Share self-studying mathematics tips

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

I just started going through Walter Rudin's Real and Complex Analysis. The hardest part for me is that he often says "A clearly follows from B," but I don't see how it clearly follows. After reading the problem in question 3 or 4 times over the span of a few days, I get it. But that takes a lot of time!

What textbooks are the best ?

NewtonsFellow said:

What are the best textbooks ?

That will be something for the following posts :)

EternusVia said:

I just started going through Walter Rudin's Real and Complex Analysis. The hardest part for me is that he often says "A clearly follows from B," but I don't see how it clearly follows. After reading the problem in question 3 or 4 times over the span of a few days, I get it. But that takes a lot of time!

Yes, Rudin is a difficult book. It's not really suitable for self-study because of these things. It's better for a class textbook so the professor can give some extra explanations. But you can of course always ask here if you have a problem with anything.

micromass said:

That will be something for the following posts :)

[emoji4][emoji2]

micromass said:

Yes, Rudin is a difficult book. It's not really suitable for self-study because of these things. It's better for a class textbook so the professor can give some extra explanations. But you can of course always ask here if you have a problem with anything.

From this post and your more extensive one, it seems you've had a lot of experience self-studying. What do you study, and why?

EternusVia said:

From this post and your more extensive one, it seems you've had a lot of experience self-studying. What do you study, and why?

Right now I am studying some probability theory and some analysis. But most of my experience comes from guiding people who self-study. So now I am just writing down my experiences.

Hey Micromass, I don't know whether this will be addressed in your textbook thread so I'll ask here just in case - which physics texts do you recommend for self-study by a prospective (i.e. undergrad) mathematician with an interest in the subject?

I am self-studying linear algebra using Sergei Treil's http://www.math.brown.edu/~treil/papers/LADW/book.pdf. I have to say that, *despite its name* (everyone has to add this one ), it is a wonderful book. I also discovered that I enjoy the abstraction in his approach, especially the treatment of vectors not as "something that has both magnitude and direction" but as elements of a set satisfying some definite axioms - a very enlightening and new approach to me. The only drawback is that there is no solution manual anywhere and in order to get feedback on the validity of my solution/proof I have to extensively search Google to hopefully find a similar problem solved somewhere (and I do not always find). Also, some more problems could be helpful.

On another note, I really like the idea of a thread dedicated to self-study. Great idea as I feel this topic should receive more attention here and in general.

Do you have experience about self-studying physics ?

Hey Micromass, I don't know whether this will be addressed in your textbook thread so I'll ask here just in case - which physics texts do you recommend for self-study by a prospective (i.e. undergrad) mathematician with an interest in the subject?

It really depends on what physics and math you already know. But as a mathematician, I have always enjoyed this book: https://www.amazon.com/dp/0521534097/?tag=pfamazon01-20 I'm sure a physicist will look at these things completely different. For example, many physicists prefer Kleppner: https://www.amazon.com/dp/0070350485/?tag=pfamazon01-20 (be sure to buy the first edition, not the later ones).

ELB27 said:

I am self-studying linear algebra using Sergei Treil's http://www.math.brown.edu/~treil/papers/LADW/book.pdf. I have to say that, *despite its name* (everyone has to add this one ), it is a wonderful book. I also discovered that I enjoy the abstraction in his approach, especially the treatment of vectors not as "something that has both magnitude and direction" but as elements of a set satisfying some definite axioms - a very enlightening and new approach to me. The only drawback is that there is no solution manual anywhere and in order to get feedback on the validity of my solution/proof I have to extensively search Google to hopefully find a similar problem solved somewhere (and I do not always find). Also, some more problems could be helpful.

On another note, I really like the idea of a thread dedicated to self-study. Great idea as I feel this topic should receive more attention here and in general.

LADW is an extremely good text. It contains about everything one should know about linear algebra, and he does it the way I would do it. Not that it matters to me, but the book is completely free which is awesome.

Why don't you post the problems here on PF? Wouldn't that be easier for you?

I agree his text could use some more problems. I like text with a lot of problems.

Do you have experience about self-studying physics ?

No, I do not. Hence why my guide is only about mathematics. Although I'm sure many tips also hold true for physics.

micromass said:

Why don't you post the problems here on PF? Wouldn't that be easier for you?

Definitely. It's just that often it takes time to write these posts. I should probably do so more often though (can I shamelessly bombard the questions section with lots of small problems?)

ELB27 said:

can I shamelessly bombard the questions section with lots of small problems?

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.

Apologies if this is slightly off-topic but what would you say helped you most in getting to grips with the nature of mathematical proof? Was there a particular class or text you can pinpoint as being of critical importance? Did it just come to you with time, experience and growth in mathematical maturity? Or were you one of those very lucky few who seem to be born with an innate understanding of mathematics and her methods? ;)

micromass said:

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.

Thanks for the good advice! I will be sure to start posting my proofs here.

IDValour said:

Apologies if this is slightly off-topic but what would you say helped you most in getting to grips with the nature of mathematical proof? Was there a particular class or text you can pinpoint as being of critical importance? Did it just come to you with time, experience and growth in mathematical maturity? Or were you one of those very lucky few who seem to be born with an innate understanding of mathematics and her methods? ;)

It's very tricky to learn it well. You can always read a proof book, but I don't like that option very much. Much better is finding somebody who is willing to critique your proofs. That way, you can start any math book (like analysis, algebra, discrete math) and start doing proofs. First you will suck, but if you keep asking for advice, then your proof abilities will get better fast. After a short while, you'll be very good at it.

Dear friends, as it was nearly 4 decades since I studied mathematics as part of my german high school and as part of the mechanical engineers study my mathematical abilities have strongly eroded and besides that mathematics has had quite a development in this time. I am surprised reading the contributions to this thread totally ignore what I consider the most valuable resource available for self study, not just in mathematics! In many countries around the world universities are making their courses available in the Internet for free. This has the advantage that you can choose a lecture from a professor whose style fits best to your personal learning preferences. For engölisch speaking people like you in this forum I would highlight the offering from the MIT in Boston through its program "OpenCourseWare". You can search through the courses offered, all for free by going to this place! I even prefer to go to this place, where courses are listed by course number, where Mathematics appears under department 18. If you go to department 18 on the left most column of the table and select it by clicking on it, you find the course numbers listed on the center column and on the bottom half of the screen a scrollable list of all the courses availble. If you focus on those that have the letters "SC" at the end of the course number you find the most complete offerings for self paced courses. To get my eroded mathematical skills up to speed I have chosen to go through the courses of "Calculus Single variable, 18.01SC and Calculus Multivariable, 18.02SC. Clicking in the course 18.01SC on the right column you see that the course is as taught in fall 2010! Clicking on the "RESULTS" offered below you get here! Same is true for 18.02SC where I offer you the link to here! Similar by the way can be found for physics courses! get a view of what the courses offer, I believe excelent videos of the lectures and assignments, excellent reading and exercises in the book to read, which is also available for free from Gilbert Strang, the professor who offers an excellent lecture about "Linear Algebra, also offered here, whose recorded course was held in 2011! Analysis 1 and 2 I prefer it following the book from "Terence tao", on whose personal page in the Internet you get download the books that are the reading for the Analysis course with honors he teaches on the UCLA! As video recorded lecture I personally prefer the recorded lecture from a german Professor, Groh, who teaches at the university in the city of Tübingen, Germany, but following the books of Terence Tao.

How many subjects do you like to self study at a given time? Do you focus on one subject or a few at a time?

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

As you might have noticed I have a couple of years I am carrying around. My main objective is my model sailboat project and within this context right now the design of a sheet control system I have developed a concept for. I want to apply the methodology of design by modeling. To do so several areas of knowledge need to be combined. So I do pursue my goals in a process were I keep learning what I feel will be of help to accomplish my goals! I do neither have the need to achieve a result within a given period of time as it would be obvious when working in the industry, nor do I have the goal to get academic titles!

Now due to the fact that I have had a successful career in the semiconductor and telecommunication industry I am used to do what is called "out-of-the-box-thinking" or applying a style known as "not-by-the-books"! So I started studying calculus to refresh my skills in this area, I look into "Linear Algebra" when I do need mathematical techniques taught there, and so on. To achieve my goals I need to combine skills from mechanical engineering, of electronics, mathematics and physics and combine this to grasp what is being teached in the context of "System Dynamics" and what is called "System Physics" to be able to use tools that help to put the relevant issues in context!

EternusVia said:

I just started going through Walter Rudin's Real and Complex Analysis. The hardest part for me is that he often says "A clearly follows from B," but I don't see how it clearly follows. After reading the problem in question 3 or 4 times over the span of a few days, I get it. But that takes a lot of time!

I have used a few of the Rudin books in classes, and I suggest that you read the exposition quickly the first time, marking but not dwelling on roadblocks like the one you mentioned above. Then as you do the exercises, go back to the exposition and study a proof further if you need some of the techniques used to solve a particular problem.

The problems are well-chosen and diverse enough to provide a good understanding of which techniques are the most important in each chapter; the exposition might give the impression that every word and detail is equal, which is not true. After doing a few of the (easier) problems, you may realize why "A clearly follows from B", or you might find that it's a question you can safely set aside for the future.

When are going to post the textbooks thread?
Take your time, I'd just like to know. [emoji2]

I will go back to school in September to get a degree in physics (bachelor + master if everything works as expected).

For the moment, I am re-learning the maths needed to not suffer too much the first year.

I am almost done with Khan Academy (everything is "mastered", I just need to finish a few exercises about series). When it is done, I'll create cheatsheets and notes in LaTeX with tips about some of the problems I had during the exercices of Khan Academy.

What should be my next step ?
Continue learning mathematics with Mary Boas' book called "Mathematical Methods in the Physical Sciences" (and update my notes) or switch to physics with the book "Fundamental of Physics" by Halliday and Resnick ?

I also have Spivak's Calculus but I think I am not mature enough with maths yet.

Thanks.

If you want something complementary to the textbook you're using you can always try to find video lectures of the subject you're studying, online. MIT OCW has some nice lecture series on linear algebra and calculus, I think. However, they seem to follow the "required" textbook for the course fairly close so it may not be a good idea if you're using a completely different textbook. I'm enrolled in a university so I don't really need to self study in the meaning that's addressed in this thread. Nevertheless, I found Prof. Strang's lectures in linear algebra on MIT OCW very informative and nice so I started to skip my own classes and watched the video lectures instead, along with getting the assigned textbook (which was the one by Gilbert Strang himself). The result of this was OK bot not excellent, I managed to get a B in the class; I was not prepared for the hardest problems in the exam but overall I think I have a decent understanding for a beginner on the subject.

My arrangement was as follows:

1) Skim through the relevant sections of the textbooks.
2) Watch a video lecture and take notes. I often paused the lecture whenever I felt the need for it, to think about stuff I didn't understand, to try to prove some statement on my own, to look something up in the textbook, etc...
3) Read my notes from the lectures, for repetition.
4) Read the relevant sections and taking notes whenever needed.
5) Do the assigned problems for the class plus some other ones that I found interesting and/or challenging.

Maybe this is just common sense stuff to do but it can't hurt to share in case someone is interested.

(BTW, sorry for any language errors, English is not my native.)

I just finished https://www.amazon.com/dp/0471827223/?tag=pfamazon01-20 (a fine recommendation from Mathwonk, if I remember correctly) a while back, and have been wondering what good follow up books there are out there; preferably ones I can buy for cheap on amazon. I found my dad's old https://www.amazon.com/dp/0070611750/?tag=pfamazon01-20, by Stein in the basement. Skimming through it, it doesn't look completely out of my league, but I'd rather hear if anyone has any recommendations on what someone in my position should read next.

I'm a big fan of Lang's "First course in calculus". It's one of my favorite mathematics books and it contains a lot of good stuff. You can also get a more serious book like nitecki: https://books.google.be/books?id=dy...MQ6AEwBg#v=onepage&q=Nitecki calculus&f=false

one suggestion on self studying theoretical mathematics is to read the statement of a theorem and then try to prove it yourself before reading the proof. I seldom succeed at finding a complete proof but I often do generate at least some idea that turns out to have a relation to the method of proof. Ot at least I clarify to myself just what is at issue. this makes the proof reading go easier and gives a mental boost. of course it takes time. in my opinion, there is no point in trying to hurry learning math - it simply cannot be done. if one hurries, one just learns it less well. so i recommend to take pleasure in whatever one does learn in the allotted time - don't try to learn a significant topic in a fixed amount of time. extra time spent thinking about math is never wasted.

I self studied a lot of mathematics. I'd finished Algebra I and II in junior high school, and then got bored in high school geometry, but had a good textbook. So, I stopped showing up to class and taught myself geometry, precalculus/trig, three semesters of Calculus, college level linear algebra, and the discrete mathematics before starting college. Later on, in college, I taught myself physics topics that weren't covered in a calculus based physics course I took in college for a month during a winter term. I also self-taught Accounting 101 (Financial) and 102 (Managerial).

I would simply go to the college book store, look at the available texts, pick the one that I liked the best after a cursory review, and started working. I must have worked out something with the administration to accommodate this unorthodox approach, but in retrospect I don't recall it being a big deal. I'd get a free period where math was supposed to be, go to the library or some other uninhabited classroom and work. I don't recall ever taking any tests or having any interactions at all with faculty of any kind, and don't recall what it looked like on my transcript, although there must have been some independent study entry and I must have checked in with a math teacher once or twice a year in brief sessions that I don't recall at all.

Since I was so disconnected from the school system in math, I worked continuously, during the year and during the summer at about the same intensity all year around. I never worried about pacing or how much progress I was making. I just worked at it regularly a few hours every week, like lots of kids practice their instruments in orchestra or learning piano every week, but without any lessons

One of the key points for me was to do every single problem in the book that had an answer in the back of the book, and to keep working on those problems until I had satisfactorily solved every single one of them and understood why before moving on. In a classroom setting, you can get by getting 80%-90% of homework correct, but when you're doing it yourself, you really need a sound foundation to build upon at every step. I also always worked out intermediate steps in the presentation in the book, even if they were skipped in the text.

I tried to find someplace, usually a library, to study quietly and studied regularly, but there wasn't any magic to it, and I didn't take all that many notes.

The hardest points are when you get stuck because some conceptual leap is not clearly elucidated in the textbook. Sometimes I'd spend a week or two puzzling over one section where I just didn't get what they were getting at until finally I got it. This is especially a problem when a textbook introduces notation without fully explaining it. I just had to tough it out, but it would have been immensely easier if I'd had someone to go to that I could have asked about the steps where I got stuck. But, somehow, I always worked it out in the end. The key is to stay calm and analytical and to keep patiently working away at each problem until you solve it.

With regard to proofs, I learned that in geometry and it never seemed difficult after that. With regard to how many topics at once, I would strongly recommend studying one mathematics subject at a time, rather than more than one.

I learned the material more solidly than most people who took ordinary classes as a result, but going back to getting classroom instruction in mathematics after not doing that for four straight years was a shock and took some getting used to in terms of the rhythm of studying, etc. I'm also sure that there are mathematical terms that I probably mispronounce because I've never heard the words spoken aloud. On the other hand, often the instructors confuse things as much as they clarify in lower level mathematics courses. This is less true, however, at higher levels (300s and up) where the textbooks are often not as polished because editors are less qualified to catch mistakes and not as many people use the books and provide feedback.

I did a few other quirky things at times. My senior year in high school, I took calculus based physics, which is easy once you've mastered calculus because it is mostly just math warmed over, but couldn't afford a scientific calculator, so I wrote the infinite series approximations for the trig functions and carried a photocopy of a log table with me instead and worked out the answers that way.

I'm sufficiently old (I graduated from high school in 1989) that I suspect that none of the books I used are around any more. I didn't watch videos or take online classes, because the Internet pretty much didn't exist then, or at least, it wasn't available to me as a high school student. I don't think I used a computer or graphing calculator for anything math related at all until my second or third year of college.

Ultimately, I ended up majoring in math because, duh, I'd already finished the first two years of coursework for majors before I set foot on campus, and made most of my money tutoring people in math and other quantitative subjects, and grading papers. Then, I sold out and became a lawyer.

ohwilleke, that was a really cool story. Thanks for sharing

I found the original thread of micromass very useful, as it contains a lot of good advice for people willing to self studying math.
Personally, I would be very pleased to see a similar thread by micromass giving advice on how to choose a good textbook when self-studying math.

I believe that the typical default lists of supposedly "good" math books do not work for everyone.
From my own experience, whenever I want to learn a new subject, I always seem to find it difficult to identify a book that, for instance, suits my needs, contains just the level of formalism that I can easily digest, contains useful exercises, and things like that.

It is usually a personal choice, and it is not an easy one to do (at least for me).
General advice in this regard would be very very welcome.

mnb96 said:

I believe that the typical default lists of supposedly "good" math books do not work for everyone.

I know very well that they don't. Like you said, textbooks are a very personal thing. I will need to think on general guidelines on how to choose a textbook. Thank you for your suggestion!

In terms of selecting textbooks, my approach is old school, but works reasonably well.

1. Go in person to your local college book store and narrow your choice to textbooks that a professor at some local college or university deemed good enough to assign to his students. This narrows your choices to typically 1-4 textbooks.
2. Exclude textbooks that don't have a significant answer set at the back.
3. Take serious time (maybe 30 to 45 minutes) examining the choices in detail and imagining yourself trying to understand concepts and do problems in a fairly early part of the book. Then choose one and don't look back.

ohwilleke said:

2. Exclude textbooks that don't have a significant answer set at the back.

That eliminates some pretty terrific textbooks and won't work for many advanced subjects.

What are your opinions of textbooks that do not have the solutions to the problems? I wanted to read through Kiselev's Geometry book 1 and 2 this summer but the book does not contain solutions to any of the problems.