Where to start self study of higher mathematics?

[Leopold Infeld]

Where to start self study of higher mathematics?

I am a student who just finished high school and i am going to go to college this fall. I got like 7 months to shape up my maths and physics because my country got only 10 years of school. I want to ask if i could self study on all the advanced topics like number theory and abstract algebra and differential equations juz by a handful of books and physicsforums and the web.

Well, the main question i wanted to know is whether it is possible to learn higher mathematics juz by self studying. I have been giving it a shot for about 3 years now and the author juz jumps with the "it is obvious that" and so on. I don't know enough of maths to understand what the author says or i am too stupid.

I hope no one laughs but i am asking for help out of despair and do forgive me if i posted it in the wrong place. Thanks.

 

[shmoe]

It's certainly possible to study advanced math on your own, just get some books and start working. The more you can work out on your own, the better off you'll be but there's no shame at all in asking for help here if you need help understanding a concept. Just trying to explain what it is you don't understand can sometimes help your own understanding.

Here's a thread with some number theory text suggestions, https://www.physicsforums.com/showthread.php?t=85248

 

[DeadWolfe]

Well, it would help if you told us what you do know.

You should know calculus fairly well before you do differential equations, but if you do, then any number of straightforward texts like Boyce and DiPrima or Zill will do.

Algebra on the other hand really doesn't have many prerequisites. Artin's Algebra works well.

And for Number theory, knowledge of a little algebra is helpful, but after that, LeVeque's fundamentals of Number Theory is good.

And as for points where the author says: "it obviously follows", these points are indeed not obvious to everyone. So take some time, think about it, write some stuff down if you have to, until you understand why it follows. I do believe it is possible to learn on your own, but it does take time and effort. The less time you spend learning this stuff the more shallow your understanding will be.

 

[daveb]

One recommendation for books, regardless of the topic, is to find ones that either have a great deal of worked out answers in the back, or a large number are online through a search. It's pretty pointless to work out a problem and not whether you did it correctly or not. Usually, university textbooks handle this best, since a good number have solutions manuals or many professors post homework solutions.

 

[neurocomp2003]

you should be able to learn Calculus, Linear Algebra, Abstract, Number THeory, Set Tehory, Graph Theory/Combinatorics, Basics of DIffential Equations and Dynamical Systems...the first bit of analysis, simple geometry, computational geometry all on your own.

The best bet is to look at some universities course webpages like MIT open course website. "ocw". And look at the assignments and topics

James Stewarts Calculus should be a good startng point. Any Number THeory book more or less will give you a good foundation in Number Theory.

 

[d_leet]

you should be able to learn Calculus, Linear Algebra, Abstract, Number THeory, Set Tehory, Graph Theory/Combinatorics, Basics of DIffential Equations and Dynamical Systems...the first bit of analysis, simple geometry, computational geometry all on your own.

The best bet is to look at some universities course webpages like MIT open course website. "ocw". And look at the assignments and topics

James Stewarts Calculus should be a good startng point. Any Number THeory book more or less will give you a good foundation in Number Theory.

The MIT site is awesome, I'm working on trying to do an independant study to teach myself some of differential equations and open course ware has practically all you need for the course including lecture videos.

 

[Leopold Infeld]

Thanks for the help.

Can someone tell me what they teach in first year american college courses? Are they hard? Thanks again!

 

[JasonRox]

It's going to be a big challenge.

Self-study isn't easy when you have no directions.

 

[benorin]

Try Multivariable Calculus on the Net, and Calculus on the Web, Online Mathematics Textbooks (links to several on-line math books in various branches of maths), and, well, click on the "Links" link in the upper righthand corner of this web page...

 

[paedrik]

Another great resource that I have found usefull is mathworld, an online math encyclopedia http://mathworld.wolfram.com/FundamentalLemmaofCalculusofVariations.html. Wikipedia is good as well http://en.wikipedia.org/wiki/Calculus. Both of these links go to an article instead of the home for some instant gratification!

 

[neurocomp2003]

First Year Canadian studies teach:
Linear Algebra-Vectors, Matrices, 3D transformations, Solving Systems of LEs
Calculus- Integration, Differentation, Applications of Both(area,length,volume, flow,rate[eg population growth] ),plotting, basics of functions,Trig/NaturalLog,Limits

2nd half Lagrange Multipliers Multivariables, Basics of Solving DiffQs, Series(Taylor) & Sequences, and some more stuff can't remember

Statistics-never took first year and I don't like stats.

 

[mathwonk]

there is a sequence of free books on advanced algebra at http://www.math.uga.edu/~roy/

 

[JasonRox]

there is a sequence of free books on advanced algebra at http://www.math.uga.edu/~roy/

They are lecture notes, so they won't be as thorough as a good text.

I'd say follow those notes while following a good text. That should be a good indication that you know what's going on.

 

[mathwonk]

have you read them? or are you making conjectures?

you are right of course that they are lecture notes, and thus have some differences from polished books, primarily fewer problems and no index, but i suggest they may in fact be superior to some books available, in terms of insight and thoroughness.

i.e. these notes are in fact more thorough than many texts, since a guiding principle the author used in writing them was never to sat anything is "obvious" or to leave any tedious calculation as an exercise.

i would be interested in your reaction to actually reading them, as i have been recommending them for some time with good responses from phd students (at berkeley, upenn, rice, uga) studying for prelims.

 

[JasonRox]

have you read them? or are you making conjectures?

you are right of course that they are lecture notes, and thus have some differences from polished books, primarily fewer problems, but i suggest they may in fact be superior to some books available, in terms of insight and thoroughness.

i.e. these notes are in fact more thoroughn than many texts, since a guiding principle the author used in writing them was never to sat anything is "obvious" or to leave any tedious calculation as an exercise.

i would be interested in yiour reaction to actually reading them, as i have been recommending them for some time with good responses from phd students studying for prelims.

Well, I do admit to only glancing at them. You recommended them in another thread, relating to those of 843, 844 and 845.

I look at the 843 lecture notes.

I was reading the beginning to just get a feel of the author, and see how he writes. I found that it's not what I would want as a beginner.

I totally agree with that it would be great for a Ph. D. student. It quickly goes through everything, which I believe why it wouldn't be good for a beginner.

I've went through lots of books at second hand bookstores and the ones at the university library. I noticed that Graduate Texts are also thorough, but also quick. You can follow at the beginning, but they will bring in new concepts faster than you can swallow them.

Anyways, that's my take on it. Maybe I should give it a read when time permits.

Of course, I did bookmark them for reference.

 

[mathwonk]

probably you are right aboit those pre PhD notes going too fast for beginners. what do you think of michael artin's book: Algebra? that was written for MIT sophomores.

It may also be a bit terse, but he really knows his stuff if you can hang with him.

 

[mathwonk]

to learn as much as possible in a finite amount of time, i recommend anything by richard courant, like "what is mathematics", or his calculus books.

 

[JasonRox]

probably you are right aboit those pre PhD notes going too fast for beginners. what do you think of michael artin's book: Algebra? that was written for MIT sophomores.

It may also be a bit terse, but he really knows his stuff if you can hang with him.

I've never read or seen Artin's book at the moment. I hear his name roaming around on PF as a recommended author for math books though. His name is certainly familiar.

I have I.N. Herstein's text for Abstract Algebra, but I'm not positive whether or not they relate. I think Herstein's text has been good so far for me. I also have Gallian's text for Abstract Algebra too, but I find it a little incomplete and that it lacks rigor in some topics (that I read so far).

 

[Aridea]

Leopold,

Listen, there is nothing wrong with asking for help. I am no math wiz myself but I'm doing the same thing as you so I think I can help.

I'm going back to school this fall for physics, and I haven't been doing really any math other than everyday kind of stuff since my first year of college; which was about 11 1/2 years ago. Anyhow I took AP calc in high school, and then my first year of college, but I literally remember nothing, I didn't even remember algebra. About 2 months ago I started trying to re-teach myself what I forgot and teach myself things I never learned.

At first I tried to use a calculus book, but that was useless, I had the same problem as you; I didn't understand everything they were talking about, so I went back to the beginning. I started with a quick review of pre-algebra concepts, then I went to algebra 1, after that I started getting into more difficult problems in algebra. Right now I'm starting to re-learn what I need for trig and calc.

Now, since you're fresh out of school you probably don't need to start as far back as I did, but begin with the basics, then move up slowly. There are also a lot of great sites on the web that help with math, one good site for help with algebra, and the base concepts that you'll need to move on to higher math is http://www.purplemath.com/.

A good claculus site, that breaks everything down piece by piece so you can start at the beginning, or go to what you are wondering about is http://www.karlscalculus.org/index.shtml .

I know it's hard sometimes to understand what the authors are trying to say when you're self teaching. The problem is that they understand what they are writing about about so a lot of times they forget to explain things better. Anyhow, I hope this helps you.

 

[alexandra]

Hi Leopold

In addition to the sites already recommended, I thought you may want to have a look at this one: http://tutorial.math.lamar.edu/sitemap.aspx.

I found the notes on this site very detailed (scroll down the page, past the 'cheat sheets'), and they also cover a variety of levels, ranging from pre-calculus algebra and basic trigonometry to multivariate calculus, differential equations, etc.

 

[Wooh]

Howdy all! I am a freshman studying Business, Computer Science, and Mathematics (the first is so much lamer than the latter two ;)). I wanted to beef up my knowledge, and I was wondering if you guys could recommend books in these two areas:

Transcendental numbers and their ilk, how it was proven Pi is transcendental, stuff like that...it just interests me.

and

Random number generation, psuedorandom number generation, statistical tests on such sequences, stuff like that. I have the art of computer programming: volume 2 in the mail, but I'm curiuos if y'all have any good sources for a more mathematical observation of the problem of random number generation.

I was going to make my own thread, but this one seemed so perfectly suited...

 

[WARGREYMONKKTL]

thanks you very much for providing me many good links for self study!

 

[Jeff Ford]

If you're looking for something that doesn't require any background, but introduces a lot of the areas in higher mathematics, Concepts of Modern Mathematics, by Ian Stewart is quite good.