Chronological Order to Study Mathematics

[tridianprime]

Hello, I am new to posting on this forum so I will give some background. I am 15 as of tomorrow and I am a mathematics enthusiast. I also consider myself a proficient java developer. Some other general bits of info are that I have an interest in physics and astronomy and I was also, two weeks ago, diagnosed with Asperger's syndrome.

I self studied A Level Maths (UK equivalent to high school) and am now embarking on self study of undergrad material. I am working through "How to prove it?" By Velleman to learn about proof writing and logic but I am also working through Spivak for calculus as I believe it will give me a thorough understanding. Following this, I intend to study linear algebra, which I have looked a little at, and ODE's simultaneously. I will most likely use Shilov and Halmos for linear algebra and then I'm not sure for ODE or, following that, for PDE. I would appreciate recommendations for this.

I have also picked up a good undergrad classical mechanics book which will be delivered soon so I can work through that for fun.

I would like to receive any feedback such as recommendations in maths, or physics, or just any ideas you may have in terms of material to learn or even books.

Thank you, Sam.

 

[kinkmode]

I wish you luck on your endeavor. At some point before ODEs and PDEs I might suggest working on some multivariable calculus. I don't know if it is in Spivak since I've never used the book.

 

[tridianprime]

I wish you luck on your endeavor. At some point before ODEs and PDEs I might suggest working on some multivariable calculus. I don't know if it is in Spivak since I've never used the book.

Oh yes, I forgot about that. I was thinking of using Apostol v.2 but that is very expensive. Do you know of any good alternatives?

 

[MrTheBard]

Tridianprime,

Spivak's book is great, so that's definitely a good read. You may want to check out Stewart's Calculus book 7E. That book is very common in the colleges / universities in the U.S.

If you're fresh out of high school, I would suggest studying College Algebra, then Trigonometry, then Pre-Calculus, Calculus, Differential Equations, Logic and Proof, Discrete Structures, then linear algebra.

That seems to be a decent order to study in. Also, be sure to sprinkle some physics in there, once you're done with Trigonometry and Calculus!

Good luck!

 

[jgens]

Oh yes, I forgot about that. I was thinking of using Apostol v.2 but that is very expensive. Do you know of any good alternatives?

The book Calculus on Manifolds by Spivak is a good introduction to multivariable calculus. It is pretty short though so you will miss out on techniques like Lagrange multipliers. There is also a book by Hubbard that some members of the forum have recommended before, and while I cannot personally vouch for it (never read it), that might be something to look into. I would also recommend looking into algebra at some point and there are some good intro texts to the subject by Dummit and Foote or Artin.

 

[tridianprime]

I've done all of the pre college material including the uk equivilant to AP calculus. Thanks, I think I'll give Stewart a go. Ill be sure to add in the discrete math as well but I thought that was normally done later on. Thank you.

 

[tridianprime]

Wow, thanks for all of the feedback so far. I was definitely going to read calculus on manifolds at some point but I thought I needed a more thorough book like you say. I will definitely study more abstract algebra at some point but probably after I've got the primary stuff I.e ODE and Linear Algebra out of the way.

 

[retro10x]

Stewart is good calculus book for solving problems, and there are some physical applications as well. The material is fairly basic so you dive right into it. There is a single and a multivariable version. I wouldn't do ODE until you understand linear algebra, as you'll get much more out of it.

 

[tridianprime]

Ok I'll do linear algebra first, thanks.

 

[verty]

Also consider this book: https://www.amazon.com/dp/0486650383/?tag=pfamazon01-20 as an induction to serious math, to me it looks well-suited for a young reader.

 

[andyroo93]

I don't know much at all about relevant books, or undergrad pure maths (although I'm starting applied maths and physics next month, which involves two pure maths modules - numbers, sets and sequences; and linear algebra and analysis), but it's cool to see I wasn't the only 15 year old teaching himself maths in his free time. I felt it was slightly "shameful" what I was doing, like a guilty pleasure or something like that.

 

[tridianprime]

That's funny. I know they're out there but I've never known of somebody my/ was my age who does maths in their spare time. What books would you recommend for classical mechanics? From your post, I wasn't sure what stage you are at.

 

[andyroo93]

As I've said, I'm not sure about particular books - I just looked things up online, looked at solutions to problems and figured out my own way of understanding things. I was going to do some classical mechanics over the summer, but then I thought I'd be covering the stuff at university in autumn anyway, so I'd just wait til then. Instead, I'm teaching myself some decision maths. I bought a couple of A-level textbooks, which is probably a waste of money (>£10 each) because they don't cover an awful lot of content - I should have splashed out a bit more on a more extensive book. I wasn't expecting to enjoy it, especially since it doesn't use calculus and stuff like that, but I was pleasantly surprised.

 

[verty]

https://www.amazon.com/dp/0393099709/?tag=pfamazon01-20
https://www.amazon.com/dp/0521198216/?tag=pfamazon01-20
https://www.amazon.com/dp/0521876222/?tag=pfamazon01-20

My recommendations.

 

[tridianprime]

Wow, thanks. I have seen French recommended before so i may get that. The other two I will look into.

 

[WellOrderedSet]

http://archive.org/details/DifferentialIntegralCalculusVolI
http://archive.org/details/DifferentialIntegralCalculusVol2
Calculus with Analytic Geometry, 2nd Ed. -- George F. Simmons
Calculus and Analytic Geometry, 4th Ed. -- George B. Thomas
Linear Algebra -- Jim Hefferon
Linear Algebra Problem Book -- Paul R. Halmos
Coding The Matrix: Linear Algebra through Computer Science Applications -- Philip N. Klein
Book of Proof, 2nd Ed. -- Richard Hammack
Differential Equations with Applications and Historical Notes, 2nd Ed. -- George F. Simmons
Ordinary Differential Equations -- Morris Tenenbaum & Harry Pollard
Applied Differential Equations, 3rd Ed. -- Murray Spiegel

 

[UltrafastPED]

Since this is self study I would make time for some number theory; does not require advanced mathematics, but does require a lot of insight and the proofs can be very tricky. Any number theory that you learn will be very useful in all of your advanced work and also in computer programming, especially numerical analysis and algorithm design.

You don't even need to buy books: http://www.numbertheory.org/ntw/lecture_notes.html

 

[tridianprime]

Ill definitely use those number theory notes especially as I'm going to delve into olympiad style maths and also IOI. Would you recommend any books to use for learning the mathematics used in IMO and also books to learn the content you need to know for IOI like algorithm analysis?

Courants books. Are they as useful for a pure approach because I heard they are more applied. Is there rigour?

 

[tridianprime]

I can't read Greek. Where should I begin with those number theory notes?

 

[UltrafastPED]

This website is a collection of notes - just skip down the list, checking for the language. Most are in English.

You should, of course, learn the Greek alphabet, upper and lower case.

 

[tridianprime]

I know the Greek alphabet. Thank you.

 

[UltrafastPED]

No need to send a message ... just hit the "thanks" button. :-)

 

[WellOrderedSet]

While Courant's book isn't as rigorous as Spivak's, it's certainly more rigorous than any of the more popular undergraduate texts. Stewart, for instance, is merely bearable: most of the proofs are there, dutifully interlarded, but they're generally useless with regard to the end-of-chapter problems. Simmons, 2nd and Thomas, 4th are similarly rigor-less, but either is better-written, and the latter is more difficult, with a reasonable introduction to linear algebra.

Calculus (Complete) by Edwin E. Moise is also quite rigorous, although it has some peculiar selections following its single-variable treatment.

Courant actually wrote another book on calculus, Introduction to Calculus and Analysis (Vols. I and II), with John Fritz. It was meant to improve upon, and therefore replace, his Differential and Integral Calculus. Nevertheless, it's an expensive book.

Another volume that you might consider adding to your library, instead, is Courant's What is Mathematics? (An Elementary Approach to Ideas and Methods). While it isn't a proper mathematics text, it isn't a coffee-table book, either. And it's affordably priced.

In the U.S., the sequence is generally calculus I-III -> lower-division linear algebra (sometimes a hybrid course, such as linear algebra & differential equations) - > lower-division discrete mathematics (or some sort of proof/transition to advanced mathematics course). Following that, a student generally branches outward, fulfilling major requirements. Contrary to popular belief, however, you can study lower-division linear algebra and discrete mathematics prior to calculus. They tend to require more "mathematical maturity," as the pedagogues say, but in-depth knowledge of calculus is typically of minimal utility in such courses.

 

[WellOrderedSet]

Another text that many people gush over is G. H. Hardy's A Course of Pure Mathematics. I've never read it, but at least a couple of editions of it are in the public domain. (Check archive.org.)