A question about mathematics textbooks for physicists

[Florian Geyer]

Hello esteemed members, I have a question with regard to what a physicist should study in mathematics.

I tried hard to find a solution to this question but I couldn't, I have just reached to some conclusions I think they are true but I am entirely not sure, thus I decided to put this question here.

Well the first conclusion I have reached is that physics students should study mathematical methods books first, why? Because they contain the core of what they need in their studying of physics.

I thought first (I think naively) that after doing this, students shall study some specialized math textbooks (which of course directed to physics not like the pure math textbooks) like for example Davis and Snider vector analysis, Debnath Linear Partial Differential Equations for Scientists and Engineers… etc. but I discovered afterwards that there is no clear (after mathematical methods boundary) because there are so many of them and some of them are very tough here is a list for some of the books I have found (which I think have some clear differences in them)

- Boas Mathematical Methods in the Physical Sciences. (Standard for undergraduates)

- Arfken Weber Harris Mathematical Methods for Scientists and Engineers. (Standard for graduate students as far as I know)

- Balakrishnan Mathematical Physics with Applications (I think just like Arfken applied to physics, thus I don’t think it should considered different from Arfken… only application to it)

- Hilbert and Courant. (I heard it is more like a pure math style)

- Morse and Feschbach (This is different from the previous in that it is more comprehensive and at the same time much tougher)

- Thirring mathematical physics (2 volumes I have not read any review about this textbook thus I am not sure where shall I put it).

- Penrose The road to reality (I don’t know a lot about this book but I think it is like a math methods book intended for those who specialize in cosmology or general relativity).

There are some textbooks which seems to be more advanced like

- Bender Orzag Advanced mathematical methods for scientists and engineers.

There are some textbooks which prefer the title physical mathematics like:

- Cahil physical mathematics.

There are some VERY THICK textbooks like

- Simon Reed methods of modern mathematical physics (I don’t know how anyone can read this)
But based on the previous I concluded that "mathematical methods" is not something we can restrict to some textbooks, and they may contain any math which can be considered useful to physics. But this also will raise the question why shall some math textbooks authors add the suffix (for physicist and engineers) if physicists have their own mathematical books which they will not be able to understand it all (for most of them).

- Is the choice between the previous based on the specialization of the students?

- If one wants to be a good theoretical or mathematical physicist, what shall he study?

One last thing I want to add, is there a difference between mathematical physic and physical mathematics? I have read a review mentioned by Prof. Gerald t'Hooft in which there is a section on mathematical methods and another for physical mathematics!!! I will leave its link bellow.

https://math.ucr.edu/home/baez/physics/Administrivia/booklist.html
Note to be considered: many of the opinions on the previous textbooks are based on some reviews and opinions I have read about on the internet. Others like my thoughts of Balakrishnan textbook is based on a 10-minute skimming and nothing more, but since I don't think this is directly related to my question I don't think I need to do some rigorous ones.

Thanks for reading

 

[Frimus]

- Penrose The road to reality (I don’t know a lot about this book but I think it is like a math methods book intended for those who specialize in cosmology or general relativity).

I would not classify this book in any way as 'mathematical methods for physicists', let alone for beginners.
BTW is there someone here who carefully read it all the way through and understood everything in it? I have got the impression that even his so called popular books are readable only by members of his league.

 

[Demystifier]

"The Road to Reality" is a semi-popular book. It is a light read for people with degree in mathematics or theoretical physics.

 

[Florian Geyer]

I would not classify this book in any way as 'mathematical methods for physicists', let alone for beginners.
BTW is there someone here who carefully read it all the way through and understood everything in it? I have got the impression that even his so called popular books are readable only by members of his league.

"The Road to Reality" is a semi-popular book. It is a light read for people with degree in mathematics or theoretical physics.

Thank you for clarifying.

 

[Frabjous]

Well the first conclusion I have reached is that physics students should study mathematical methods books first, why? Because they contain the core of what they need in their studying of physics.

You can spend forever learning the math and never get to the physics. Determine which physics you want to learn and learn the required math. Notice that one “learns” physics topics multiple times, so one will generally need to improve ones mathematical sophistication with each iteration. Also notice that physics texts “teach” a lot of math, so the maths do not always need to be addressed independently.

 

[gleem]

Most of my math education came from individual math courses. I went back and added the total number of pages of all the courses covered in Boas' book. She covered a few topics that I didn't have. Her book did it in 800+ pages. The total of my math course actually somewhat less is about 2000.

Boas' book table of Contents is below. The checkmark is what was covered in my math courses.

• 1. Infinite series, power series √
• 2. Complex numbers √
• 3. Linear algebra √ plus matrix algebra, coordinate transformations
• 4. Partial differentiation √
• 5. Multiple integrals √
• 6. Vector analysis √
• 7. Fourier series and transforms √
• 8. Ordinary differential equations √
• 9. Calculus of variations
• 10. Tensor analysis
• 11. Special functions √
• 12. Series solutions of differential equations; Legendre, Bessel, Hermite, and Laguerre functions √
• 13. Partial differential equations √
• 14. Functions of a complex variable √
• 15. Probability and statistics.

A math methods course tends to be 6 credit hrs. but SUNY Geneseo covers it in 25 classes (Hrs?) in 14 weeks. My courses totaled 21 credit hours given over about 5 semesters, 105 classes

So were the physics students in the 60's that slow that they needed so much more time to absorb the content or are the present students that much more capable? Or were we wasting our time with irrelevant mathematical content? I don't know but it is hard for me to think that I would have been as capable with a 25 or even 50 classes covering all that material. Also, consider the additional problem assignments the individual classes should have had. Now I might have not needed all the math that I learned but since I had no way of knowing its value to me in my future endeavors, As a physicist I had it in my toolbox just in case.

Determine which physics you want to learn and learn the required math.

The problem is that you may not decide until you are about to graduate. Grabbing the first train out of town may not take you where you want or should be.

Also notice that physics texts “teach” a lot of math, so the maths do not always need to be addressed independently.

Not well. They usually present a distilled cookbook presentation. Luckily didn't have to depend on them.

 

[vanhees71]

- Boas Mathematical Methods in the Physical Sciences. (Standard for undergraduates)

This I don't know, but it seems to be a standard reference in the US.

- Arfken Weber Harris Mathematical Methods for Scientists and Engineers. (Standard for graduate students as far as I know)

A bit superficial.

- Balakrishnan Mathematical Physics with Applications (I think just like Arfken applied to physics, thus I don’t think it should considered different from Arfken… only application to it)

This I don't know.

- Hilbert and Courant. (I heard it is more like a pure math style)

One of the best and most useful books ever written on the subject. E.g., it delivered the mathematics ready for use by Schrödinger when discovering the wavemechanics formulation of modern quantum mechanics!

- Morse and Feschbach (This is different from the previous in that it is more comprehensive and at the same time much tougher)

Very complete and utmost useful for more complicated mathematical problems in physics. I'd not recommend it as a first read though.

- Thirring mathematical physics (2 volumes I have not read any review about this textbook thus I am not sure where shall I put it).

This is more a textbook on mathematical physics than a math-methods book for physicists. It is very brief but includes a lot of gems for advanced theorists (it's 4 volumes by the way).

- Penrose The road to reality (I don’t know a lot about this book but I think it is like a math methods book intended for those who specialize in cosmology or general relativity).

is not a textbook but a quasi-popular book.

There are some textbooks which seems to be more advanced like

- Bender Orzag Advanced mathematical methods for scientists and engineers.

There are some textbooks which prefer the title physical mathematics like:

- Cahil physical mathematics.

There are some VERY THICK textbooks like

- Simon Reed methods of modern mathematical physics (I don’t know how anyone can read this)

These I don't know either.

 

[Florian Geyer]

This I don't know, but it seems to be a standard reference in the US.

A bit superficial.

This I don't know.

One of the best and most useful books ever written on the subject. E.g., it delivered the mathematics ready for use by Schrödinger when discovering the wavemechanics formulation of modern quantum mechanics!

Very complete and utmost useful for more complicated mathematical problems in physics. I'd not recommend it as a first read though.

This is more a textbook on mathematical physics than a math-methods book for physicists. It is very brief but includes a lot of gems for advanced theorists (it's 4 volumes by the way).

is not a textbook but a quasi-popular book.

These I don't know either.

I am very grateful for your review of the previous books. I appreciate your generosity. Your insightful reviews will undoubtedly be invaluable to me in my future studies.

Regarding the textbook of Thirring I am not sure, but I think the English editions I know are both by Springer, the first in five volumes, and the second in two volumes, called Classical mathematical physics (vol. 1), and quantum mathematical methods (vol. 2) which I think have the same content as the first edition in 5 volumes.

 

[Florian Geyer]

This is more a textbook on mathematical physics than a math-methods book for physicists. It is very brief but includes a lot of gems for advanced theorists (it's 4 volumes by the way).

May I ask you what is the difference between mathematical physics and theoretical physics? I have a basic understanding of the difference between theoretical physics and mathematical methods based on a short video made by Prof. Elliot Lieb (who worked, by the way, with Thirring as far as I know). However, the difference between mathematical methods and mathematical physics is beyond my understanding.

 

[WWGD]

Anything I've run into by Baez is excellent. I've learned plenty of it as a Mathematician.

 

[jasonRF]

There are some textbooks which seems to be more advanced like

- Bender Orzag Advanced mathematical methods for scientists and engineers.

Is more advanced than Boas, but less advanced (and much easier reading) than books like Courant and Hilbert, Morse and Feschbach. According to the authors, part of the book can easily be used for undergrads. The entire book is a reasonable read if you have taken a solid course in complex analysis. Includes a lot of good stuff that often gets taught piece-meal in various physics/engineering courses: solutions to ODEs near irregular singularities, WKB approximation, approximate solutions of nonlinear ODEs, asymptotic expansions of integrals, perturbation theory, multiple-scale analysis, etc.

 

[vanhees71]

May I ask you what is the difference between mathematical physics and theoretical physics? I have a basic understanding of the difference between theoretical physics and mathematical methods based on a short video made by Prof. Elliot Lieb (who worked, by the way, with Thirring as far as I know). However, the difference between mathematical methods and mathematical physics is beyond my understanding.

This is a difficult question. I think the difference is in the level of mathematical rigor applied. Mathematical physics tries to formulate everything in a mathematically rigorous way with formal proofs, while theoretical physics uses math simply as a language to express the physical theories and uses it as a tool to solve equations to predict physical phenomena without taking too much interest in formal proofs.

"Mathematical methods" usually discusses the mathematics needed by physicists (not only theoretical but also experimental physicists by the way) at this practical physicists' approach, i.e., it aims at teaching you more the practical methods used to solve equations than to proofing all theorems with the utmost level of rigor or under the minimal assumptions in full generality. I'd say it's a bit like the difference between "Calculus" and "Analysis".

 

[WWGD]

If I may suggest, all the Math you need can be learnt from a full course in Advanced Calculus.

 

[vanhees71]

If it includes sufficient linear algebra and (at least for theorists) a minimum of Lie-group and representation theory ;-)).

 

[WWGD]

If it includes sufficient linear algebra and (at least for theorists) a minimum of Lie-group and representation theory ;-)).

Yes, I meant it in this broader sense, not just going through the sequence of Calc1 through Calc 3. In Advance Calc you learn Linear Algebra as part of the definition of Derivative as a linear map.

 

[mpresic3]

What level of physics students are you addressing in your post. Freshman physics in college study calculus often as a co-requisite. The study physics concurrently. In a sense they are studying math first, although when I studied physics, physics was at 9:00 am, and Math was at 11:00.
If you are speaking to physics graduate students, often Math Physics with Arfken/Weber is often taught concurrently with quantum and electrodynamics, or other graduate courses. It would not be a popular shift to teach these physics courses sometime after Linear Functional Analysis at the level of Reid/Simon.
In addition, a walk through any physics department will take you through researchers in biophysics, geophysics, astronomy/astrophysics. etc. Do biophysicists applying physics to telomeres, or studying photosynthesis, or the lens of the eyes (using tensors and stresses etc), need to know topological spaces in the same way as theoreticians studying general relativity. For that matter, do theoreticians studying numerical solutions need to know the same mathematics and methods as the professional mathematician studying solutions to hyperbolic differential equations?
Graduate students are encouraged to get to research with the bare minimum of coursework which is deemed necessary, like it or not. I certainly wanted to take more courses, but my research advisor reminded me that the PhD is a research degree. We have to get on with it.

 

[Florian Geyer]

Graduate students are encouraged to get to research with the bare minimum of coursework which is deemed necessary, like it or not. I certainly wanted to take more courses, but my research advisor reminded me that the PhD is a research degree. We have to get on with it.

Illuminating.
Thanks for addressing this point of view.

 

[Frabjous]

It’s a question of trade offs.

Phillip Morse, 1961

 

[bhobba]

In my opinion, the book, as mentioned by others, is Boaz. I suggest it even for mathematicians, followed by Hubbard, who adds rigour (if that is your thing):
https://matrixeditions.com/5thUnifiedApproach.html

Thanks
Bill

 

[AndreasC]

Simon Reed methods of modern mathematical physics (I don’t know how anyone can read this)

Imo it is one of the most readable of the ones you cited lol lots of mathematical methods books rank very low on readability.

These texts have very different purposes. Personally I don't really like the concept of most of these books. They're too big, they're often a hodge podge, and very hard to learn from. I think it's better to just learnt the physics and then look up specific things as you go along.

Paradoxically I find that the older ones are more useful. The reason is that they contain some "obsolete" stuff that you can't easily find on a Google search, but sometimes you need them. For instance I recently encountered "dyadic Green functions". This is something that currently lacks a wiki page and people don't talk much about it any more, and I guess dyadics are generally out of fashion. But there it was in the final chapter of Morse and Feshbach. Unfortunately its coverage there was confusing to me so I ended up looking for another source. Similarly, I know Hilbert and Courant have a good coverage of integral equations, which is not that common in some modern sources. I've read very little of it but anything written by Hilbert (and Courant!) is bound to contain some gems.

Bender and Orzag is a GREAT book and possibly even greater are Bender's lectures on mathematical methods which are on YouTube. HOWEVER, it's a different concept from the traditional "mathematical methods" books. It's a book that pretty much specifically teaches you some approximate methods of solving differential equations, especially asymptotics etc.

Reed and Simon is also COMPLETELY different, it's a rigorous, almost "pure" math book, specializing in techniques of functional analysis. They are also very good books, but very different in scope. You need math background to read them and they are often taught in math courses, less so in physics courses. But they are well written and not a hodge podge.

My suggestion for what to read right now is none of it lol. If you know advanced calculus you are good to start learning physics. Then as you progress you may want to learn more about specific subjects, at which point you will read things specifically about them rather than these massive tomes. The one book that at some point I would suggest everyone to read is Bender and Orzag, because it contains some stuff that you often learn in physics courses, but never learn it properly or systematically. But that's a book for later. Reed and Simon is also great but for specific people.

 

[Florian Geyer]

Thank you for considering my thread.

I have understood a lot since I have published my thread, the reason for asking this thread, like most of mine is to get some guidance (Thankfully I always get good guidance in this forum), since I think this is the most important part I lack in my community. For a naïve person who has no experience in physics and math subjects the titles of the textbooks may cause a lot of confusion to him, thus he can really tell nothing form the title of the textbook only, rather you have to dedicate time for serious readings about the textbooks before you study any one of them.

Regarding my current level, yes I agree I need to work a lot to improve it, I am working hard to change this.

The points you mentioned about some of the advanced textbooks is stimulating and thought-provoking, but I think it will take me a long time to reach their level, then I will don my best with them.

Again thank you for this rich reply.​

 

[Frabjous]

For a naïve person who has no experience in physics and math subjects

While I have successfully completed a Bachelor's degree in physics, I often feel that my foundational knowledge isn't as deep as I'd like.

That's right, I have done that to add some privacy and freedom to my activities in this forum.

You really need to stop the lies about your background.

 

[Florian Geyer]

You really need to stop the lies about your background.

Can you explain to me more why do you think I lie about my background?

 

[Vanadium 50]

Because you have said that you have no experience in physics and you also say you have a degree in physics. They cannot both be true.

 

[WWGD]

Maybe he's saying his experience is limited, given he only has a Bachelors.

 

[Florian Geyer]

Because you have said that you have no experience in physics and you also say you have a degree in physics. They cannot both be true.

But the real level of my degree is lower than that of AP!!
You have also told me that my level back then was lower than that of the most advanced student who start their undergraduate study.

You can refer to my first thread in this forum.

 

[Vanadium 50]

That wouldn't have been a bad guess, had the OP not said that he's posting false information about his backgroimd. We aren't guessing - he's telling us that. It's his choice, but I don't see how he can give us false information about his background and then complain our messages are over or under his head or otherwise don't fit his situation, whatever it really is.

 

[MidgetDwarf]

The vector calculus book by snider is terrible. Its at a low level, readable, but there are better text that reward the reader.

Ie., Marsden Vector Calculus [at the same level, but more coverage,details], Hubbard and Hubbard higher level than those two, but readable.

For PDE, i like Bleaker.

 

[WWGD]

I always suggest when available, possible, people drop by a college or otherwise library with a Math section and browse through books in the area of interest, see which ones feel right for them. Usually things like a carefully laid out book , with notation, glossary, is a sign of a carefully-written book.