Order of learning Mathematics

MPKU

As a 4th level student of University majoring in Pure Mathematics & Physics, I think set theory would be a good approach, of course that and calculus. You can even try studying Real Analysis while doing Calculus, infact, it is probably for the best. However, with a foundation of higher mathematics(proof writing, etc.) this will probably be a huge challenege, but you certainly seem up to that :). I always found set theory enjoyable, and not very hard to learn at all; I'm sure you've already went over some in algebra (i.e. finding domain of specific functions). As for books, I'm not so sure..

There are great videos on the web of tutorials and lessons on Advanced Mathematics, just search for them, not to say you can learn math just by watching someone else do it; if basketball players watched a video on how to play basketball, that wouldn't make them good players, so you've got to get out there and try the problems.

Something I can't stress enough is to learn the theory and implications of the topics you study, not just the application (although that is a very important part of why we do mathematics). Learning the theory of calculus and WHY we study calculus will help you so much in your studies.

nlsherrill

After elementary algebra/trigonometry/geometry/pre-calculus I think a good progression is(in order):

Calc 1
Calc 2
Calc 3
Differential Equations
Linear Algebra
Partial differential equations(not required, but I recommend)
Set Theory
Abstract Algebra
Real Analysis
....

I think there are a few main things to point out about a good math route. First of all, it doesn't need to be strict, but you surely need to take the courses I listed minus PDE's. I recommended PDE's solely because there is so much more you learn in that class(at my UNI) than ODE's that I think every mathematician should see. For example, you really get to see how linear algebra is the basis for it all. Also, I think you should put off real analysis until you have taken set theory and abstract alg. This is because the jump from set theory proofs to real analysis proofs is quite a large one, so as much practice with this beforehand would be best.

I don't really think the order of taking calc 3, linear, and ODE's really matters that much. I took linear and ODE's at the same time and that was fine, although I didn't really see their connection until PDE's. I think you can take ODE's after calc 2, just make sure you know partial derivatives first.

micromass

I agree but I would place Linear algebra before Calc 3 and Diff eq. Those subjects tend to make more sense with a knowledge of linear algebra.