Self-Studying Mathematics: Where to Begin?

[Winzer]

So, for engineering, I am in collge and studying calc II, will go onto III, diff eq. ,linear Algebra, and eventually Analysis .
However I really want a good knowledge in all mathematics. So I am stuck what I should study on my own. Should I start with number theory?

 

[JohnDuck]

Many books on number theory may be beyond you at this point (I'm not really familiar with the literature, though). Real analysis, abstract algebra, and point-set topology would be a good starting place.

 

[Winzer]

Shouldn't real analysis be studied after I finish the regular calc?

 

[JohnDuck]

Not necessarily. Introductory courses in real analysis develop all the ideas you get in "regular" calculus courses with a focus on proofs and theorems rather than problem solving. You don't need the skills developed in the latter per se, though they may be useful at times.

 

[Winzer]

what is point-set topology?

 

[JohnDuck]

It's a good starting place for learning about topology, because it doesn't require much background (just some basic notions from set theory) and it develops ideas which will pop up again and again in more advanced maths. http://en.wikipedia.org/wiki/Point-set_topology"

 

[Pseudo Statistic]

Not necessarily. Introductory courses in real analysis develop all the ideas you get in "regular" calculus courses with a focus on proofs and theorems rather than problem solving. You don't need the skills developed in the latter per se, though they may be useful at times.

Indeed-- the typical first real analysis course is done concurrently with multivariable calculus here, so your ability to put up with the theorem-proof approach to math would matter more so than your calculus knowledge/techniques when it comes to learning real analysis.
Isn't it possible to take real analysis concurrently with calculus or linear algebra over there? (Depending on the course material, it might help to have a linear algebra background)

 

[Winzer]

Well I go to an Engineering Uni. and they have things pretty laid out ciriculum wise so I don't think I would be able to. I will have to check.
I am just in Calc II by the way

 

[JohnDuck]

The subjects I previously mentioned should be within your means. It's the kind of stuff math majors fresh out of the standard calculus sequence take, and at an introductory level they make minimal use of calculus (except real analysis--it is calculus).

 

[Winzer]

What is the difference between real analysis, and analysis?

 

[Pseudo Statistic]

Real analysis deals with $$\mathbb{R}^n$$ and analysis deals with everything including $$\mathbb{R}^n$$, i.e. inner product spaces, normed linear spaces, metric spaces, ... , $$\mathbb{C}^n$$.
...or so I think.
Anyway, as far as you being in Calc II is concerned, I don't think real analysis is too much of a stretch, as everything is developed from scratch (the field axioms) and so it's more about your proof/proof-following skills than your knowledge.

 

[MathematicalPhysicist]

Many books on number theory may be beyond you at this point (I'm not really familiar with the literature, though). Real analysis, abstract algebra, and point-set topology would be a good starting place.

if youre not familiar with the literature then how could you know that it's beyond him?
most first courses in number theory require a minimal background in maths, perhaps induction is all one needs to know beforehand, even this is sometime recalled again in the course\book.

anyway, there are quite a lot elementary number books, i used burton's, but there are more ofcourse.

 

[Winzer]

Real analysis deals with $$\mathbb{R}^n$$ and analysis deals with everything including $$\mathbb{R}^n$$, i.e. inner product spaces, normed linear spaces, metric spaces, ... , $$\mathbb{C}^n$$.
...or so I think.
Anyway, as far as you being in Calc II is concerned, I don't think real analysis is too much of a stretch, as everything is developed from scratch (the field axioms) and so it's more about your proof/proof-following skills than your knowledge.

Thanks Pseudo Statistic for clarifying that. yes I would like to gain and refine skills in proofing so real analysis sounds right up my alley, plus it will give more insight into Calc. I will have to go to my Uni.'s library and check one out.
Should I check out intro to real analysis, or just real analysis?

if youre not familiar with the literature then how could you know that it's beyond him?
most first courses in number theory require a minimal background in maths, perhaps induction is all one needs to know beforehand, even this is sometime recalled again in the course\book.

anyway, there are quite a lot elementary number books, i used burton's, but there are more ofcourse.

Ok so maybe the study of real analysis and some elementry number theory will do me good, thanks.
Should I get a book like intor to number theory?

 

[tronter]

If you want to learn intro topology, study Munkres Text.

 

[radou]

Start with google to see what you're interested in and what you can handle. I don't know what engineering field you're into and what you're really interested in, but numerical analysis has wide applications in most engineering fields, so, if you have a solid analysis/linear algebra background, you could get into that.

 

[Brad Barker]

i'm not sure if you really need any more math after linear algebra for engineering. any other math topics will be taught during courses.

(heck, my brother never had an actual course in linear algebra, and he's almost done with his masters degree.)

 

[^_^physicist]

I would frown upon number theory for engineering, but this is just because I don't know of many applied mathematics texts that make use of number theory (so I could be totally wrong).

In my personal opinion picking up multi-linear algebra and differential geometry would be a great place to start. You might be a little over your head in a differential geometry class depending on the approach and might need a class on PDE's first--but if you are going into any engineering that will require continuum mechanics differential geometry and a solid knowledge of tensors will be of use.

 

[Winzer]

Ok so here is the thing, I am majoring in BS. Engineering Physics, and I might later double with an applied math BS.
So yes I will be studying a lot of applied math: Calc I-III, linear, math physics, ODE's & PDE's. However, just for self interest and better knowledge of mathematics-especialy calculus- I woudl like to get into some pure mathematics.

 

[Pseudo Statistic]

Thanks Pseudo Statistic for clarifying that. yes I would like to gain and refine skills in proofing so real analysis sounds right up my alley, plus it will give more insight into Calc. I will have to go to my Uni.'s library and check one out.
Should I check out intro to real analysis, or just real analysis?

You should probably read the first few pages of each book, either from the library or from http://books.google.com, and decide which suits you better.
Since you'll be checking out of a library, you might want to try Spivak's Calculus. However, it lacks a few things like functions from $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$ and the calculus of such functions.

 

[Winzer]

You should probably read the first few pages of each book, either from the library or from http://books.google.com, and decide which suits you better.
Since you'll be checking out of a library, you might want to try Spivak's Calculus. However, it lacks a few things like functions from $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$ and the calculus of such functions.

I believe I own a Spivak equivalent:One-Variable Calculus, with an Introduction to Linear Algebra SECOND EDITION vol I - Apostol.