Studying Share self-studying mathematics tips

[Hellmut1956]

Good afternoon. I am planning on studying computer science or a math major, haven´t decided yet. I am passionate about programming, mathematics, pysichs and logic. I struggled at mathematics (3s and 4s out of 20, yes that bad!) because i didn´t see the beauty of it and now after becoming passionate, i am quite satisfied with my skills (got 16 out 20 in the national high school exam), but i could do much better. By the way i didn´t made any Math subject, so my exam performance was my final grade. I learned all the math by my self using Khan Academy, Explicamat (Portuguese website).

I am passionate about math, i took the liberty to dig deep and create insights, which most schools don´t do, the main reason, students fail miserably in the national exam, which tests students logical and analytical skills. I did so much better, despite self-learning, because i understood the concepts, didn´t just memorize formulas.

Since i am taking an engineer course quite similar to computer science or even a math major, i will be taking integral and differential calculus, complex analysis, discrete mathematics, linear algebra and calculus-based pysichs, i really need a deep understanding of the material covered in high school. I feel like i can to much better, so i am devising a plan to cover high school math material with more rigour, proofs included, so to speak, increasing my math maturity.

Why i am doing this? I don´t want to faill those math classes in the first year already. I want to be the best, i am willing to work to achieve such massive goal and for that i need the basics well developed just like a building a house.

I was thining of reading Basic Mathematics by Serge Lang. I don´t want some silly plug and chug exercises ( i had enough), i am looking for problem-solving exercises, word problems, proofs, logic, foundations, etc.. Will that book provide me such needs?

Short story: I want to develop a mathematics mind set and the foundations necessary to study harder subjects. What do you recommend me?

Thanks in advance.

Hi Bruno

Due to other reasons to do with my hobby I have to acquire the knowledge as given in a math bachelor, as well as bachelor physics. So first issue was to teach myself mathematical thinking and so I found an offer from the university of Heidelberg were for free the lecture were offered as videos. Talking to the professor he told me that he bases his course on the 2 books about Analysis from Terence Tao and his course with honours. The books I found legal and free as pdfs at the homepage of Terence Tao, Analysis I and II. What I liked about his approach was that he spends comparatively a lot of time to teach mathematical thinking and prove thinking by using the natural numbers and moving from there. So the kind of statement, "as it obvious..." becomes none existing. I can highly recommend this book in english as the teaching at the german university is in german!

As nearly 4 decades have passed since I studied mathematics at high school and at my study for mechanical engineering, I soon found out that I had to refresh those topics teached at high school. So i found the courses of Calculus from MIT, OpenCourseware, 18.01 and 18.02, Single and Multiple variable calculus using the also free pdf book from professor Strang very useful.

 

[Franco_Carr14]

Whenever I study Mathematics, I always find myself highly irritated, I feel like I always have to remind myself of what I have already learned to be put in the right mind set, I can't just read a book without thinking about this stuff because I feel like I maybe losing knowledge. I'm always looking for a mindset before I read, but I find it a very arduous task.

 

[Hellmut1956]

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!

 

[Franco_Carr14]

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!

If I understand correctly, I should first develop the basics? That's great advice! I think I feel irritated because of my difficulty to concentrate while reading, but that's more of a personal problem, not unless you are willing to spare some advice for reading.

 

[Hellmut1956]

Well, I would say it is an iterative process of reading, then applying the reading to some problem hopefully available in the book you read and verify if what you think you have understood fits to solve the problem. An example of a good learning book is the one about calculus 1 from Gilbert Strang that is made available for free in the material accompanying the course about Calculus single variable from the MIT in teir free offering within OpenCourseware available in the internet. Here the link to the course supported not just by videos of the lectures given at MIT, but also uses the book from Gilbert Strang. You might see that as part of this course even the Assignment lectures are recorded as videos.
But in general it is to say that between believing to have understood something while reading it and getting the ability to apply it is a way to go. That why iterations in which the "already understood" text of a book should be reread. Happens to be that you catch new facets of the topic read a couple of times with exercises and a couple of days between each run!

 

[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.

 

[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.