Where to go from here?

[hkfranz]

Hi everyone. My name is Hans, and I'm new here.

 

I'm an undergraduate student at the University of Michigan studying electrical engineering and performing arts technology. Throughout high school I was very interested in mathematics, but now that I'm in college and studying other things I don't have the time to take more advanced, formal mathematics courses. The last courses I took were in multivariable calculus and linear algebra. I'm still interested in continuing my study of math, though.

 

So here's my question to the community:

 

Where do I go from here?

 

I'd like to get some books and continue to learn, but I'm not sure which areas of mathematics I'd be ready to tackle.

 

Can anyone recommend to me any interesting fields? They should fit somewhat these requirements:

 

- Understandable to someone who has taken courses through multivariable calculus and differential equations.

- Interesting outside of a practical sense (i.e. NO statistics!).

- Quality and quantity of available resources.

 

I'm not ready to give up on math yet!

 

Thanks!

[lurflurf]

Lots of interesting subjects are availible to studens with the equivelant of one year of college math and you have had two! You should try and decide where you interest lie. A good place to start given your requirements is one of the many introduction to applied math books (they are often titles something like advanced math for physics and enginearing). Such Books normally require a year of calculus, they include material on vector calculus, odes, and linear algebra (some of which you can skip if you know it); next they deal with some finite calculus, numerical methods, pdes, fourier series, integral transforms, complex variables, and calculus of variations. These books have a variety of interesting material and if one of their areas especially apeals to you you can find a nice book that deals with one of the areas in geater depth. Vectors, Tensors and the Basic Equations of Fluid Mechanics (Dover Books on Engineering) by Rutherford Aris is a nice book that assumes a year of calculus (if tensor or fluids interest you). Also you could start on algebra or calculus (ie those that presuppose your level of knowlege).

[timo]

It´s hard to tell what you´d like. However, apart from the standard answers that would be topology or differential geometry I´d say you might want to take a look at general algebra and perhaps group theory in particular. In fact, that´s what I (personally) consider "real math" in contrast to stuff like differential equations or linear algebra which every engineer or natural scientist learns.

[hkfranz]

Thanks for the responses!

 

As per Atheist's comment, I think I'm a lot more interested in what you call "real" math - I already have enough applied math to deal with as an engineer. I've been studying discrete mathematics (used for signal processing and such) lately, and even that is a bit too "applied" for my recreational tastes.

 

So I think I'll start looking into group theory (is this the same as set theory, of which I occasionally hear mention?) and maybe some topology.

 

Also, lurflurf's suggestion of "algebra or calculus" intrigues me. What sort of things do more advanced levels of algebra and calculus cover? How are these different from the precalculus algebra and differential/integral calculus I've taken?

 

Again, thanks!

[timo]

Is this [group theory'] the same as set theory, of which I occasionally hear mention?

From what I know: No. A friend of mine did his Diploma Thesis in Set Theory (if that really is what I´m thinking of, atm - I´m not entirely familiar with the english terms for the different branches of math). From what I´ve seen there, Set Theory is really, really abstract. I once took a look in one of his books and shyed away when I read the first Theorem which startet with something like "Let T be a complete theory ...". Well, the definition for a complete theory wasn´t that hard to understand when he explained it to me (but I forgot it anyways) but this sentence still left an impression.

Group Theory is much more illustrative in comparison to that as it deals with symmetries and their representations a lot.

[lurflurf]

In introductory Algebra one is usually concerned with algebraic structures and relationships between them. In previous work you have seen specific examples of groups, fields, rings, vector spaces, modules, division algebras, lie algebrasand others. Now you can learn about these stuctures in general, learn new specific examples, and how different stuctures relate. For example since log(x*y)=log(x)+log(y) for real numbers x,y>0 we see that in some sense multiplication of positive reals is doing the same thing as addition of reals. Group theory is a part of algebra, but it is incorrect that it is not used in applications. Group theory is used alot in chemisty, physics and enginearing. Though the perspective is different, they are less interested in general groups and are more interested in groups thar repersent symetry and the matrix repersentations of such groups.

 

In calculus many new results are given and known results are proven. In many intro calculus books there are lots of phrases like "for proof consult a more advanced book" or "this step is valid but consult a more advanced book to see why". Those type of questions and more are answered. Also more general integrals are studied such as the Lebesgue Integral, Lebesgue-Stieltjes Integral, and Riemann-Stieltjes Integral. Of great interest is what happens when the order of certain limiting processes are interchanged. For example conditions under which a sum can be integrated term by term. The epsilon-delta limit proofs are used more than in most intro courses. Differential forms are often introduced as a new way to view vector calculus. Special funtions are studies such as the gamma function, hypergeometric functions, and functions arising as integrals of certain elementary functions such. There is often some topology which is the study of properties of sets that are invarient under continuous deformations. Sometimes there is some complex variable theory where calculus is done using complex numbers. In general calculus is considered from a more theoretical perspective and less time is spent finding integrals and derivatives of elementary functions.