# I want to learn something new.

I'm a math/physics major, and I just finished my freshman year of college. I have almost 4 months off, so I decided I'd like to teach myself some subject in the realm of math. I've taken vector analysis/calc III, linear algebra, and advanced calculus.

Advanced calc. at my university is, as I understand it, supposed to be a somewhat rudimentary version of real analysis. We used Rudin's text (Principles of Mathematical Analysis), and went through chapter 6 (on integration), skipping sections liberally (ie, we completely ignored Lebesgue integration).

So, what would you recommend I do with myself? I'm looking for something interesting, that doesn't necessarily have to build on what I've already learned (chaos theory, I believe, is like this [am I right?]). I really enjoyed Calc III, so I was thinking about something like fluids, but I'm not sure if I have the background for that.

Any help would be appreciated, I have no idea where to start looking, what books to look into, etc.

Edit: I'll be taking probability and ODE (diff eq's) in the fall, if this is of any consequence.

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double post

Start studying those ODEs, they are the basis of mathematical physics.

chroot
Staff Emeritus
Gold Member
How about differential geometry or group theory? Both have direct applications in physics, and both are very interesting in their own right.

- Warren

Crosson said:
Start studying those ODEs, they are the basis of mathematical physics.

Is there any advantage to start teaching myself ODE now? I figured there would be no point since I'm going to take the course in the fall.

chroot said:
How about differential geometry or group theory? Both have direct applications in physics, and both are very interesting in their own right.

- Warren

Sounds good, I'll look into those two.

Thanks for the responses, if anyone has any more ideas I would be grateful.

Is there any advantage to start teaching myself ODE now? I figured there would be no point since I'm going to take the course in the fall.

Advatage #1: You will get an easy A in your course and be able to concentrate on other studies.

Advantage #2: You will get a head start on learning something that would take many lifetimes to learn.

Advantage #3: You have already taken linear algebra. LA is a very elegant viewpoint from which to study linear differential equations, but this is angle is usually not explored in a first course on ODEs. If you begin studying ODEs now, by the time you actually take your course you will have a much more sophisticated perspective of whats going on in class.

Advantage #4: In mathematics, you don't understand things, you just get used to them (Von Neumann). So the sooner you expose yourself, even a little, the better off you are.

Icebreaker
Crosson said:
In mathematics, you don't understand things, you just get used to them (Von Neumann).

I don't agree, but it seems that is certainly the philosophy behind math classes for non-science students. A friend of mine is taking "calculus for commerce", in which they don't even learn the delta epsilon notation for a limit. They are learning it a la "1/x^n goes really small when n is infinity, so let's call it zero!"

In mathematics, you don't understand things, you just get used to them (Von Neumann).

I think Neumann's quote is about how symbolic manipulation is at the root of what we call "understanding".

A friend of mine is taking "calculus for commerce", in which they don't even learn the delta epsilon notation for a limit.

The idea of a limit is directly useful to an economist, but the epsilon-delta formalism is only indirectly useful for increasing mathematical maturity. I think it is possible to understand limits without the epsilon delta notation. (As many great mathematicians did).