Share self-studying mathematics tips

[faiziqb12]

Yes, Rudin is a difficult book. It's not really suitable for self-study

difficulty is due to low IQ and matametical intution
dont forget that the book is meant for you and not the professors

 

[bacte2013]

Franco, if I understood you right, you are aware of the fact that mathematics requires to be tuned to mathematical thinking. About the course of Analysis 1 from Terence Tao of the UCLA he himself comments that by following his scheme in his course with honours his students start with less abstract and new concepts like those dealing with natural numbers i.e. to learn the mathematical thinking and its application to solve the mathematical proves. So his students the first couple of weeks advance less fast than those students of the "normal Analysis 1" course but later they catch up and pass those students due to have had the learning of mathematical thinking and its application to tasks!

The books like "Analysis I" by Terrence Tao and "Numbers and Functions" by R. Burn focus on teaching the construction of real number system and developing how to apply the real number system to the real analysis. Both books are incredibly strong books, but I think first few chapters from both books are enough to devel the mathematical thinking and the understanding of real number system. Another good book, but one I do not like that much, is "The Real Numbers and Real Analysis" by Ethan Bloch. He has a same philosophy as Tao and Burn, but Bloch's treatment already assumes the mathematical maturity from prospective readers, and he also does everything quite rigorously.

 

[Hellmut1956]

Well, I believe and it is my personal opinion that most of us probably always will always have room to improve mathematical thinking. But Analysis and Linear Algebra are fundamental basics. So far I have reached the opinion that all of the mathematics you learn as part of a bachelor study besides learning mathematical thinking are just learning a toolbox so to be able to really deal with mathematics. This even applies for at least part of the master study courses. Once you are through your mathematical toolbox will help you to know which tool in the bos of techniques you will have learned is applicable to a specific question you might deal with!

 

[Primrose]

Hello, I am self-studying mathematics in English. Just finished high school and Rudin is so much painful. However, at Uni my courses are in French. Do you think I should look for French textbooks?

 

[micromass]

Hello, I am self-studying mathematics in English. Just finished high school and Rudin is so much painful. However, at Uni my courses are in French. Do you think I should look for French textbooks?

Reading math in English is a skill you're going to have to master eventually. Most advanced books and advanced papers nowadays are English. When you write a paper to publish it, you will have to do it in English. When you have to give an international talk, it will have to happen in English. So you're going to have to get good in communicating math in English anyway.
So if you really feel uncomfortable with English language books, then sure, go search for good French books. But know that there is a huge variety of good English analysis books out there, while there are not so many French books.

 

[Primrose]

Thank you so much Micromass. I will do my best to master both.

 

[emma watson]

Are you self-studying mathematics? Do you have any questions on how to handle it? Anything you want to share? Do so here!

Mathematics is a vast and constantly expanding discipline, with numerous major subject divisions such as algebra, geometry, analysis, topology and hundreds of subdivisions. Just as with languages, different branches of mathematics may have different degrees of usefulness to you, or different aesthetic qualities in terms of the beauty of their central ideas.

So which should you select? To sharpen your focus on just those areas that might be of interested and relevance to you.

 

[Ahmad Kishki]

Certainly, but don't like post 10 questions at once. Only post like 3 questions at once and more questions if they get resolved.

In my opinion, proofs can be learned best by letting somebody critique your proof. So ask somebody to rip apart your proof completely. It is really the only way to learn. Watching somebody else's proof doesn't teach you much. Computational problems are very different though.

please rip apart this proof for me: https://www.physicsforums.com/threads/closed-set-proof.830944/

i am self studying real analysis fro Understanding Analysis by Stephen Abbott, and i must say, i am having the time of my life!

 

[Ahmad Kishki]

I am currently self-studying 6 subjects at a time. But I'm a bit extreme. I think 3 should be a decent number.

wow! i can barely manage 1 subject! but i get so consumed mentally in the subject, i just can't think of anything else. How do u manage 6 subjects?

 

[ohwilleke]

I have much advise, but I don't really know what you're looking for. But here's some things I would have liked to hear:

1) Get in touch with the profs. Many profs are more approachable than you think (while some are absolutely not!). Get to know them, go to office hours, ask questions, etc. And I don't (only) mean to talk about the class, but talk about other things in physics/math too.

2) Don't be discouraged by your class mates. While in undergrad, and while teaching undergrad I have seen many classes with bright students, but with an atmosphere that is very bad. Many would care about the grades only, and others openly disliked the courses. This reflected on the entire class. Don't let yourself be discouraged by them.

3) Don't care about your grades (only). I have mentioned this before in point (1), but there is more than grades. Grades =/= understanding (although there is a correlation). Focus on understanding the topic, not only on getting good grades.

4) Think before you ask a question. Don't just go to a prof and start asking a lot of questions without first thinking about it for a long time. Of course, if you REALLY don't know, then ask the prof and don't be afraid to do so. But it is worthwhile to think things through first.

5) Be sure to have fun too. Life isn't only about learning.

This is really good advice, especially (1), (4) and (5). Even if you have no formal affiliation with a university, most profs truly enjoy spending a little time with an earnest young math scholar who is talented and is asking thoughtful questions. Most of the biggest names in mathematics and physics (e.g. Nambu, Einstein) had a few people that they had this kind of relationship with, and it does wonders for the isolation you can feel working away for whole courses with little human interaction as well. And, many famous people in these field (e.g. Emily Noether, Oliver Heaviside, Leonhard Euler, and Srinivasa Ramanujan) were on the student side of these kinds of relationships at some point in their lives. These kinds of people can also make excellent references for college or graduate school.

 

[ohwilleke]

Emma Watson's observation flows from something more fundamental. Math is a very mature discipline. There is almost nothing in the mathematics curriculum even up to the 500 level graduate curriculum (with a handful of isolated exceptions such as fractals and certain kinds of optimization problems in linear algebra) that wouldn't have been familiar to someone like Euler, hundreds of years ago.

Physics isn't quite as mature, but it is close. Classical electromagnetism is about 125 years old, and classical mechanics, Newtonian gravity and first year calculus are about 350 years old. Even pure General Relativity hasn't changed much in the last hundred years, although there have been some advances in cosmology and our understanding of black holes based upon it. Obviously, there have been some new discoveries made in physics more recently, mostly in high energy/quantum physics, optics and condensed matter physics. But even there, the Standard Model is more than 40 years old, except for the fact that neutrinos have mass and the precision with which some of the constants have been measured.

Unless you are studying a field that is very new (e.g. string theory), it isn't important to get hot off the presses texts. Pedagogy most certainly hasn't made any great strides in the last four or five decades (although it does feel a bit lame and depressing to read a book that boldly wonders if man will ever make it to the Moon, or still thinks its trendy to call black holes "frozen stars").

 

[bacte2013]

I have been reading following two books, and I would like to take this chance to recommend them to others.

"Foundations of Analysis" by E. Landau
"A Concrete Approach to Classical Analysis" by M. Muresan.

Landau's book is great to learn the number systems and their construction. He basically give clear proofs to even trivial properties of the numbers. This book is great read before jumping into the analysis texts. I found Muresan a good complement to Rudin as he provide different approach to the proofs and thought-process behind many proofs and definitions. Professor Micromass, I would like to hear your opinion about them if you read them before.

 

[micromass]

Landau is a very good book. It is a classic for good reasons. The book "Real numbers and real analysis" by Bloch is somewhat similar in approach to Landau, but covers more.
I don't know the text by Muresan, but it seems to have some cool and nontraditional topics.

 

[bacte2013]

I really like Landau too. I read portions of the Bloch but I did not like it as much as Landau since Bloch is not concise and clear as Landau (personal opinion and taste). I really regret not reading Landau earlier since I had been facing difficulty with the number systems and their rigorous construction when studying the Rudin and Apostol. Now I finished reading Landau, I have better ideas about how to construct the number systems and implement them to the proofs.

Do you have any recommendation for the introductory books about mathematical logic? I would like to investigate this topic, but I am not sure which will be a good place to start.

 

[Ssnow]

For mathematical Logic I suggest '' Mathematical Logic '' Joseph R. Shoenfield , I think is the best.

 

[micromass]

Check out Enderton or Ebbinghaus
https://www.amazon.com/dp/0387942580/?tag=pfamazon01-20
https://www.amazon.com/dp/0122384520/?tag=pfamazon01-20

 

[bacte2013]

Check out Enderton or Ebbinghaus
https://www.amazon.com/dp/0387942580/?tag=pfamazon01-20
https://www.amazon.com/dp/0122384520/?tag=pfamazon01-20

Hello Professor Micromass, have you read the books "Analysis I-III" by Herbert Amann/Joachim Escher or "A Course in Mathematical Analysis" by Garling? While browsing my university's library, I saw them and they look very interesting. Both books are from European universities, so I thought you might know them. If you do, how are they compared to the mathematical analysis books like Rudin and Apostol? I just reserved them but did not yet take them.

 

[Hellmut1956]

I just found an most interesting video from a mathematics professor at Stanford that at about the second half speaks about how this freely available lectures will change the world:



Prof Keith Devlin from Stanford University speaks about the future of studying and the effect on the presence universities due to this lectures available for free in the Internet. I have failed to convince people in this thread to switch their thinking about "Self-studying mathematics" by asking for the proper book and to make the rational analysis of the benefits of that offering that has become widely accepted since universities like Stanford and MIT do offer those courses for free in the Internet. Have a look at the video!

P.S.: I found this video while investigating about the author of the book: "Introduction to Mathematical Thinking"

 

[rduarte]

What would be a good text for someone who wants to understand limits better? Lang's First Course in Calculus seems to be lacking on that topic.

 

[micromass]

What would be a good text for someone who wants to understand limits better? Lang's First Course in Calculus seems to be lacking on that topic.

Yes, Lang is severely lacking there. Now to fully understand it, you will need an analysis book. But depending on the rigor, there are several options.

On the rather elementary level, I recommend Keisler: https://www.math.wisc.edu/~keisler/calc.html Keisler covers two very different approaches to limits: the standard epsilon-delta approach, and the infinitesimal approach. Both approaches really help understand the concepts.

Somewhat more advanced, there's good books like Nitecki's calculus deconstructed and Apostol's calculus. Those are somewhat closer to being analysis books, but they still qualify as calculus. After that, there's analyis.

 

[rduarte]

Yes, Lang is severely lacking there. Now to fully understand it, you will need an analysis book. But depending on the rigor, there are several options.

On the rather elementary level, I recommend Keisler: https://www.math.wisc.edu/~keisler/calc.html Keisler covers two very different approaches to limits: the standard epsilon-delta approach, and the infinitesimal approach. Both approaches really help understand the concepts.

Somewhat more advanced, there's good books like Nitecki's calculus deconstructed and Apostol's calculus. Those are somewhat closer to being analysis books, but they still qualify as calculus. After that, there's analyis.

Keisler's looks great. Thanks!

 

[KSG4592]

I plan to use Serge Lang as my first calculus book, so this is good to know. Keisler looks like a good book as well - got to love free books!

 

[deskswirl]

I have failed to convince people in this thread to switch their thinking about "Self-studying mathematics"

I like and utilize this modern option. Carl Bender's lectures on Mathematical Physics are great (https://www.perimeterinstitute.ca/video-library/collection/11/12-psi-mathematical-physics). I don't have the patience to sit down and read his text on Asymptotics and Perturbation Theory.

But I haven't totally given up on printed texts as I am also studying Bressoud's A Radical Approach to Real Analysis using Mathematica to plot things of course

 

[AlexOliya]

Hi guys! So I am a university student software engineering major. I absolutely love math and passionate of becoming someone who is fluent in math but not majoring in it education-wise that is.
I would like to know where to start from (imagine giving advice to someone who is an undergraduate in mathematics and doesn't listen in class...)
(by the way I am not zero in math I was a A student in high school but only limited to high school math)
Thanks
Alex

 

[micromass]

Hey AlexOliya. Do you want to discuss this on facebook with me? Feel free to PM me if you do.

 

[AlexOliya]

Hey AlexOliya. Do you want to discuss this on facebook with me? Feel free to PM me if you do.

Is it okay if we discuss this via email? I don't have a Facebook unfortunately.

 

[micromass]

OK, unfortunately. What math do you know already? Where would you like to get eventually mathwise?

 

[AlexOliya]

OK, unfortunately. What math do you know already? Where would you like to get eventually mathwise?

I know High school math tops and a bit of calculus due to the first semester in university and I would like to study such that I could be like a pure mathematician.

 

[micromass]

OK, then you should start by studying calculus and linear algebra. I would recommend the free calculus text by Keisler since it also introduces infinitesimals rigorously. For linear algebra, I recommend you the free text "Linear algebra done wrong" by Treil. Finally, it might be useful to go through Euclid's Elements. Getting through that book is an amazing experience and will teach you a lot of mathematics.

 

[AlexOliya]

OK, then you should start by studying calculus and linear algebra. I would recommend the free calculus text by Keisler since it also introduces infinitesimals rigorously. For linear algebra, I recommend you the free text "Linear algebra done wrong" by Treil. Finally, it might be useful to go through Euclid's Elements. Getting through that book is an amazing experience and will teach you a lot of mathematics.

I will do that. By the way I was studying Calculus I by Apostol. What would you add on that? Should I continue or change plans?

 

[micromass]

Oh, you should probably continue with Apostol then if you enjoy it.

 

[zigzagdoom]

Would anyone know of a hard computational multivariable calculus book (i.e. not a real analysis type proof based book).?

Stuff with hard integration questions or deeper algebraic manipulations would be especially useful. Primary aim is to use the text to study mathematical methods.

Thanks!

 

[mathwonk]

As to the treatment of limits in Lang's First course, I may be wrong, but there is something there, in a somewhat non traditional presentation. I no longer have my copy, but as I recall he assumes in the text that it is possible to define the concept of a limit of a function f(x) being equal to L, as x approaches a, so that the usual rules hold. Then he uses those rules to deduce theorems quite rigorously from that assumption. His stated opinion is that most students do not need to know how limits are actually defined using epsilon and delta, or how to prove the assumed properties from that definition, but for those who do, he does so in an appendix. So one could presumably begin the usual theory of limits by reading that appendix, and if you already have the book, I suggest trying that. As a crude estimate that appendix is 20 pages long, as compared say to the roughly 25 page section on limits in Apostol. Unfortunately I cannot see on amazon search whether in that appendix Lang gives the proofs of the non trivial intermediate value and extreme value theorems (which Apostol does include), but earlier in the book he says he will omit them, since they "belong to the range of ideas" in the appendix. Needless to say one cannot really come to grips with the definition of a limit and continuity unless one sees them used to prove something non trivial.

 

[Loststudent22]

Yes, Lang is severely lacking there. Now to fully understand it, you will need an analysis book. But depending on the rigor, there are several options.

On the rather elementary level, I recommend Keisler: https://www.math.wisc.edu/~keisler/calc.html Keisler covers two very different approaches to limits: the standard epsilon-delta approach, and the infinitesimal approach. Both approaches really help understand the concepts.

Somewhat more advanced, there's good books like Nitecki's calculus deconstructed and Apostol's calculus. Those are somewhat closer to being analysis books, but they still qualify as calculus. After that, there's analyis.

What about Spivak Calculus that seems to be recommend online a lot also for a soft introduction to analysis.

 

[MidgetDwarf]

Would Lay, " An introduction to Analysis," combined with Sherbet: Introduction to Analysis, are suitable books for some someone with no proof writing skills and as a a self study with no instructor/ help? My end goal is to be a Mathematician (Pure).

Or are there better intro books in your experience.