Studying Share self-studying mathematics tips

[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

 

[micromass]

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.

If you're completely on your own and not in any classes relevant to pure mathematics, then send me a message. I might be able to help. This goes for everybody reading this.

 

[Stephen Tashi]

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.

Given the resources on the internet, it's possible to find the answers to most of the questions about math and physics that do have answers. The aspect of a mentor or study partner that is missing is the motivational aspect. There's a big difference between "I I'll study chapter 12 tonight" and "I'll study chapter 12 tonight, because I'm going to meet Ludwig tomorrow at lunch to talk about it."

However, it is a statistically rare experience to have a mentor or a study partner. So if your education is dependent on motivation from a mentor or study partner, you'll probably have a tough time.

We can learn things from interacting with people we don't respect or don't like. For example, the worst coder on the staff may be the best at getting the fax machine to work. A poorly written textbook may have some profound insights scattered in its pages. Such educational experiences are not an organized curriculum, but they are always available.

 

[L-x]

Hi all,

I'm a physics graduate now working as a maths and physics teacher in a sixth form college. I loved my degree, though wish I'd had the work ethic I do now when I was studying full time. I took a theoretical physics course in my 4th year (syllabus here http://www-thphys.physics.ox.ac.uk/...012/Theoretical_Physics_Option_home_page.html), and passed the exam, but to be honest I feel like I learned processes without fully grasping their understanding. I felt this was especially true for the parts of the course in which i was most interested: looking at field theories. This was because at no point in my course up to that point had I ever done any group theory, and I didn't (at the time) have the motivation or though to teach myself some from scratch. I can't help but feel that with a good mathematical grounding I'd have been able to see the beauty in the course that I know is there, so I'd like to try to develop it now, but I'm wondering if anyone can give me some advice on where to start.

Should I first learn about sets, rings, and (mathematical) fields? Or is there an introductory group theory text which would be sufficient to get me far enough that I could make another attempt at looking at field theories?

Thanks in advance!

 

[zahero_2007]

I would like to share an observation that works for both mathematics and theoretical physics. If you're going to be extremely good at some subject , Let's say Quantum field theory or algebraic topology , the only way is to work out everything independently on your own. You know some basic tools & tricks and play with them in order to solve problems with varying levels of difficulties.You must invent your own problems as well. You don't really learn by reading a textbook. You learn by trying to rediscover these insights in the textbooks.

 

[PlanetGazer8350]

Hello, I am a 12 year-old boy, and I am very interested in physics and maths. For maths, would you recommend me to keep learning by myself following the order of topics for example, in each year in KhanAcademy? Or should I learn it in a different order? For physics, for the moment, I don't want anything that requires too many math, I first prefer to have a good math base that can be used for physics, and then gradually go applying it to my physics learning. So in physics, what order should I follow of topics, Eg: should I follow a certain order in each field, quantum mechanics, motion, etc (If it includes some maths in does not matter, but I prefer it to explain the concepts more strongly)? If you know about any books about physics, that mainly and strongly explain concepts (if it includes some math it doesn't matter), could you please tell me? It would be very helpful.
Currently, I am to self-studying about math, physics (mainly concepts), computer programming (before I was a bit confused and advancing little, as I was with many, now I am with just Python) and electronics. Do you think I should do them all at once, each one a smaller amount but at once, or 1 mainly at a time (and the rest but MUCH less time, so I keep thinking about them, and don`t forget), for a certain period of time, and then go changing which one I learn mainly and which one I revise in less time?

 

[Hellmut1956]

Dear young friend. I did share your interest in physics and I was not bad at school. i think the way the Khan academy proceeds in physics is a good option and I would be you, I would follow their path. The same applies to mathematics. So far to the rational way to proceed.

Emotionally physics was and is for me a fascinating science and dealing with it opens our sense for what our current science is finding out about it. Forget about quantum physics and relativity theories from your learning of physics. Look for good videos in youtube and find some that speak in a more general public kind about those fields of science. That puts you in touch with those topics and you can start to reflect about what you find interesting.

With mathematics the way it is taught at school and which you will need to master your tests in mathematics at school the Kahn academy gives you solid information to learn. But as with physics, where there are topics that you need at school in which give you a starting point, mathematics offers at least equally fascinating topics that equal in opening your eyes for a whole new way to see the world around you. i f you allow me, I would to tell you a bit about what I mean!

I did finish my studies at a technical university in 1979. That was then the last time I had to deal with mathematics in the way I did learn until then! So nearly 4 decades later the science about mathematics has also been developed very heavily. The mathematics as you learn to the end of high school really is not mathematics but is learning to apply formulas to solve equations. The top of this king of "mathematics" goes and includes calculus. Terms like differentiation and integration are ones that you can search via google and find information about it.

I like very much the current definition about the science of mathematics. Mathematics deals with finding "structures"! There is a course from the Standford University which can be taken for free here: "Introduction To Mathematical Thinking" This course has the purpose to help students to make the transition from the kind of mathematics they have been taught until then and to the kind of mathematics the students are required to learn and apply at the university. Do not let yourself be intimidated. Its even more! As you have not yet been spoiled to think the traditional way mathematics was done until about 200 years ago, you will probably have it easier to grasp this "Mathematical Way of Thinking! If you listen to the lecture of prof. Kevlin that you can see as videos, if you listen to the videos were he very detailed explains interesting topics as help for solving the challenges to think the mathematical way, you could have it easier than older ones to capture what is being taught.

 

[PlanetGazer8350]

Thank you very much for this advice, I will take it into account, and definitely check out the course on 'Mathematical Way of Thinking' . Currently, I am also in another Coursera course of Stanford University in 'Introduction to Logic'

 

[Greg Bernhardt]

Yes I want know How can I be an Expert in Trigonometry ?

Study and practice

 

[fisicist]

Hello, I am a 12 year-old boy, and I am very interested in physics and maths. For maths, would you recommend me to keep learning by myself following the order of topics for example, in each year in KhanAcademy? Or should I learn it in a different order? For physics, for the moment, I don't want anything that requires too many math, I first prefer to have a good math base that can be used for physics, and then gradually go applying it to my physics learning. So in physics, what order should I follow of topics, Eg: should I follow a certain order in each field, quantum mechanics, motion, etc (If it includes some maths in does not matter, but I prefer it to explain the concepts more strongly)? If you know about any books about physics, that mainly and strongly explain concepts (if it includes some math it doesn't matter), could you please tell me? It would be very helpful.
Currently, I am to self-studying about math, physics (mainly concepts), computer programming (before I was a bit confused and advancing little, as I was with many, now I am with just Python) and electronics. Do you think I should do them all at once, each one a smaller amount but at once, or 1 mainly at a time (and the rest but MUCH less time, so I keep thinking about them, and don`t forget), for a certain period of time, and then go changing which one I learn mainly and which one I revise in less time?

When I was in your age, I liked Jay Orear's Physics and the Feynman lectures a lot. In general: You cannot learn physics without math. So if you want to learn physics, you must learn math first, namely vectors+matrices, analysis (differentiation and integration) and later vector calculus (the gradient, divergence and curl operations and the theorems of Gauss and Stokes). The Feynman lectures cover all this, Jay Orear expects you to know differentiation.
Don't waste your time trying to learn physics without math or with as little math as possible. It doesn't work/will give you a pseudo-understanding.
I'd recommend you start with differentiation (by the way, I don't think the explanation that Feynman gives is very good; I don't think I'd have understood it there if I hadn't known it before). I would simply start here (I learned it first from the math formula reference book that we used at school, so not really a big difference): https://en.wikipedia.org/wiki/Derivative (sections 1.1, 1.4, 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3)

 

[Christina Lin]

What are the most reliable (preferably free) online courses for algebra 2, pre calc, and calc A, B, and C that give students a deep understanding of the topics? Would I be better off with textbooks or a paid online program?

I would like to self-study the math necessary before I dive into physics, however, I would like to not break the bank in the process. I am currently enrolled in ALEKS MAT 130 and MAT 170 for algebra 2 and pre calc by ASU, but I’m not sure if they were intended for students who are getting exposed to the material for the first time. Furthermore, I’m not getting an understanding of the topics when I take algebra 2 from MAT 130. It is simply straight-up memorization. In this case, should I go for a paid online course or textbook? I don’t know about MAT 170, but if it’s anything like MAT 130, I don’t think it would be for me. I am also enrolled in Calculus 1A: Differentiation, Calculus 1B: Integration, and Calculus 1C: Coordinate systems and infinite series by MITx on Edx. I have yet to take anything from it, but it seems promising. If anybody has experience with it or suggestions for a source that provides a deeper understanding of calculus, please let me know.

 

[IGU]

ALEKS is useful for review, to check if students have any obvious holes in their knowledge, but it's not actually useful for learning anything well. If you want the best courses for students who love math and are good at it, check out Art of Problem Solving. My kid preferred the books to the online courses because he could go at his own pace.

In any case, not free. Also not particularly expensive, especially the books. But really, really good.

 

[Christina Lin]

ALEKS is useful for review, to check if students have any obvious holes in their knowledge, but it's not actually useful for learning anything well. If you want the best courses for students who love math and are good at it, check out Art of Problem Solving. My kid preferred the books to the online courses because he could go at his own pace.

In any case, not free. Also not particularly expensive, especially the books. But really, really good.

I've heard a lot of great stuff about AoPS! I will definitely check them out, and thanks for the quick response.

 

[PlanetGazer8350]

This website (http://www.openculture.com/math_free_courses) may help, all of the courses are free. Another very good website is https://openstax.org/subjects/math.

 

[geologist]

I'm interested in learning math, partially for its own sake, but mostly because I am very interested in learning scientific computing (emphasis in environmental modeling, e.g. climate, groundwater). For that I'd like to be able to cover precalculus material (algebra, trig) and basic calculus (limits, differentiation, basics of integrals).

Currently I'm going through Precalculus demystified and the Brilliant.org Algebra practice problems, which should take me through mid-May. I think two months should be a reasonable time to review algebra and trigonometry, I don't feel that it's terribly difficult (challenging, but not bang my head against the wall).

From there I'm planning on taking couple computer science online courses through edx.org (Introduction to computer science and programming using Python and Simulation and Modeling for Engineering and Science), which don't require more than a reasonable aptitude for math, probabilty/statistics, and some programming (all of which I have). Once I'm completed with these I plan on going through Elementary Calculus (with supplemental problems, e.g. brilliant.org and Paul's online math notes).

 

[geologist]

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

I know this is a bit of an old post, but I often find myself wanting to study more subjects than I should. How many hours per week do you partake in self-study?

 

[PlanetGazer8350]

I am currently learning on math, physics (mostly conceptual), artificial intelligence using tensorflow in python, and sometimes I research and keep on learning on arduino, electrical engineering, space exploration and the geology of different planets or celestial bodies, with occasionally some other small topics. However, I sometimes feel I don't have enough time to do it all at the pace and depth that I would like. I usually try to organize myself weekly or bi-weekly, as a week may sometimes not be enough for me to cover or do what I want to do, learn, or keep researching. I also try to focus on math, as it will then let me learn even more on other topics such as physics and electrical engineering. If I try to do it the other way round, with math as something secondary, when I encounter something that isn't mainly conceptual, it is very difficult for me to understand it. In addition, I think that truly learning about physics, for example, is not about the formulas at all, but the concept itself, and why the formula is the way it is, and how it has been constructed through a series of experiments. Sometimes, previously, I didn't focus much on learning math, but physics concepts, or electrical engineering, but almost always complex math appeared when researching on something, which is why I turned it the opposite way round.

With the number of hours each week, it really depends on which week I am on, or what I must do, even though I wouldn't really want to that thing, and rather just focus on what I truly like and interests me. To try to more or less balance what I do, with my primary focus on math, I may be 1 or 2 weeks just covering a reduced number of topics, to focus on a smaller number of topics at a time, and then I may switch some for another time period, maintaining some topics I consider very important to keep on learning further on other subjects.

 

[Terrell]

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

Please show me your ways, master.

 

[Hellmut1956]

@PlanetGazer8350: I can very much identify myself with the kind of activities you are pursuing and why! I am now retired and to keep me busy and to have my days properly organized and to satisfy my curiosity I do exercise the muscle between my ears! I started to reactivate my hobby from my days at school, model building. As my financial situation is not anymore as it used to be i decided to switch from building planes to build a sale ship from scratch. Work with wood, lamination with epoxy, later building using aluminium the project of the sail ship model became more a path that brought me in touch with many technologies. As I used to work in the semiconductor industry and later in the telecommunication industry it became evident to me that using my own build electronics would open many new ways to realize projects related to the sailship. While over the years my workshop became more and more sophisticated, milling machine and lathe purchased based on the more mechanical aspects of my project, electronic has become my main technique. Investigating my own way to combine a stepper motor to control a pulley that controls the sheet that define how much a sail can open and wishing to do it the way it was done in early 20th century i run in conflict of this part of the project with naval modelers, experts in sailboats as they claimed this was impossible due to the friction of the sheets in the pulley and in their path through the body of the sailboat. Discussing it with the community of physics they said that friction was neglectable.. So I decided to model the system that was to control my sails. Doing a lot of research, Matlab and similar tools had no price for individuals and so where financially impossible to acquire, I decided to go with Mathematica from Wolfram and their tool "SystemModeler" that uses the language Modelica. I wanted further to compare the "quality" of my Modelica models by using the ability of the Wolfram software to collect data from physical experiments and to improve this way the quality of my models.
Somewhere along the road of this activities I realized that mathematics are the language required for Physics, Electronics and Simulation. Suddenly I found a course MOOC, from Terence Tao that build the mathematics starting with the theory of numbers. The book used was available for free from Taos personal website. I fall in lough with mathematics! I even decided that it would be worthwhile to study mathematics at the "Technical University Munich". I wen to an introductory event and had the opportunity to talk with one of the mathematics professors. he told that since recent decades mathematics is viewed from the perspective of studying "structures". I had no clue of why it happen to be this way. Soon I found fascinating courses for physics, cosmology and mathematics that demonstrated the power of approaching mathematics by studying structures. Even a professor that gave a course available for free in the internet from the University of Erlangen went through the whole theoretical field of physics up to beginning master level using more and more sophisticated models of mathematical structures. Even there is a fantastic introductory course available for free as MOOC from the Stanford University named. "Introduction to Mathematical Thinking". A very worthwhile course to take. In his first lesson he defined mathematics as the science of structures!. I do not need to mention that also electronics is a science the uses mathematics. Also artificial intelligence uses statistical methods.
I have taken the effort to present how I got into this trying to confirm your opinions and so be able to pass the message to you: science is advancing so fast in many fields that even me who is 24/7 available for this studies have come to the conclusion I need a method to combine the curiosity of the fields mentioned here with the chance to benefit from what I am learning, tools for the science disciplines that affect my project of a sail model ship build from scratch in my very own way.
So I study mathematics in depth enough so that the software Mathematica takes the job to solve equations. To have my mathematics skills well enough advanced that I can capture the concepts I meet. This results in an iterative way to advance my mathematics studies.

 

[ZHHuang]

I use Understanding Analysis, by Stephen Abbot, because Rudin's book is too difficult to read. I found Understanding Analysis is thoroughly explained. It is an introduction to analysis, so the book does not contain the concept of metric space and Euclidean N-space. Self-studying is extremely time-consuming. It costs me 7 months to learn 7 chapters.

 

[Tatsuya]

Hi guys,
I'm interested in geometry and topology. What should I study if I want to be able to study topology? and in what sequence? My highest level of math education is high school and right now I can't even remember conic sections... thanks

 

[S.G. Janssens]

Hi guys,
I'm interested in geometry and topology. What should I study if I want to be able to study topology? and in what sequence? My highest level of math education is high school and right now I can't even remember conic sections... thanks

Why are you interested in topology? In order to give advice, it helps to know.

If you are a beginning mathematics student at a university, a typical sequence would be:

calculus - analysis - (metric) topology - (general) topology,

where the latter two may be one course. (I took them separately.)

On the face of it, general topology requires very little background beyond naive set theory. I write "on the face of it", because I actually think it makes little sense to jump to general topology right away. To see where definitions come from and to gain essential intuition, I would recommend first taking a rigorous (but not necessarily long) course in single-variable analysis, treating such topics as: the triangle inequality, convergence and continuity.

The nice thing about this approach is that a lot of analysis books also treat elements of general topology, at least superficially. If you find this too much of a detour, consider beginning with a very elementary topology book, make a start and see how far you get.

Topology is broad: subfields such as algebraic topology and differential topology (with which I am not familiar) but also topological vector spaces build upon general topology. These subfields have considerably more prerequisites than general topology proper.

 

[Tatsuya]

Why are you interested in topology? In order to give advice, it helps to know.

If you are a beginning mathematics student at a university, a typical sequence would be:

calculus - analysis - (metric) topology - (general) topology,

where the latter two may be one course. (I took them separately.)

On the face of it, general topology requires very little background beyond naive set theory. I write "on the face of it", because I actually think it makes little sense to jump to general topology right away. To see where definitions come from and to gain essential intuition, I would recommend first taking a rigorous (but not necessarily long) course in single-variable analysis, treating such topics as: the triangle inequality, convergence and continuity.

The nice thing about this approach is that a lot of analysis books also treat elements of general topology, at least superficially. If you find this too much of a detour, consider beginning with a very elementary topology book, make a start and see how far you get.

Topology is broad: subfields such as algebraic topology and differential topology (with which I am not familiar) but also topological vector spaces build upon general topology. These subfields have considerably more prerequisites than general topology proper.

Thanks S.G. Janssens. I'm into topology because I'm interested in the 'shapes' and 'forms' like klein bottle and triple torus etc., which is the same reason I like geometry. I want to know the theories behind them and how to create various forms with the knowledge. Also I used to watch a TV show about maze-solving using topology and I got hooked. The pattern and analytical approach to solve the problem seem fascinating - I'm a fine art student if that helps and sorry if I've used wrong math terms. cheers

 

[S.G. Janssens]

Your question is interesting because one could perhaps think about topology as a way of capturing the essence of shape without being "hindered" by geometry. (See the paragraph halfway this page, for example.)

If you want to make your interests mathematically precise (for example, the Klein bottle can be realized as a "quotient space", which is a precise way of "gluing" objects together) and you are really mostly curious about topology and geometry without caring so much for additional mathematical "baggage", then I would suggest:

1. A good course on sets, propositions, relations and functions. (This is typically the first course that 1st-year mathematics students take in my country.)

2. A course on elementary general topology, ideally also introducing you to some geometric topology. Croom's little book "Principles of Topology" may be a good choice. There may also be full online courses that are worthwhile.

It will require investment of time and energy. The reward will be that you can understand at a much more precise level what actually fascinates you.

 

[Tatsuya]

Your question is interesting because one could perhaps think about topology as a way of capturing the essence of shape without being "hindered" by geometry. (See the paragraph halfway this page, for example.)

If you want to make your interests mathematically precise (for example, the Klein bottle can be realized as a "quotient space", which is a precise way of "gluing" objects together) and you are really mostly curious about topology and geometry without caring so much for additional mathematical "baggage", then I would suggest:

1. A good course on sets, propositions, relations and functions. (This is typically the first course that 1st-year mathematics students take in my country.)

2. A course on elementary general topology, ideally also introducing you to some geometric topology. Croom's little book "Principles of Topology" may be a good choice. There may also be full online courses that are worthwhile.

It will require investment of time and energy. The reward will be that you can understand at a much more precise level what actually fascinates you.

thank you very much S.G. Janssens! this is really helpful and now i have a clearer idea of what to do! cheers

 

[mathwonk]

a remark, motivated by requests for free books. a friend of mine wrote a math book and decided to try to publish it so as to have some income from it. she had to revise it many times over several years to satisfy the publisher, putting her research career on hold, but the result was a much better book that was ultimately recognized as the best text in its area in the country. as examples of calculus books, the best ones by all accounts are spivak, apostol, and courant, none free. so the moral is that the best books cost money, the contrapositive being that the free books are not the best. sorry about that, but those who write the really carefully polished books do deserve something to live on from that effort. my friend mike spivak lives essentially entirely from proceeds of sales of his calculus book, and he is not a rich man. so for the best results, try to pay your way, is my suggestion.

my algebra book, notes for math 843, 844, 845, free on my website, is apparently good enough that i once received an email from Wiley Interscience publishers asking me to submit it for publication, but I never did want to take the time to make it ready for publication. there are also many other much better, and more polished, books freely available online, such as Sergei Treil's Linear Algebra Done Wrong, at Brown, but many of the best still cost, and are worth, a certain price. Not everyone can afford to donate their labor and knowledge, as people do here on PF. Please try, when possible, to make an effort to support people who make a real contribution to learning.

later edit: There is one case of counterexamples to my principle of better means costlier. In the case say of the famous calculus book of George B. Thomas, the newer ones with added names like Hass, and Weir, cost 10 to 20 times more than the original ones by Thomas himself from about 1953, and are far inferior, in my opinion; and I have taught from at least 4 versions over a teaching career spanning 40 years. One of the newer ones, Thomas and Finney, 9th edition, is also available for less than $5, and is far superior to those other even newer ones costing $60 and up, but I like the 1953 version by Thomas himself, at about $5 used from abebooks. Thomas/Finney is easier to read, but the original Thomas has expert insights on how calculus is used by engineers, that the newer books omit.