Share self-studying mathematics tips

[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.

 

[TheDemx27]

For first year calculus I'd recommend "quick calculus" 2nd edition. It was designed for autodidacts, so I'd recommend checking it out.

 

[Obliv]

hi, I'm trying to learn linear algebra a bit before I take the course formally at my school. I picked up Axler's book "linear algebra done right" and have been formally introduced to vector spaces (although I have already studied them prior in physics). I learned that vector spaces are a module-like algebraic structure and that fields are a ring-like algebraic structure. Should I go learn sets,groups, algebraic structure from abstract algebra before I continue with linear algebra or does it matter? Axler doesn't formally define fields and any other algebraic structure yet (not sure if he does later on) Just wanted to know if anyone had some insight on what to do :p

 

[micromass]

hi, I'm trying to learn linear algebra a bit before I take the course formally at my school. I picked up Axler's book "linear algebra done right" and have been formally introduced to vector spaces (although I have already studied them prior in physics). I learned that vector spaces are a module-like algebraic structure and that fields are a ring-like algebraic structure. Should I go learn sets,groups, algebraic structure from abstract algebra before I continue with linear algebra or does it matter? Axler doesn't formally define fields and any other algebraic structure yet (not sure if he does later on) Just wanted to know if anyone had some insight on what to do :p

Nope, it's not necessary to learn about all these algebraic structures.

 

[Obliv]

Nope, it's not necessary to learn about all these algebraic structures.

I just want to make sure my foundation is being built correctly. Will I learn abstract algebra at some later point in my physics/math education? I know they have very important applications in modern physics.

 

[micromass]

I just want to make sure my foundation is being built correctly. Will I learn abstract algebra at some later point in my physics/math education? I know they have very important applications in modern physics.

If you're serious about building a foundation, then you will learn it at some point. Usually, people get comfortable with vector spaces first and then move to other algebraic structures. But this is not a hard rule. It's definitely possible to do groups and rings before vector spaces. In your case though, since you're preparing for a course, you should probably not investigate into algebraic structures.

 

[Obliv]

If you're serious about building a foundation, then you will learn it at some point. Usually, people get comfortable with vector spaces first and then move to other algebraic structures. But this is not a hard rule. It's definitely possible to do groups and rings before vector spaces. In your case though, since you're preparing for a course, you should probably not investigate into algebraic structures.

Thanks for the advice. It's still months from now and I really can't bring myself to learn any more 'special cases' before learning about the big picture/architecture of something. I think the rigor will benefit me regardless :p

 

[bill2018]

Hello , Micromass. There is something that I have some trouble with when self-studying mathematics but I think I could not state precisely what my problem is but I will try. I would like to use advanced mathematics to understand string theory
The problem is that I'm not exactly clear about how to best approach an entirely unfamiliar mathematics discipline by reading textbooks. What I always try to do is to formulate questions myself & try to answer them independently using guidance from reading certain parts of mathematics textbooks but I have found that this is not a good approach for some one having his first exposure to some discipline. What I think is better approach is to choose a textbook & Start reading from chapter 1. However , If I try to do this in a topic such as algebraic geometry , I try to understand everything I read to the deepest possible level , so I think that I should begin with a commutative algebra & a category theory textbook to understand more clearly what is going on in modern algebraic geometry. Usually , reading a textbook in mathematics takes a very long time and I find that I got bored quickly before I could reach the most interesting parts which are usually situated near the end of the book.

I have a question. Does anyone read mathematics textbooks (in algebraic geometry , algebraic topology or differential geometry) by starting from chapter 1 and read linearly until he reach the end of a book ? Or should one selects whatever part he finds interesting and then read it along with any required prerequisite readings from other textbooks ?

 

[micromass]

I have a question. Does anyone read mathematics textbooks (in algebraic geometry , algebraic topology or differential geometry) by starting from chapter 1 and read linearly until he reach the end of a book ? Or should one selects whatever part he finds interesting and then read it along with any required prerequisite readings from other textbooks ?

Sure, a lot of people read a book starting from chapter 1. The other extreme also happens: people who just read those parts of the book that they think will be useful. I have personally done both of them. And I am still doing both of those things. It really depends what you want to get out of something. If you merely want to prove something and find a useful technique, then you might not need an entire book. Just reading one proof would suffice already. On the other hand, if you want to get a good grasp of something like algebraic geometry, then you'll need to read a book from chapter 1 (in the case of algebraic geometry, that would need multiple books).

 

[Marcus-H]

I've been studying mathematics on my own for awhile using khan academy and textbooks, one thing I'm really struggling with is how the concepts are related.. is mathematics a unified field? Also I'm having great difficulty determining if I have gaps in my knowledge, if somebody could give a rough outline of the order they learned mathematics in I would be immensely grateful. Thanks

 

[micromass]

I've been studying mathematics on my own for awhile using khan academy and textbooks, one thing I'm really struggling with is how the concepts are related.. is mathematics a unified field? Also I'm having great difficulty determining if I have gaps in my knowledge, if somebody could give a rough outline of the order they learned mathematics in I would be immensely grateful. Thanks

Mathematics is an extremely unified field. I understand that this might not be all that apparent if you're rather new to it, but the connections should become clearer gradually.

As for gasps in knowledge. If you're studying high school math, then you should check out ALEKS which can pretty accurately determine that. You can also check out several online exams (just google them) or check problem books. Of course you can always ask people on this forum to test your knowledge, I would be happy to do that.

As for the order in which to learn mathematics, one such order is in my insights articles. You might find that useful.

 

[Sahil Kukreja]

Hi, I want to self learn combinatorics .Its basics have been taught to me in high school, but my basics are not clear and problems go haywire. I have tried a lot, but i still fail in understanding it. How can i proceed further?
Thanks.

 

[Sahil Kukreja]

^Please Reply, its been more than one day. Thanks!

 

[IGU]

Hi, I want to self learn combinatorics .Its basics have been taught to me in high school, but my basics are not clear and problems go haywire. I have tried a lot, but i still fail in understanding it. How can i proceed further?
Thanks.

If you love math, then you can't do better than Art of Problem Solving for this sort of thing. Check out Introduction to Counting & Probability.

 

[Perry]

I enroll as a freshman in physics on september and i have already started studying calculus using thomas finney's textbook. I guess my question is, because the time left is rather limited, should i solve all the practise exercises or should i be more selective? Right now, i am studying the applications of derivatives.

 

[ohwilleke]

I enroll as a freshman in physics on september and i have already started studying calculus using thomas finney's textbook. I guess my question is, because the time left is rather limited, should i solve all the practise exercises or should i be more selective? Right now, i am studying the applications of derivatives.

I am personally an advocate of solving all the practice exercise. Doing this gives you a solid foundation and guards against thinking you've mastered it when you really haven't. You'll cover all the material sooner or later in your formal classes. Your goal is to have the foundation that you start from be superior to your peers (which insures that you'll grasp the material being taught better than the median student to which instruction is being targeted). You have the early basics of your current class and its prerequisites down solidly while everyone else is grasping to recall what they learned the previous year.

 

[ohwilleke]

Hi, I want to self learn combinatorics .Its basics have been taught to me in high school, but my basics are not clear and problems go haywire. I have tried a lot, but i still fail in understanding it. How can i proceed further?
Thanks.

You might want to locate a more basic textbook in discrete mathematics or probability to work from (aimed at lower division undergraduates). You have probably missed a few basics between HS and your current study. Also, when in doubt, focus on marking sure you are clear on all notation and terms. In my experience mistakes concerning these issues are most common and most vexing.

 

[ohwilleke]

If you're serious about building a foundation, then you will learn it at some point. Usually, people get comfortable with vector spaces first and then move to other algebraic structures. But this is not a hard rule. It's definitely possible to do groups and rings before vector spaces. In your case though, since you're preparing for a course, you should probably not investigate into algebraic structures.

This is tough. Physicists usually learn "special cases" as part of a mismash course of advanced math for physicists (it was called "applied analysis" at my college). A big picture course is usually taken by mathematicians, usually called "Abstract Algebra" but the quality and rigor of those courses vary widely. Avoid math courses in Abstract Algebra primary targeted at educators rather than mathematicians. Also, this field, in general, has a very steep learning curve - expect to take it slowly but steadily as there are a lot of big, novel, weirdly named concepts that have to be mastered one after the other before anything makes sense.

 

[Hellmut1956]

As it been correctly written in contributions in this thread, preparing for the courses at the university is always worth to do. It is also correct that between grasping the concepts taught and being able to apply the to solve assignments and even more important to learn to think the way the academics require. there is a relate course from the Stanford University, called "Introduction to Mathematical Thinking" taught by professor Keith Devlin that can be taken for free and will start again on September 16th, but you can start right away, as you can access the course items. I even bought his book about the same topic as a eBook. Here he states the reason for the difficulty many students encounter is due to the difference how topics are taught at universities and how we used to learn at school. At school you learn methods to solve assignments for the different topics, at university you learn to understand why something is and how to apply it to solve problems. But why should I explain in my poor words what the prof. does in this video!

 

[Sangam Swadik]

Can anyone tell me the best books on elementary maths ( Numbersystems ..basic arithmetic and algebra ) ?

 

[micromass]

Can anyone tell me the best books on elementary maths ( Numbersystems ..basic arithmetic and algebra ) ?

You're going to have to be way more specific if you want help.

 

[Hellmut1956]

I can tell you to look for the courses of Calculus Single Variable and Calculus Multivariables from MITs OCW MOOC offering. I have found both self-paced courses excellent and very helpful to refresh my decade old mathematics studies as a preparation to take the Linear Algebra Course presented by Prof. Gilbert Strang. Prof. Strang has written an excellent book that represents the "readings" for both Calculus courses. The book is for free and legal as a pdf.

The combination of the excellent book from Gilbert Strang and the 2 equally brilliant OCW courses I have supplied you the link to did get me more up to speed so that I guess I know more today then I did know as a high school freshman in Germany! I did also investigate what is a basic knowledge required for any math intensive study and I came the conclusion they are:

1. Linear Algebra: Here the course from Gilbert Strang is excellent, the video recordings of his lectures are those of real expensive MIT courses!

2. Analysis I and II: I have selected a video recording of the lectures from a professor Groth from the University of Tübingen, because I did like his way of teaching. His course builds upon 2 books written by Prof. Terence Tao from a university in California. He help the courses for Analysis 1 & 2 as course with Honor and his books are the readings for it. I remember quite a time ago I searched for his personal webpage and there it was possible to download free and legal the 2 books.

Having had some talks with a mathematics professor from the technical university of Munich, Mathematical institute, during an "Information Event"! I praised both the course from Prof. Groth and Prof Terence Taos way to address Analysis. I did like that both follow a very stringent methodology starting from the "Number Theory". The message I got from him was, that after a french anonymous group of mathematicians did work over decades on bulding the whole mathematics starting from the number theory today "structures" were the approach of mathematics. When you watch and listen to the video recording Dr. Keith Devlin I showd in my earlier contribution, a YouTube, he defines that mathematics is the study of structures. It toook me about a year to investigate what the Prof from the Munich University meant when he taught about structures and I found an exciting course from a professor, Dr. Schuller from the technical university of darmstadt and head of an institute there. The book on which he builds his course on theoretical mechanical physiscs was build upon diverse kinds of topologies, as Dr. Kevlin says, the Mathematics of "Closeness and Position", the book is called "Gravitation" by "Charles W. Misner, Kip S. Thorne John Archibald Wheeler". I do not remember from where I did download the PDF of this 2 Volume book, for free and legal.

What I did learn by then was that mathematics has undergone a revolutionary development in those nearly 4 decades since I was at the university. Dr. Kevlin expresses this too! So restarting my competence in Mathematics resulted in more than just refreshing my former knowledge from my days at the university. The courses of the Bachelor in Mathematics is really a combination of learning to think as a mathematician as Dr. Kevlin course presents and getting a toolbox of mathematics. Real mathematics in my personal opinion is the key competence to work in todays technology fields. I do regret to have studied mechanical engineering. There I was taught that mathematics is not a competence to understand, but to know to which basic formula styles a problem can be mapped to and the apply the established methodology! I also would have choosen to be 20 years old today and delve into the sciences the way it is done in the 21 st. century and having all those opportunities that MOOC courses open!

 

[mathemasochist]

Would you say that it's worth it to take notes if you can just find all the material in the book anyway?

 

[JorisL]

Definitely, you have to digest what's written and taken notes helps with this.
For the same reason you should solve as many problems as possible.

In a subject with a lot of proofs I like to summarise the idea behind the proof.
I also list the big steps.

 

[remote]

Can anyone give me advice about how to find mentors or study partners?

I'm a computer programmer, and I'm trying to learn some math and physics on my own. It seems pretty hard, since there's no one I can really go to with questions.

 

[dkotschessaa]

Can anyone give me advice about how to find mentors or study partners?

I'm a computer programmer, and I'm trying to learn some math and physics on my own. It seems pretty hard, since there's no one I can really go to with questions.

Hey, I saw you posted this on my birthday, and it didn't get a reply. So I feel it is my duty to answer. ;)

You have some options. There's plenty of online communities, like this one, and there are online tutors and mentors, but not usually free.

I know that with some of the MIT open courseware stuff (ocw.mit.edu) there is something where you can create an online study group for a particular class. There are also math courses on coursera (coursera.com) which always have forums accompanying the class.

For in person, your nearest college or university, even if you don't want to enroll in classes, is likely teeming with people that wouldn't mind making a few bucks doing some math or physics mentoring. If you visit the campus you'll see flyers hanging up with people offering. Of course, also, not free.

I'm not sure about in person study groups. I've often wondered if starting a meetup.com group would work here (I joined a couple for technical ventures, but there's nothing for math. I've never checked for physics). Of course the people at your nearest college or university would all be studying for a class, but it would probably be terribly awkward to say "I'm not in your class, but can I join you guys?" :)

-Dave K