Mathematical Reference: Overview of Fields & Formulations

[einKI]

Hi

I am searching for some kind of mathematical "reference".

My math knowledge is mainly from a five year technical school and some basic
university courses. I can solve equations (linear and differential), use vectors
and matrices, use geometry in 2/3 D, use probability and statistics, ...

But when I want to train myself additional knowledge I simply don't get it.
I think I lack knowledge especially in the formulation of mathematics (set theory, ..). When I read something about say Hilbert space I simply don't understand the equations and words used.

So what I need is not a book (tutorial, ...) that teaches me linear algebra once again but one that gives me a overview of the fields of mathematics with the tools and formulations used.

If anybody know something similar please give me a link
thx
einKI

 

[Crosson]

Here are good web enyclopediae:

www.mathworld.com

www.wikipedia.com

Mathworld is smaller then wikipedia, but does not have the reliability issues that wikipedia does.

Its interesting that you mention linear algebra, because this can be a good bridge to more abstract areas of math. Do you know what a vector space is?

Have you ever looked at powerful application of linear algebra to the solution of differential equations? Maybe your technical school discussed Sturm-Liouville theory.

Another way to build on your present knowledge is to dive into Nonlinear Dynamics, a subject that starts with questions about differential equations and leads to topology and analysis.

What interests you in math?

 

[einKI]

Hi

Its interesting that you mention linear algebra, because this can be a good bridge to more abstract areas of math. Do you know what a vector space is?

Well yes and no. Yes because I have used that stuff many times and No because it wasn't even mentioned that that is called a vector space.
And that's the main problem I think we did do much calculation and to less
math.
For example when I read about Hilbert Space there was a reference about
Euclidean spaces so I looked it up only to discover that is what we called
a vector we not even mentioned that there can no anything else than
finite dimensional vector spaces and therefore doesn't use the right
description.

Have you ever looked at powerful application of linear algebra to the solution of differential equations? Maybe your technical school discussed Sturm-Liouville theory.

well the name sounds alien to me but since in wikipedia Bessel's equation is
listed as an example and we solved this i suppose yes it was just not called
Sturm-Liouville once again.

Another way to build on your present knowledge is to dive into Nonlinear Dynamics, a subject that starts with questions about differential equations and leads to topology and analysis.

maybe that's a good idea to start with. Will try it

What interests you in math?

well up to a certain degree it's scientific curiosity or however you will
call it but since I am going to study computer science and physics soon
I am very interested in formulation of quantum mechanics, relativity etc..

thx
einKI

 

[neutrino]

You may want to refer to the list for maths textbooks.
http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html#background
It's mainly for general relativity, but it also includes topics which are essential learning for physics, like linear algebra, DE's, analysis, etc.

For a quick reference, I second Crosson's recommendation of mathworld. Planet Math is also another useful site. http://planetmath.org/

 

[Crosson]

Well yes and no. Yes because I have used that stuff many times and No because it wasn't even mentioned that that is called a vector space.

That is a loud and clear NO. One of the hurdles to you achieving your goal, is that you have to be willing to go back to things you learned long ago, and relearn them in a new (more abstract) language.

In abstract linear algebra, vectors are not defined as ordered n-tuples of coordinates, they are defined by their algebraic properties:

http://mathworld.wolfram.com/VectorSpace.html

The euclidean vectors you studied are a special case of a vector space. Another example is the set of real numbers, where the "addition" operation is normal multiplication, and the "multiplication by scalar" operation is exponentiation. This example is not important, but it serves to show how different an abstract vector space can look from "physics vectors".

well the name sounds alien to me but since in wikipedia Bessel's equation is
listed as an example and we solved this i suppose yes it was just not called
Sturm-Liouville once again.

Imagine a high school student who knew how to plot y = x^2. You ask them what a function, they say they don't know what it is but wikipedia says x^2 is a function, so they presume the learned the same thing.

The example is not perfect, and I don't mean to mock, rather I hope to excite you by alluding to the scope of the unknown.

I am guessing you know about Fourier series, Bessel series, Legendre Series, Hermite, Laguerre, etc. Did you ever wonder why these series work? Why any continuous function can be represented Bessel/Hermite/Fourier series? What do all these named classes of functions have in common? They are all solutions to second order linear ODEs --- Sturm-Liouville theory studies the connection to linear algebra that shows how ODEs of this type have solutions which form complete basis for a very large set of functions, viz. all those whose derivatives are discontinuous on at most a set of Lebesgue measure zero.

Sturm-Liouville theory is also critical to understanding non-relativistic quantum mechanics, in which the fundamental question is "how does the wave function evolve with time"? For the linear problems which are exactly solvable, their is a complete answer to the question in terms of an expansion into a Sturm-Liouille type series of special solutions. This mathematical gem is buried deep under all of the standard PDEs stuff you might have learned with QM, but I haven't seen even an allusion to it in a QM book.

 

[einKI]

That is a loud and clear NO. One of the hurdles to you achieving your goal, is that you have to be willing to go back to things you learned long ago, and relearn them in a new (more abstract) language.

Well I think that summarizes it pretty well. And I am definitely willing to learn
old things in a new way.
The question remains how I should start.
Is it a good idea to work mathworld or wikipedia or whatever from set theory
, linear algebra, etc... up or is there any book which I should read?

thx
einKI

 

[Crosson]

Learning math is like learning any foreign language: your best bet is total emmersion.

Think of yourself as someone who took a year of french at an American school. Then you move to france and hang around people who speak french all day --- you can hardly understand them, but you are able to catch a few snippets, and each of these snippets is precious because you are desperate for human contact, desperate to get out of the world that is half nonsense. Then, when things start to click, there is a snowball effect and it gets easier and easier until you reach the point where you learn new french vocabulary by pulling a french novel off the shelf, that is, you never again return to the world of nonsense.

This is how I advocate learning mathematics!

Go to a university library and borrow all the books that seem interesting to you, that tickle your curiosity. If you can understand the first sentence, see how far you can go in a single sitting. If it gets too hard/boring skip some pages and try a different section, if all the sections are too hard/boring return the book. You are not trying to memorize, you are not trying to understand (enjoying yourself means you understand well enough, details don't matter at this point), you are just trying to expose yourself.

Then when you are exhausted from textbooks go read www.mathworld.com randomly, following links that seem interesting to you and exploring categories that sound interesting. The reason I recommend mathworld over wikipedia for someone at your level is because mathworld relates better to the maths you already know, and because the categories are easy to browse (wikipedia is really best if you already know the name of what you are looking for).

After doing this for a while, you will have boosted your mathematical maturity enough to read textbooks like newspapers, and learn those things which interest you in their full detail.