What are you reading now? (STEM only)

[Auto-Didact]

Read "The Quantum Story" and report back. ...
Is particle physics better than condensed matter physics? Does such an attitude reflect a problem?

Will do.

I think you are getting off topic.

To get back on topic, currently reading:

For my current theoretical research in dynamical systems, I'm doing a comprehensive literature review of the mathematical literature going back three centuries. Curiously, any which way I approach the topic, it noticeably brings me back to one single source again and again.

As a result, I decided to finally pick up a book that was recommended to me awhile ago: Euler: The Master of Us All, by William Dunham, 1999.. I'm currently still reading it.

In any case, during this literature review, I've changed my mind about the rankings of who is the best mathematician of all time, which previously had Gauss on top. I think the opening quote of this book summarizes both my current opinion as well as the book quite well: "Read Euler, read Euler! He is the master of us all." - Pierre Simon Laplace

 

[vanhees71]

I’m reading “ Zen and the art of motorcycle maintenance”
View attachment 258194

May not be exactly within boundaries of (stem) category ... classic tho

Hm, I'd rather apply the natural laws as known by engineers and mechanics than some Zen Buddhism to maintain my vehicles

 

[martinbn]

Hm, I'd rather apply the natural laws as known by engineers and mechanics than some Zen Buddhism to maintain my vehicles

You don't know the book, but it is not about Zen nor motorcycles.

 

[PeroK]

Hm, I'd rather apply the natural laws as known by engineers and mechanics than some Zen Buddhism to maintain my vehicles

One aspect of the book is the contrast between two characters: one of whom rides a motorbike that requires a lot of maintenance and loving care and attention; and the other rides a BMW that never breaks down. I think I know which bike you'd prefer!

 

[vanhees71]

As a theoretician, I'd prefer the BMW. You know, it's a desaster when theoreticians try to do something practical or even physics experiments (remember poor de Haas, who was persuaded by Einstein to get the "right" gyrofactor of 1 rather than making the discovery that it's different; today we know it's about 2, and this was discovered shortly after Einstein's and de Haas's publication) ;-)).

 

[Mark44]

I’m reading “ Zen and the art of motorcycle maintenance”
View attachment 258194

May not be exactly within boundaries of (stem) category ... classic tho

I found this so interesting when I first read it back in about 1975, that I wrote notes in the margin, something I don't usually do. A reprint or new edition came out in 2000, so I bought a copy for my bookshelf.

 

[Mark44]

I'm reading "The Mapmaker's Wife, A True Tale of Love, Murder, and Survival in the Amazon," by Robert Whitaker. The book chronicles the quest by a group of French academicians in 1738 to measure the length of one degree of arc of the Earth's circumference at the equator. At the time, Rene Descartes and other scientists believed that the Earth was a prolate spheroid, sort of like a football (an American football). Newton's calculations led him to believe that shape was that of an oblate spheroid.

I'm about halfway through the book now, and the group, led by Charles Marie De La Condamine, has made very accurate measurements and maps of the area around Quito, which was a city in the Peruvian Viceroyalty, long before the existence of separate countries such as Ecuador and others.

The group in the book made significant advances in astronomy, mapping, and botany (bringing knowledge of cinchona, the source of quinine, as well as rubber, back to Europe), and numerous other areas.

The mapmaker's wife of the title, traveled 3,000 miles from Quito across dangerous passes in the Andes and down the Amazon to rejoin her husband, one of the Frenchmen in La Condamine's party.

 

[vanhees71]

I just got Weinberg's newest textbook. A gem again, as expected:

https://www.amazon.com/dp/1108415075/?tag=pfamazon01-20

 

[Mondayman]

I just got Climbing the Mountain, a biography about Julian Schwinger by Jagdish Mehra. It is the counterpart to The Beat of a Different Drum, the biography of Feynman. I'll probably reread these books when I have a basic understanding of QED.

I also have The Maxwellians by Bruce Hunt. It's about the development of Maxwell's equations as we know them, going from Maxwell's Treatise to the work of FitzGerald, Heaviside, Lodge, and Hertz. Electrodynamics is by far my favorite subject, and I can't wait to tackle Jackson and Landau.

 

[Klystron]

I picked up a thin mathematics text on a whim from the library, "Infinite Ascent" by David Berlinski. Despite a roguish take on historical greats, the book is readable and informative. Each brief subject -- geometry, analytical geometry, calculus, etc. -- follows the typical curriculum we learned in school but with an altered perspective.

 

[Mondayman]

Just picked up a new book on Feynman and another on Dirac. This is my current STEM list:

Dirac: A Scientific Biography*
Feynman and His Physics*
The Beat of a Different Drum*
Climbing the Mountain*
Subtle is the Lord*
J. Robert Oppenheimer
The Genius of Science
QED and the Men Who Made It*
Inward Bound*
Oliver Heaviside
The Maxwellians
QED
Lost in Math

The ones with * are technical and don't shy away from the mathematics.

 

[Auto-Didact]

I'm reading Peter Woit's QM textbook:
Quantum Theory, Groups & Representations

Gotta say, I'm enjoying it way more than I had anticipated. This is probably my favourite textbook on QM.

 

[Auto-Didact]

I'm reading Peter Woit's QM textbook:
Quantum Theory, Groups & Representations

Gotta say, I'm enjoying it way more than I had anticipated. This is probably my favourite textbook on QM.

For some context, I originally learned QM via Griffiths in conjunction with the Feynman Lectures. Over the years I have reviewed a few other standard texts (Sakurai, McIntyre, Shankar, Ballentine and Nielsen & Chuang). All of these have their strengths and weaknesses, but none of them seem to be able to achieve what Woit already manages to largely achieve in the first few chapters.

In my opinion, with respect to being able to convey, not merely mathematical technique and physical theory, but also a deep understanding of the subject matter at an unexpectedly unified and sophisticated mathematical level, Woit's book seems to be superior to all of them. This seems to be especially true when approaching the matter from the perspective of an advanced undergraduate intending to go into either mathematical physics or theoretical physics (with a double major in mathematics and physics).

The reason for this praise is that this book is not merely the first book on QM that I have read so far, that has actually achieved such a sense of clarity for me - similar to what MTW did for GR for me - but so far the only book on QM that I have ever read that has achieved this; i.e. no single other textbook - not even more advanced texts on QFT, nor (post)graduate level monographs on other interpretations of QM, not even Landau & Lifshitz - were able to achieve that. This is simply the closest thing I have ever seen to a legitimate first principles treatment of the subject, from the perspective of foundations (NB: opposed to many illegitimate first principles treatments... I won't name names).

More colloquially, I feel like so far all other books have made me merely accustomed to QM, while this is the first time I feel that a single author in a single work has actually managed to treat QM as a form of an application of pure mathematics, and that in a format understandable for undergraduates. Is my perspective on this book typical or generalizable? I don't know; all I know is that Woit seems to have achieved something which as far as I can tell no one else in history has.

 

[vanhees71]

I don't know this book. What difference in treatment makes it so unique for you? Is it only about "interpretation" or is it about the "no-nonsense mathematical and physics aspects"?

 

[Auto-Didact]

I don't know this book. What difference in treatment makes it so unique for you? Is it only about "interpretation" or is it about the "no-nonsense mathematical and physics aspects"?

Nothing to do with interpretation, the book virtually makes no statements on interpretations. The difference instead is that it seems to be written with a specific audience in mind instead of generically for physics students, i.e. it is tailormade for the 21st century (aspiring) mathematical physicist, mathematician or mathematics-oriented theoretical physicist instead of for any (aspiring) physicist more generally.

By 'mathematics-oriented' I am not so much speaking about mathematical skill, specific content (e.g. specific mathematical topics/theories) or chosen methodology (e.g. axiomatic or numerical methods), but instead about the overall writing style and presentation of the material. Canonical texts in foundational physics where the research has reached a certain stage of maturity has a specific presentation style - usually as a consequence of being tidied up by the mathematical physics community; both Newton's Principia and MTW are written in this style.

On the other hand, so far all textbooks on QM I have read are specifically not written in this mature style, but instead written in a distinctly schizophrenic 'half mathematics, half physics' style, which unfortunately has become quite characteristic of texts on QM since von Neumann & Dirac. More concretely, the typical style of QM texts pretends to be a rigorous mathematics text but then with unjustifiable caveats, i.e. a distinctly non-foundational style; this actually indicates that the field of research in question is still quite premature.

In contrast, the style Woit utilised in writing his book is very different than the typical QM textbook style, but more importantly also doesn't quite resemble modern pure mathematics texts. It instead resembles quite closely modern applied mathematics texts with one major difference: the mathematical content actually belongs to 'pure mathematics' and not to 'applied mathematics'. This particular mixed style i.e. 'application of pure mathematics' was - prior to the 20th century - the characteristic style of mathematical physics, which is precisely why I love this book; the sophisticated mathematics is just icing on the cake.

 

[weirdoguy]

I've read half of the book some time ago, and I have similar feelings about it. Woit introduces everything through Lie groups and algebras and makes it in a very clear and rigorous way. For more mathematically oriented readers this book is a pure gold. It was so easy to read for me. I've finished "Group theory in a nutshell for physicists" by Zee last year and I feel like I've learned more from the first few chapters of Woit's book than from whole Zee's book..

 

[vanhees71]

Nothing to do with interpretation, the book virtually makes no statements on interpretations. The difference instead is that it seems to be written with a specific audience in mind instead of generically for physics students, i.e. it is tailormade for the 21st century (aspiring) mathematical physicist, mathematician or mathematics-oriented theoretical physicist instead of for any (aspiring) physicist more generally.

By 'mathematics-oriented' I am not so much speaking about mathematical skill, specific content (e.g. specific mathematical topics/theories) or chosen methodology (e.g. axiomatic or numerical methods), but instead about the overall writing style and presentation of the material. Canonical texts in foundational physics where the research has reached a certain stage of maturity has a specific presentation style - usually as a consequence of being tidied up by the mathematical physics community; both Newton's Principia and MTW are written in this style.

On the other hand, so far all textbooks on QM I have read are specifically not written in this mature style, but instead written in a distinctly schizophrenic 'half mathematics, half physics' style, which unfortunately has become quite characteristic of texts on QM since von Neumann & Dirac. More concretely, the typical style of QM texts pretends to be a rigorous mathematics text but then with unjustifiable caveats, i.e. a distinctly non-foundational style; this actually indicates that the field of research in question is still quite premature.

In contrast, the style Woit utilised in writing his book is very different than the typical QM textbook style, but more importantly also doesn't quite resemble modern pure mathematics texts. It instead resembles quite closely modern applied mathematics texts with one major difference: the mathematical content actually belongs to 'pure mathematics' and not to 'applied mathematics'. This particular mixed style i.e. 'application of pure mathematics' was - prior to the 20th century - the characteristic style of mathematical physics, which is precisely why I love this book; the sophisticated mathematics is just icing on the cake.

Sounds like a good idea to get this book.

On the other hand, of course the problem with quantum mechanics (I talk about non-relativistic QM, which is mathematically well formulated) is only partially the mathematics. Of course there are subtleties like that observables are represented by self-adjoint densely defined operators not by hermitian ones, the treatment of unbound operators with continuous spectra etc. All this is nicely solved also mathematically rigorously by formulating QM from the very beginning via rigged Hilbert spaces ( see, e.g., the books by Galindo and Pascual). The real didcatical problem is the physics, because you have to forget the classical paradigm completely. While Dirac's book (for the wave-mechanics approach the pendant is Pauli's "Handbuchartikel") is still among the best presentations for physicists (mathematicians will find it to lack mathematical rigor), von Neumann's treatment is outdated (though mathematically rigorous) since the rigged-Hilbert space formulation is as rigorous but much simpler, and the interpretational part is not even wrong ;-).

As many discussions in this forum show, the "no-nonsense approach", which is simply accepting that nature behaves non-deterministic at least as far as our ability to observe it, one rather discusses pseudoproblems of some philosophers.

A bit less known is that you have the same phenomenon with general relativity. I was very surprised that the (in)famous hole argument is still debated. It was based on an incomplete and also phenomenologically wrong predecessor theory by Einstein, and after Einstein (and at the same time Hilbert) has found the correct generally covariant theory, this apparent problem has become completely obsolete, and the physics interpretation of general covariance, which simply is a local gauge symmetry, has been given by Einstein already in 1916. This week, I've seen an entire volume of a physics journal (I forgot which one) dedicated to the "hole argument"...

 

[martinbn]

My personal impression is that Woit's book is very disappointing. It looks like many other books written by physicists. It is full of elementary undergraduate mathematics and trivial computations. It makes me wander who the intended reader is. If you already have studied some maths then the book has mostly things you know. If you haven't, it can look impressive but each topic only scratches the surface and it is not nearly enough to learn anything properly. May be the goal is to give a glimpse or to develop an interest. Why does a book on representation theory and quantum mechanics need a chapter on linear algebra and one on Fourier analysis presented in a way that suggests the reader has never seen that before!

 

[martinbn]

Just out of curiosity, what do people that like Woit's book think about van der Waerden's book " Group Theory and Quantum Mechanics"?

 

[vanhees71]

I don't know Woit, but van der Waerden's book is simply a masterpiece. I once found it in my university's physics library and just started reading it. The librarian had to remind me that the library is about to close in the evening ;-)).

 

[Auto-Didact]

My personal impression is that Woit's book is very disappointing. It looks like many other books written by physicists. It is full of elementary undergraduate mathematics and trivial computations. It makes me wander who the intended reader is. If you already have studied some maths then the book has mostly things you know. If you haven't, it can look impressive but each topic only scratches the surface and it is not nearly enough to learn anything properly. May be the goal is to give a glimpse or to develop an interest. Why does a book on representation theory and quantum mechanics need a chapter on linear algebra and one on Fourier analysis presented in a way that suggests the reader has never seen that before!

As I said before, the intended audience seems to be the aspiring mathematical physicist or mathematician interested in physics, i.e. undergraduate physics students who already know early on they want to do mathematical physics or mathematics-oriented theoretical physics with QM foundations as their main intended research field. Not being able to recognize this is essentially an incapability of being able to distinguish what is necessary from what is sufficient, i.e. the result of simply not having been trained to do foundational research.

Whatever you may personally find about the level of the mathematics in Woit's book (e.g. representation theory, Lie groups, Clifford algebra, geometric quantization, etc) this material treated by Woit certainly isn't standard curriculum mathematics for QM at the undergraduate level of the standard physics curriculum. Students should not be assumed to have picked such subjects up tacitly e.g. by osmosis of interaction with older peers or staff or assimilation into some school of thought; instead it should be available directly inside the curriculum e.g. actually integrated into a special track, instead of merely placed within the electives and then left as a game of chance of the sufficiently interested or sufficiently lucky students picking it.

As for why linear algebra and Fourier analysis are treated in the book, that should be obvious: to make the book fully self contained in order to ease the transition to go beyond these mathematical tools in later chapters by literally replacing them and/or integrating them at a conceptual level - from the perspective of pure mathematics - with more advanced mathematical tools from pure mathematics which have not necessarily found applications yet. This is largely the same reason that differential forms and exterior calculus are taught in MTW in order to logically make the way for the spinor calculus and Regge calculus.

More generally speaking, to optimize the education of students aspiring to go into mathematical physics, it seems to be preferable that they not be taught textbook QM in the manner of typical physics curricula, but instead get taught mathematical QM immediately as soon as possible in a special track, when the other students are getting taught within the standard curriculum; the development of such a special track requires a textbook such as that of Woit and a professor such as Woit capable of seeing this bigger picture, i.e. preferably a mathematical physicist. It is the 'one size fits all' mentality of education what has been detrimental to progress in foundations of QM and so progress in theoretical physics more generally: the 'one size fits all' perspective is neither necessary nor sufficient for doing mathematical physics.

In order to illustrate this, at the universities where I work, in mathematics education seen over many decades there have arisen specialized tracks in the undergraduate mathematics degree that teach the math freshmen calculus in three different ways, formalized into three separate curriculum tracks: 1) standard calculus for the 'applied math' oriented students together with the physics students, 2) analysis for the 'pure math' oriented students, and 3) analysis via differential forms for the 'pure math' oriented students who actually want to do research in the theory of analysis.

Moreover, it has actually been demonstrated that in the long run of decades, encouraging mathematics and physics students early on to generalize their basic knowledge (e.g. calculus I, II & III), not just into some special case (e.g. tensor calculus) but more broadly in a general sense is good for science in the long run e.g. being able to see the theory of analysis as an incomplete theory in mathematics which can be extended in several non-equivalent ways, which can each subsume entire branches of applied mathematics and physics. The sufficiently interested and skilled students may actually automatically reinvent several branches of higher mathematics without necessarily realizing that they have done so and then typically meet their reinvention back later in their career within the vast literature.

Systematically teaching in such a manner - i.e. not merely in order for the students to be able to quickly master techniques and pass exams - can quickly give the students learning analysis a very mature and intimate perspective of mathematics, at least if they are receptive and also able to follow without getting lost. In practice this requires world class educators and then still most students are incapable and feel lost and yearn to get boxed in again, but not all students, i.e. this manner of teaching automatically has as a side effect the development of an inquisitive mindset at a sophisticated level in a few students.

By continually challenging the students who want to get 'boxed in' within a certain course - in order to safely maximize their mastery and so their grades - in a similar manner during their entire degree, the hope is then that they still develop such a mindset in the long run. All in all, this more challenging method of education contributes not only to the stimulation of charting new territory by those who are able to follow, but also increases the likelihood of acceptance and application of known but unconventional mathematics in order to overturn existing physical theory; this is of course the main goal of mathematical physics, with the best historical example the development of Lagrangian and Hamiltonian mechanics, simply by pushing against Newtonian mechanics and calculus as being necessary and sufficient.

 

[vanhees71]

I think the one topic which is not covered well enough by the standard university curriculum is Lie-group theory. You indeed learn it by "osmosis" or when you get interested enough by reading textbooks. The main problem with the math literature is that it is usually written in the "Bourbaki style", which is not easy to translate to the physicists's needs. A pretty good book on this level, also readable for the math-inclined physicists is Hassani, Mathematical Physics.

Here older books are much better. Some may lack mathematical rigor, but that's fine for physicists who want to apply it.

The simplifying trick is to just discuss matrix groups (subgroups of GL(n)) and also there the most simple ones, appearing in physics like SO(n), SU(n), SO(1,3), SL(2,C), ISO(3), Poincare group, maybe also symplectic groups and also some finite groups (e.g., for crystallography).

I learned representation theory from vol. 3.2 by Smirnow, van der Waerden, Hamermesh and some other sources. Also the corresponding parts of Weinberg, Quantum Theory of Fields vols. 1+2 is fine, but it's pretty brief. So it's better to have some pre-knowledge to better understand it. A very good physics book is also Sexl, Urbandtke, Relativity, groups, particles.

 

[Dr Transport]

I think the text by Wu Ki Tung covers Lie groups fairly well for a budding high energy theorist.

https://www.amazon.com/dp/9971966573/?tag=pfamazon01-20

 

[CJ2116]

I Just started working through Understanding Analysis by Stephen Abbott. I took a year of real analysis in college about 8 years ago using Rudin and struggled immensely. I was kind of curious how I would fair coming back to it years later, with enough detachment from my experiences years earlier. So far, in working carefully through the first chapter I already have a way better understanding of what a proof is, common approaches to them and how to write one.

I'm also reading Analytical Mechanics by Nivaldo Lemos. I'm finding it good to have two different subjects going at the same time in case I get frustrated on one of them!

 

[member 587159]

I Just started working through Understanding Analysis by Stephen Abbott. I took a year of real analysis in college about 8 years ago using Rudin and struggled immensely. I was kind of curious how I would fair coming back to it years later, with enough detachment from my experiences years earlier. So far, in working carefully through the first chapter I already have a way better understanding of what a proof is, common approaches to them and how to write one.

I'm also reading Analytical Mechanics by Nivaldo Lemos. I'm finding it good to have two different subjects going at the same time in case I get frustrated on one of them!

Isn't Rudin-> Abott an immense downgrade? If you had a course with Rudin,then you should know what a proof is and Abott should be too easy. What kind of course was it?

 

[CJ2116]

Isn't Rudin-> Abott an immense downgrade? If you had a course with Rudin,then you should know what a proof is and Abott should be too easy. What kind of course was it?

In theory, yes it is a huge downgrade in terms of how rigorous it is. A lot of the issues I had were that I took the course sequence as a sophomore with only calculus and linear algebra under my belt. I definitely didn't have too much mathematical maturity at the time and survived it by just brute force memorization of definitions/theorems. It was a normal course - I did end up with C grades at the end of both sequences!

It's funny because I can still pull a lot of my old course textbooks off of my shelf and more or less remember the topic and how to solve some of the problems, but during and after the course with Rudin I still felt like I just couldn't internalize anything in that book!

 

[member 587159]

In theory, yes it is a huge downgrade in terms of how rigorous it is. A lot of the issues I had were that I took the course sequence as a sophomore with only calculus and linear algebra under my belt. I definitely didn't have too much mathematical maturity at the time and survived it by just brute force memorization of definitions/theorems. It was a normal course - I did end up with C grades at the end of both sequences!

It's funny because I can still pull a lot of my old course textbooks off of my shelf and more or less remember the topic and how to solve some of the problems, but during and after the course with Rudin I still felt like I just couldn't internalize anything in that book!

Yeah, that's the effect of Rudin's books. They are written for someone who already has an introduction to the topics, not to learn from.

 

[Borg]

Currently reading a free online book Kalman and Bayesian Filters in Python that comes with a complete Github repo. I have to come up with a topic for AI courses that I'm taking and this looks interesting.

 

[Mondayman]

I found three physics books that look pretty good. I ordered the first two.

Night Thoughts of a Classical Physicist by Russell McCormmach. About a physicist used to the classical world and the development of quantum mechanics.

Deep Down Things by Bruce Schumm. Book about particle physics.

Nuclear Forces: The Making of the Physicist Hans Bethe by Silvan Schweber. Book about Hens Bethe's work. Typical Schweber, very technical with lots of mathematics.

 

[CJ2116]

I've been buying a lot of math books the last few months (John M. Lee's Introduction to Smooth Manifolds and Introduction to Riemannian Manifolds, Gadea Et al. Analysis and Algebra on Differentiable Manifolds, Munkres' Analysis on Manifolds, Axler's Linear Algebra Done Right)

Two that I'm currently reading and have found very helpful are:

Lara Alcock's How to Think About Analysis: I wouldn't say that there's anything particularly deep here in terms of mathematical content, but what I've found extremely helpful are the chapters on the studies of how students learn (or don't learn) real analysis. I've started using some of them, such as creating flow charts between definitions and theorems and also keeping them in a separate and easy to navigate notebook for quick study/reference.

Daniel Velleman's How to Prove It: I've been working in data science/web development/programming for about 8 years and his approach of treating proofs like a structured computer program really clicked with me. I'm not too far into this, but like it so far!

 

[mathwonk]

@CJ2116: I just took a look at that part of Abbott's real analysis book that is visible on amazon and compared it to baby Rudin. Abbott was at first much too verbose and chatty for my taste, but when I located and read some of the definitions and proofs of theorems, such as countability of the rationals, and uncountability of the reals, I found it equally as rigorous as Rudin, but far more clear and user friendly, while giving essentially the same arguments. So I do not consider Abbott much of a come - down at all in terms of rigor, just a vast enhancement in clarity.

Both of them e.g. define a function as a vague "rule" or "association" between elements of sets, neither one making it entirely precise by introducing the notion of the graph. The nice proof of uncountability of the reals via the shrinking intervals property of the reals, is one I had not seen before, and is in both. I thought the explanation of it in Abbott was much clearer and equally rigorous as the same argument in Rudin. Both arguments for the countability of the rationals seemed about equally, and not totally, rigorous in both, but again clearer in Abbott. The nice avoidance of the problem of repeated occurrences of the same rational in the set of pairs of integers was beautifully dealt with in Abbott, if slightly colloquially, and also in essentially the same way in Rudin, with a bit less clarity, but still without complete precision in my opinion. (In Rudin the slightly incomplete part is in the proof of the result 2.8 that an infinite subset of a countable set is also countable, where he asserts but does not prove that his construction gives a bijection. In Abbott the slightly colloquial part is where he argues that a countable union of finite sets is countable by just displaying them side by side, in order of index, rather than giving an inductive definition of the enumeration. Some may prefer Rudin's more precise inductive definition of his bijection, but since he omits the proof that it actually is a bijection, he seems to become guilty of the same incompleteness of rigor. ... Aha, I have read further in Abbott and found exactly the same inductive definition for the same lemma of Rudin, as an exercise in Abbott. So what Rudin omits, Abbott gives explicitly as an exercise. So which is more rigorous, omitting an argument without comment, or assigning it as an exercise? And neither of them even state the crucial well ordering property of the integers, although both use it in this proof.)

OK here is a comparison for you to assess the relative clarity and ease of reading of essentially the same proof in both books: compare the proof of Thm. 2.43 in Rudin, with Th. 1.56, pp 28-9 of Abbott.

I have not read anywhere near all of Abbott of course and may have gotten my favorable impression from only a small, possibly unrepresentative, sample. It may also be that Abbott's arguments look rigorous to me because I know how to fill in the details. I still feel, even as a retired senior mathematician, that Rudin is not a welcome source for either learning or even summarizing the material. For me, with over 40 years teaching and research experience past my PhD, I thought that by now Rudin would indeed read like a useful quick summary of the facts, but even today it was still less readable even for me, than Abbott. In particular, a summary should be clear. I have heard from people who like Rudin, but I never recommend it to anyone for learning analysis.

To be fair, I have written a book with similar failings myself, on linear algebra, where my goal was to see how briefly I could cover the material. I managed to provide more theoretical "coverage" than is in many introductory linear algebra books, and in only 15 pages, and posted links to it here. (I later expanded it to 120 pages, still short by usual standards.) I think Rudin also was trying to see just how succinctly he could summarize the material, whereas Abbott was trying to teach it. Abbott went too far, for me, in his voluble writing style, but students seem to like it, in contrast to Rudin.

My point is, even if Abbott's proofs look less formal than Rudin's, in my opinion they are reasonably rigorous, and I suspect you will learn more from it than you did from Rudin. One difference is the coverage is greater in Rudin. I.e. in a bit over 300 pages, Abbott covers only the material in the first 2/3 of Rudins 330 pages. But for that one variable material, I recommend Abbott.

Please forgive me for always jumping into every discussion of Rudin on the "against" side. I had to teach out of that thing, maybe that is the source of my frustration. But I like Dieudonne', which is even more condensed, but I think much better and more insightful. I.e. I learned things from it, (even if my class may not have).

One more remark, this time in Rudin's favor. The fact that Rudin has been around for so long, and his arguments are solidly rigorous, means that later writers can borrow from his material in substance, and work at improving the presentation. So in a way, many later analysis books, may be something of reworkings of Rudin, or if not, at least they had Rudin available for consultation if they chose. Of course Abbott says explicitly that it is the book of Bartle to which he owes his own education. I am not as familiar with that one. The books appeared in roughly 1964 (Rudin), 1983 (Bartle), and 2001 (Abbott), so each of the last two authors at least had the option to have access to Rudin.

One reason I realized this was the fact that Abbott has the same argument, via intersections of shrinking compact sets, for uncountability of the reals, as Rudin does, and I thought I had not seen it before. Actually I realize after thinking about it, it is basically the exact same idea as Cantor's diagonal argument that I had seen in high school, where he constructs a decimal that is not anyone of those in a given list. Namely you determine a real number by a sequence of things, and you arrange successively each subsequent one of those things so as to rule out the nth listed number, eventually ruling them all out from your list. It does not matter whether the sequence is a sequence of decimals digits or a sequence of compact intervals. In fact a sequence of decimals can obviously be viewed as a sequence of intervals, with the nth one of length 10^(-n). I.e. the decimal number π = 3.141592653589793..., is the intersection of the intervals [3,4], [3.1,3.2], [3.14, 3.15],... So even though I hadn't seen it phrased this way, there is no new idea other than Cantor's original one. Thus it was not necessary to have read Rudin to come up with this argument, which is probably very old.

 

[jasonRF]

I recently picked up the 6-volume Lectures on Theoretical Physics by Sommerfeld. Vanhees71 has recommended them on so many occasions that when I saw a complete used set for < $60 (US) I couldn't resist.

I am an electrical engineer, and in the near future am most interested in the volumes on Electrodynamics, Optics and Partial Differential Equations in Physics. I'm starting with the portions on electromagnetic waves, HF waves on imperfect conductors, waves on wires, radiation above a ground plane and diffraction. Of course some of this material was originally developed by Sommerfeld, so it is interesting to read his take on it. I do love how some of these older physics texts are full of material that is often in the domain of the modern engineer. While reading his discussion of the impedance of plane and cylindrical conductors I couldn't help but think back on when I first worked through this material as a 3rd year engineering student. Eventually I will probably go through at least portions of the other volumes; my graduate work was in plasma physics so I'm particularly curious about his treatment of kinetic theory, as well as the theory of waves, shocks and turbulence in fluids.

So far I am really appreciating how straightforward the presentation is - no unnecessary abstractions or generalizations, using the required math but no more, and clearly communicating his physical and mathematical reasoning. He actually takes time to discuss the results he derives and offers his insight into the underlying physics. I expect hours working through sections of these books will be time well spent.

Jason

 

[CJ2116]

@mathwonk

Thanks for the detailed response! Before I bought this book, I read through a few reviews of people who learned analysis from this, but it's helpful (and reassuring!) to get perspective from a mathematician who has taught this subject for a long time time.

 

[weirdoguy]

So, I just got my hands on this book: Constructing Quantum Mechanics by Anthony Duncan and Michel Jenssen. Duncan wrote a wonderful book on QFT (The Conceptual Framework of Quantum Field Theory) and in its first two chapters he goes through some aspects of history of QM/QFT with quite a lot of detailed derivations. I lived for it then and now I'm excited to see even more of this

 

[Dr Transport]

I picked up two gems for ~$25, Theory of Electromagnetic Waves by Kong and Variational Techniques in Electromagnetism by Cairo & Kahan.

Both look pretty good, Kong treats electromagnetic materials in a moving frame immediately, which gives insight into materials becoming bianisotropic immediately. I think it's going to take time for me to digest it but worth the investment.