Share self-studying mathematics tips

[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.

 

[micromass]

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.

If you have no proof writing skills, then it is very dangerous to do analysis completely by yourself. I really recommend you to find somebody who can help you. The danger is that you will write proofs that are wrong, inefficient and ill-structured. This happens to everybody. If you have no help/tutor/instructor, then you will not receive the feedback necessary to really master analysis. Compared to linear algebra, calculus or geometry, analysis is very very subtle and it is devilishly easy to make mistakes somewhere. If nobody criticizes your proofs, then you will not learn efficiently, or even worse: you will learn wrong things.

That said, if you really don't find anybody to help you, then you should find books which make the transition as smoothly as possible. Lay is a decent book. I think there are better books out there. But if you're completely on your own, then books like this will serve you well.

Good luck!

 

[Hellmut1956]

Even as evidently nobody takes the time to see what i have answered, maybe somebody someday will see what I am writing. The link to the introduction video of the Stanford university I have given above addresses the issue that is the key difference between doing mathematics as it is taught and learned at high school and thinking as a mathematician, as it is required to deal with university mathematics. I will not summarize what he writes in his book or lectures in the recorded introduction to mathematical thinking. That sources are superior to whatever I could summarize. Addressing another difference between the doing mathematics as it is taught at high school and thinking mathematically, as it is required to really embrace mathematics of a university level, it is engineering mathematics the other perspective on mathematics. It took me very long to get to understand the justification of the engineering kind of mathematics I was confronted with while studying mechanical engineering. At school I was used to understand the mathematics to apply to a problem and so my path to the correct solution was fully documented in my answers in tests. Nevertheless I only got a fraction of the points that I would have had to receive by answering correct and showing the path to my solution. The response I got when I asked why I got so few points was the following. You received the points by getting the right results and showing how you got there. You did not get the points to recognize to what basic type of equation the problem could be modified to and you did not get the points by proving that you knew how to apply the standard method. I was angry and demotivated!
Years later I found the answer to why the engineering style was justifiable! An engineer's work has to follow "by the books" methods so that QA could be fully applied and possible liabilities could be refuted. So each kind of dealing with mathematics has its justification!
So I have spend and am still spending a lot of efforts to train myself in mathematical thinking and have clearly realized that basically all of the mathematics courses taught for a bachelor degree and part of what is taught as part of the master are just courses to get you the toolset to apply when thinking as a mathematician, and/or as a physics to be able to recognize the patterns in a problem you are dealing with and be able to pursue a prove.
Keith Devlin says that what mathematics of the 20th and 21st century are is to identify patterns, opposed to what was done in the prior milleniums that was doing mathematics. I am getting a glance of what it means while learning courses on mechanics as it is traditionally taught, I do learn by seeing how using the diverse kind of topological manifolds for the same topic and I have started to look into "System Physics", as taught by the swiss professor "Werner Maurer" following the Karlsruhe didactics. I started to get aware of this structures and patterns of modern mathematics that each has its own perspective while dealing with the same topic. From a informal conversation I had with a mathematics professor at the technical university of Munich, mathematics institute, professor Brokade, this was a couple of years ago, I told him that I was happy to learn the mathematics by following a rigorous path starting with the set of numbers and starting to learn the right thinking by following lectures from a german professor from the university of Tübingen whose course followed the Analysis course from Terence Tao, UCLA and whose 2 books can be downloaded legally and for free from his personal webpage. His answer was that he felt that in the last decades mathematicians were leaving the path as the referenced professor Terence Tao does and were a famous group of french mathematicians had been working for decades to get the complete range of mathematics by following such a rigorous process and looking into the structures. At that time I had no clue what he meant and so I started to investigate this. So I learned about this french anonymous group of mathematicians and where they run into a blockade. But I not only found out about what the structure topic is about, but as I wrote a few lines above, I was able to see that the same physical field could be viewed getting correct verifiable results, but using a different kind of "structure" to describe the topic.
So, as one of you wrote in this thread, I would expect to become a pure mathematician will require to develop consciously the skills of mathematical thinking and in consequence the applying of the toolset available for proving stuff. As Keith Devlin also wrote and says in his lecture about "Introduction to Mathematical Thinking", today's mathematics can be very abstract and the results to be in conflict with our intuitions and that the language mathematics is the only way to describe and grasp those often none intuitive abstract patterns and use rigorous mathematical proving to verify that a result is valid!
This is to my personal opinion and judgement what often leads to the "questions" raised in threads where somebody is trying to apply intuitive thinking and think about consequences by following deductive thinking of an analogy used to express what can only mathematically be correctly expressed and consequences deducted from such concepts need to be mathematically presented, otherwise it is just "nice chatting"!

 

[tomwilliam2]

Hi all,
I'm interested in studying maths by myself (or in a group, but without a teacher) because it fascinates me. I'm not really sure I would want to become a mathematician (I have a decent job in an unrelated field) bug I would like to understand maths better and get to some really advanced stuff.
I completed a degree in Physics and Maths a few years ago, but am already finding rusty patches in my knowledge. So does anyone have any advice about how to delve a little deeper into maths, which books or resources to use, how to approach it, etc.? I didn't do much pure maths in my degree, but enjoyed differential equations, coupled systems, etc. I did a tiny bit of number theory but don't feel confident writing proofs and all that.
I recently read "Love and Math" by Edward Frenkel and felt inspired to look into Galois groups and sheaves, but I need some easy access stuff first I think.
Any advice welcome
Thanks

 

[micromass]

Can you tell us which math you know very well, what math you want to revise and what your eventual goal is?

 

[tomwilliam2]

Most of the maths I know solidly has direct application to physics: ODEs, PDEs, quantum wave equations and operators, etc. I don't have much pure maths apart from a basic grounding in number theory and analysis. I suppose that would be a good place to start, but I would eventually like to know a lot about sheaves, symmetry groups, Lie algebras and other things that sound interesting but which I don't yet know much about.

 

[ohwilleke]

Most of the maths I know solidly has direct application to physics: ODEs, PDEs, quantum wave equations and operators, etc. I don't have much pure maths apart from a basic grounding in number theory and analysis. I suppose that would be a good place to start, but I would eventually like to know a lot about sheaves, symmetry groups, Lie algebras and other things that sound interesting but which I don't yet know much about.

It sounds like you should start with an Abstract Algebra book aimed at math majors (as opposed to physicists, which are too practical, and as opposed to teachers, which are too dumbed down). I'm afraid I don't personally have a good recommendation for a particular study source, but that is the topic you should probably pursue.

 

[lostinthewoods]

I will be in the middle of no where for 9 months with limited to no internet access. I need to study or have a companion item(s) while I take my calculus courses. Any recommendations? I was thinking calculus of dummies would fit the bill, as the trigonometry version broke down every problem I needed.

 

[ohwilleke]

I will be in the middle of no where for 9 months with limited to no internet access. I need to study or have a companion item(s) while I take my calculus courses. Any recommendations? I was thinking calculus of dummies would fit the bill, as the trigonometry version broke down every problem I needed.

Is weight a consideration? Some of the better calculus texts add a lot of pounds to a backpack or duffle bag (I used one that weighed about 15 pounds before considering a binder for notes and problems; this was a real drag as I biked around town with other stuff as well), so if weight is a consideration and you have access to reliable electrical power at least intermittently, a text that you could get in a Kindle edition might be seriously worth considering as an option. (Kindle's are much more power thrifty and have a wider array of title choices than Nooks).

 

[lostinthewoods]

I have more information about the area I will be in. I will be able to use a kindle. I brought up the dummies series because of the break down of majority of the subject. Kind of like a tutor in a book. Please give recommendations.

 

[lostinthewoods]

I googled calculus books or something of the sorts and ran into a forum. The people there gave a link to paul's online math notes.

http://tutorial.math.lamar.edu/download.aspx
It has calculus I to III, to include a section for differential equations. Sharing my finds, as I desperately seek resources, before I am stuck in "the land of the lost" for 9 months.