Re: Self-education for a late-onset math geek

On Apr 30, 12:50 pm, DanEsch <daniel.a.e...@xxxxxxxxx> wrote:
On Apr 28, 1:07 pm, DanEsch <daniel.a.e...@xxxxxxxxx>
wrote:

Do you want to explore different branches on maths of
undergrad level,
or you have something more specific in mind? Do you
feel you enjoy
doing maths in general and want to learn more?

The latter. Independent reading is showing me overlaps between elementary real analysis, elementary number theory, introductory abstract algebra - enough that I'm not sure which string to unravel first.

What do you mean by 'basic knowledge of calculus'?
Did you take one or
two courses of single-variable calculus? (Normally
the second covers
covers things like improper integrals, some advanced
techniques of
integration, a bit of Taylor Series and Taylor
Polynomial and may be
kind of a gentle introduction to diff equations.)

Single-variable in high school, multi-variable in college - the problem of course being that the single-variable stuff wasn't appropriate prep for the mulitvariable course I did take.

Very well, then, your situation is fairly closed to mine, so I can
share some of my experiences with you and hope that will help.
Apparently you did some exploration of 'post-calculus' things, such as
Real Analysis, Number Theory and a bit of Abstract Algebra. I did the
same thing, but by actually taking the introductory courses on
undergrad level. My major was Computer Science, in fact, but I took
many math courses on undergrad level as well, practically everything
you can think of: Calc I-II-III, Diff Equations, Linear Algebra, Intro
to Complex Variables, and also Probability and Stochastic Processes.

As I was taking the Calculus, I was always curious where all those
Theorems came from. Well, the foundations of Calculus is exactly what
(Real) Analysis is all about. I took introductory course on Analysis
on undergrad level, liked the whole thing and proceeded taking
analysis on grad level; normally introductory course on analysis are
either one or two courses, I'm finishing the second one now. So if you
feel like learning the proofs of many theorems that are introduced in
Calculus without proofs (or without sufficient proofs), the notion of
"delta-epsilon" definition of the limit etc etc is your cup of tea,
most certainly learn analysis. The interesting thing about Analysis is
that it really has no prerequisites. Well, that's not entirely
correct. Two prerequisites are (i) Logic and proofs and (ii) basic set
theory. What you are doing is mostly proving things. Very different
from Calculus where you mostly compute some things. I can recommend
the book by Lay, "Analysis with Introduction to Proofs", at least the
first several chapters deal with logic, proofs, and set theory, I've
found them very useful. The whole book, unfortunately, is not entirely
suitable for learning Analysis on your own. We used it as a textbook
when I took my undergrad course but professor kept using another book,
Analysis by Kirkwood, I have that one as well. What I would recommend:
read about logic, proofs, and set theory, Lay is completely adequate.
Then you can still use Lay, but you may need some alternative
resources as well. The one I highly recommend: 'Interactive Real
Analysis', online.

http://web01.shu.edu/projects/reals/reals.html

It start with discussing sets. It does not discuss proof techniques. I
would say, Lay plus this aforementioned resource should get you going.
That's plenty to learn. Really a lot. By the end of this cycle you'll
have some better idea of why integration and differentiation work, and
much much more. If you study thoroughly, that should keep you amused
for many months to come.

Now you have mentioned abstract algebra. Well, I took one course of
abstract algebra, again, on undergrad level, to get my feet wet, so to
speak, and if I decide to enroll into grad program (and I probably
will), I'll have to take two courses of Abstract Algebra, just to pass
the qualifying exam. But seems like I have no taste for that staff. A
bit too abstract, for my taste. Interestingly enough, seems like the
field of Analysis does not use lots of abstract algebra except for
one: vector spaces. That brings me to the second subject I highly
recommend to learn: Linear Algebra. Think of it as a branch of
Abstract Algebra which is entirely focused on Vector (or Linear)
Spaces. The knowledge of Lin Alg becomes extremely important when you
move to Multivariable Calculus, and more advanced things. For
instance, in Analysis II we discussed Fourier Analysis and suddenly we
ended up using the concepts of Vector Spaces, and quite extensively
so. So, once again, get into Linear Algebra. The topic may appear
highly specialized. This is not the case. Vector Spaces are probably
the *most* important algebraic structure used in Analysis. (Of course,
one may argue that the most important algebraic structure is group,
but I'm talking from analytical point of view.)

I'm not sure which resource to recommend, though. You probably
familiar with ocw.mit.edu. It has a video of all lectures on Linear
Algebra given by Professor Strang, with his textbook. I happened to
take that course (at MIT, not online), and I really didn't like the
book because it stresses the techniques of manipulating with matrices
but does not give the big picture: the fact that any finite vector
space is isomorphic (structurally identical) to vector spaces in
Euclidean Space, so the staff you are learning is *extremely
important*. Later I retook the course, the second course I took was
quite different in a sense that it did provide the broader view. The
textbook we used was by Otto Bretscher, Linear Algebra, I liked that
textbook, many people don't. It does discusses some broader topics,
like Inner Product Spaces. Well, take a look at the MIT online course
anyway. (You'll need a textbook to do the exercises and as a way to
backup the video lectures.)

So, to summarize, what I would recommend, start exploring both
Analysis and Linear Algebra. Note that the basic analysis keeps all
the development in R^1 (single-dimensional) space. At some point you
probably want to review your multivariable calculus; I would suggest
to do Linear Algebra first.

I never took Number Theory, so I can't recommend anything in this
respect. As it was suggested, it may be fun. For analysis and algebra
you need some basic facts from Number Theory, you probably can learn
on a fly. I do have a textbook by Jones and Jones which I find very
readable. (I'm thinking of learning some Number Theory before I start
seriously taking Abstract Algrebra on a grad level, it's nice to know
some facts like "little Fermat Theorem" beforehand.)

I did take differential equations but this is in the real of applied
mathematics. Unless you plan to use it, don't bother. Interesting
topic, but not much fun.

Other things to explore: the staff which sometime is presented as a
part of Calc III or sometime as "Vector Analysis", and mostly things
like Line and Surface Integrals, Grad, Div, and Curl; Green, Gauss and
Stock's theorem. The way this topics are presented in Calculus is with
very strong applied flavor (mostly electrical engineering). However
generalization of these concepts led to another interesting flavor of
analysis: analysis on manifolds. So see if you can learn Vector
Analysis on your own as well. I'm not sure which resource to
recommend, though.

A final note. In mathematics passive learning (that is, just reading
the textbook) does not work. You have to do exercises, and do as much
as you can. That's only one way to insure you understand what's going
on. It's a lot of hard work. I guess the fun is when you did finish
the proof or something like that and you look at all the work you've
done and you are tired by happy. Like climbing the mountain. Finally
you are there, at the summit. You have to be really dedicated and
heavily driven just by being curious about math. I guess ultimately
you *would* want to take a few courses in school, not just studying
everything on your own.

Good luck.
.

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