What's the Best Self-Study Plan for Advanced Math After Basic Courses?

[tamtam402]

I have taken 3 calculus courses (First one was Diff. Equations, second one was integrals, and the third one was a mix of both), a statistics course and a linear algebra course.

I'd headed into engineering, so my "pure" math education is stopping here. Maths interest me a lot and I'd like to keep learning, I want to teach myself what I could see if I was to head into an undergrad math program.

Can someone "plan" me a smart order to learn this stuff, with recommended books if possible? I want to learn this stuff for real, I don't want to read books and skip over everything without doing the recommended problems, so I don't need 5000 recommendations. 2-3 books that ressembles something an undergrad would learn would be perfect. Also, what's the best way to learn how to write proofs? Is there a book on that? I think that will be the hardest part to teach myself without a teacher, aye?

 

[tamtam402]

Bump. No one got recommendations on the next logical step for my math education? I'm trying to learn what a math undergrad would learn at school, at a much slower pace since I'm about to attend university in an EE program.

What's the next step after calculus 3 and linear algebra? I was told this is the point where undergrad math majors start learning how to write proofs in a much more rigorous way; is there any good books to learn how to do that?

 

[Diffy]

Every school has different syllabuses, teachers and courses. Your best bet is to talk with someone in the math department at the university you are in (or were in) and get recommendations from them. This will be more relevant to you personally in regards to continuing your education rather than advice from us.

In regards to proofs. The next book you pick up in order to progress your mathematics education do this: Read up until the point at which it says THEOREM, read the theorem and try to prove it yourself without reading the authors proof. Once you think you have done it, read the proof in the book and compare. You can only get better at doing proofs by practicing and reading them.

 

[tamtam402]

Every school has different syllabuses, teachers and courses. Your best bet is to talk with someone in the math department at the university you are in (or were in) and get recommendations from them. This will be more relevant to you personally in regards to continuing your education rather than advice from us.

In regards to proofs. The next book you pick up in order to progress your mathematics education do this: Read up until the point at which it says THEOREM, read the theorem and try to prove it yourself without reading the authors proof. Once you think you have done it, read the proof in the book and compare. You can only get better at doing proofs by practicing and reading them.

I wanted to do that, until I realized that teachers don't always use textbooks at the university level. OR, if they use textbooks, they have to be complemented by notes given in-class by the professors. I came here for advice because I figured I'm not the only person in the world that wants to learn higher level mathematics without attending university, and there HAS to be complete books out there. What I mean by complete books are books with theory, exercices, and if I'm lucky, some of the exercices should have the answers available.

Now that I'm done with calculus I, II and III and linear algebra, I could give pointers to someone trying to learn this stuff, ex: you have to learn about derivatives before you can learn about integrals! Here's a good book:

Knowing that you have to understand derivatives before understanding integrals is obvious, when you've already covered and learned the subjects. As a guy who's knowledge stops at Calc 3, I don't know what I "can" or cannot learn at this point. That's why I need pointers. Math has a lots of branches, and while it "makes sense" to someone that has more knowledge, someone in my position doesn't know a logical order to learn this stuff. Of course there's more than 1 path, but I couldn't even find one valid path by myself.

As for proofs, but I've seen people here mention stuff like: "You can disprove this by giving one example that shows the statement is wrong, it's called proof by <X>". Stuff like that. Basically, I think there's some theory in regards to proofs, no? Isn't there a "basic" set of tools that I can use and apply to any mathematical proof? The kinds of proofs we see in college are pretty basic, and we skip over some of the theorems given in the books.

Sorry about my sub-par english by the way, english is my second language, although I'm good enough to read and understand books in english.

EDIT: would teaching myself physics be a good way to learn some higher level maths? Ex: Advanced classical mechanics I is the first university level mechanics course given at my physics university. Would the mathematics needed in that course be covered in an Advanced classical mechanics I textbook? That might be a valid way to teach myself more mathematics, no?

 

[Liquid7800]

Hello,

I really respect what you are going for...I am going for a pure math higher education goal, and I realize you need to somewhat specialize sometime, so I want to also learn very specific subject areas along the way---and to do that I may have to do it on my own.

In terms of a math 'pedagogy' I recommend looking at the progression here as well as the resources:

http://ocw.mit.edu/courses/mathematics/"

I think this has everything you may want or need as they have (for some not all):

Lecture notes
Projects and examples
Image Galleries
Selected lecture notes
Projects (no examples)
Online textbooks
Assignments and solutions
Exams and solutions
Multimedia content
Assignments (no solutions)
Exams (no solutions)


I am going through these myself and the only thing I really needed was a desire and proper order of study...hope these help you.

 

[stevenb]

What's the next step after calculus 3 and linear algebra?

What about re-exploring calculus from a differential forms point of view?

"Advanced calculus: A differential forms approach", by Harold M. Edwards

https://www.amazon.com/dp/0817637079/?tag=pfamazon01-20

 

[Liquid7800]

I agree with stevenb, I didnt even 'know' about differential forms approach until I started playing around with the operations and then was led to the area(s) of advanced calculus

 

[tamtam402]

Thanks for the suggestions guys! As for the mit course, do I have to purchase the recommended books to follow the courses?

 

[General_Sax]

Yes, you do. I'm selling all the books + a bridge for only $189.99!

;p