like the books by Frankel, Nakahara, and Szekeres, but there is another possibility.

Another book worth looking at is Differential Geometry and Lie Groups for Physicists by Marian Fecko,

https://www.amazon.com/dp/0521845076/?tag=pfamazon01-20.

This book is not as rigorous as the books by Lee and Tu, but it more rigorous and comprehensive than the book by Schutz. Fecko treats linear connections and associated curvature, and connections and curvature for bundles. Consequently, Fecko can be used for a more in-depth treatment of the math underlying both GR and gauge field theories than traditionally is presented in physics courses.

Fecko has an unusual format. From its Preface,

A specific feature of this book is its strong emphasis on developing the general theory through a large number of simple exercises (more than a thousand of them), in which the reader analyzes "in a hands-on fashion" various details of a "theory" as well as plenty of concrete examples (the proof of the pudding is in the eating).

The book is reviewed at the Canadian Association of Physicists website,

http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf.

From the review

There are no problems at the end of each chapter, but that's because by the time you reached the end of the chapter, you feel like you've done your homework already, proving or solving every little numbered exercise, of which there can be between one and half a dozen per page. Fortunately, each chapter ends with a summary and a list of relevant equations, with references back to the text. ...

A somewhat idiosyncratic flavour of this text is reflected in the numbering: there are no numbered equations, it's the exercises that are numbered, and referred to later.

Personal observations based on my limited experience with my copy of the book:

1) often very clear, but sometimes a bit unclear;
2) some examples of mathematical imprecision/looseness, but these examples are not more densely distributed than in, say, Nakahara;
3) the simple examples are often effective.

 

[homeomorphic]

Ultimately, I want a book that shows the math for itself but then applies it later on to physics.

That's not always the natural order of doing things, which is one reason why I think it is a bad idea to separate math and physics too much. Sometimes, it's more natural to start with the physics concepts and use them as inspiration for the math, and sometimes it's more natural to develop the math first and then observe that it can be applied to physics.

Looking at the table of contents of The Geometry of Physics, I do think it might not be the best place to start at your level. Personally, I prefer to work my way up the ladder slowly, rather than to have abstract concepts thrown at me that might be unmotivated, given my current level of knowledge. A famous mathematician (who also happens to be my thesis adviser) once told me that most people start with the most complicated case first, and that's wrong. You should start with the simplest case, first.

In this case, I would study the differential geometry of curves and surfaces before learning anything about manifolds.

If you want to learn physics, you shouldn't get ahead of yourself. At your level, I think the two subjects that lie next in your path are electromagnetism and classical mechanics at a bit higher level than what you have seen so far. I would focus on those before moving on. Electromagnetism, you can learn now. For that, maybe try Purcell or Griffiths--haven't read either. I used some EE book. This is one of the cases where I think it's more natural to do the physics, then the math. If you understand electricity and magnetism, it will provide good motivation for things like differential forms.

If you want to understand more advanced classical mechanics more visually, that's a case where it might be more natural to do a lot of math first, at least if you followed my tastes, which happen to very, very extreme on the visual side. Some bits of it stand well on their own without much mathematical dressing, but ultimately, to understand the meaning of things like Hamilton's equations, I think differential forms are essential. Probably, I would recommend cobbling together an understanding from various sources in a somewhat complicated way, not just one source. Chief among these sources being Mathematical Methods of Classical Mechanics, by Arnold (the geometrical insight really shines through in the later chapters--you might get the wrong impression from some of the earlier ones).

The good news is that visualization can take you very far into math and physics, but the bad news is that it gets to be a very abstract sort of visualization. In particular, you have to develop your higher-dimensional intuition.