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sal
sal is offline
#7
Sep29-08, 12:02 PM
P: 78
I talk too much, and this post has gone 'way overboard. I won't be responding in this thread again, unless some very specific question (with a short answer!) comes up.


Quote Quote by Ulagatin View Post
Hi Sal,

I am very ignorant of analysis, and so I would love to hear what it is
all about
"Analysis", at least where I came from, was multivariate calculus
"done over again, right". But I suspect the term may cover quite a
bit more than that depending on where you are and who's using it.

Real (as opposed to complex) analysis is calculus in Rn,
with everything proven rigorously as you go; epsilonics are the order
of the day. A few things are a little hard to grasp at first, such as
the definition of continuity which may seem to have no apparent
relation to what "continuity" means in calculus-1 (a function is
continuous if and only if the inverse image of any open set is open).
Typically one doesn't go nearly as far into techniques for solving
particular integrals and such in an analysis class, as one does in
basic calculus; the issue is more with definitions than with solutions
to numeric problems.

(Some schools kind of merge calculus and analysis. I think Harvard
may do it that way.)

Quote Quote by Ulagatin View Post
I was certainly not aware that
this is where tensors are from.
In my experience, almost nobody who has not actually learned tensor
calculus has a clue what a tensor is

Quote Quote by Ulagatin View Post
Now, regarding tensors, are they as scary as many people say? Or is
this a percieved image with no realism to it?
You are driving too far ahead on the road. There are a few things you
should be aware of.

1) You should get through calculus, linear algebra, and all the other
things which come up in a good two-semester post-calculus algebra
course before you worry much about tensors. (A half-semester of
linear algebra is just an introduction, please note.)

2) Very few people who start out double-majoring in math and physics
actually get two degrees, one in math, and one in physics. Don't
bet the (psychological) farm on getting a double degree; odds are it
won't work out that way.

The reason (2) is true is mostly because it's a wide world and it's
hard to see what the possibilities are from the vantage point of a
high school classroom. Math and physics stand out, everybody knows
about them, they're really "techie", they have a certain caché, and a
lot of people think it would be really "cool" to be in both fields.
Maybe they seem like "ritzier" fields than some pedestrian old
engineering discipline, for instance. And so lots of folks enter
university thinking they're going to be math/physics majors all the
way.

But when you get to university you find, first, that there are an
awful lot of fields with interesting work, and with extremely
challenging work -- theoretical ("pure") math and physics have
no monopoly on being difficult to master or having cool,
sophisticated mathematical underpinnings!

And, second, you learn a fact about the world: A degree in
theoretical math or theoretical physics is excellend preparation for
... well, teaching other people to do it, at the university level, if
you happen to be brilliant and if you find a tenure slot somewhere. If
you can't find a tenure track post at a university, you can always,
um, paint houses or something. To put it bluntly, it's nice work
... if you can get it.

Quote Quote by Ulagatin View Post
Mathwonk states in his paper on algebraic geometry (that I briefly
skimmed through) that this subject is related to the study of geometry
of polynomials, or as he now likes to think, the study of geometry of
rings (or was it groups?). This sounds interesting, but as with a
large chunk of higher mathematics, I am again very ignorant of the
subject. I am aware that groups and rings are important concepts in
abstract algebra, but that is all.
Mathwonk can be seriously obscure when he puts his mind to it, IMHO.

Quote Quote by Ulagatin View Post
I will have a closer look at solid state physics, as it is an area of
theoretical physics that I have not looked at. I think, at my age, it
would be better for me to see the "big picture", perhaps, something
that I haven't been doing.
Solid state physics is seriously cool, very important, deep,
difficult, complex ... and not what I would call theoretical physics,
at least not in the same sense that, say, astrophysics is
"theoretical". In fact, I would describe it as "Applied" with a
capital A. As I said, it's solid state physics which makes your
computer go. If you don't know what I mean when I say that, then you
definitely have something to learn about solid state physics, and
about electrical engineering, as well.

Entering an applied field is not "death", and what's more there's more
demand for people who work on things which have applications than for
people who don't.

Quote Quote by Ulagatin View Post
I am guessing, because higher-level geometry and analysis are "pure"
branches of mathematics, they are on a completely different level,
than say, partial diff. eqn's?
The term "level" is undefined in this context, so I can't really
answer that.

Quote Quote by Ulagatin View Post
Is it not Fast Fourier Transforms (or FFTs) that govern the rendering
of computer graphics from the graphics pipeline?
No.

Quote Quote by Ulagatin View Post
Is it correct that
Fourier transforms are essentially methods to solve differential
equations?
No. You are thinking of Laplace transforms; they are not at all the
same thing.

Fourier transforms move a function from configuration space (where it
has, like, values and stuff) into phase space (where you get to see
the frequency spectrum).

They are used in quantum mechanics, where each quantum state could be
said to be one component of the Fourier transform, and they're used in
signal processing, because the transform of an acoustic signal is
exactly its frequency spectrum, and the transform of an image is its
spactial frequency spectrum. But AFAIK polygons get blasted into the
frame buffer in your graphics card with no use of Fourier transforms.

Quote Quote by Ulagatin View Post
It was these types of questions that I liked solving, and as a
sidenote, geometry is my favourite branch of mathematics so far.
OK time for the bucket of ice water. Sorry, I apologize for this in
advance. I'll start with a little background, and I expect it'll
rapidly become obvious what I'm leading up to.

When I was an undergrad I took a semester of organic chem, and I
really loved the "carbohydrate game", as they called it in that
course, which was looking at how carbohydrate molecules could be
mushed around from one form to another. And when we finished that
unit of the course, I asked about further information or opportunities
to study that sort of thing, because it was far and away the coolest
bit of chemistry I'd ever encountered ... and they told me "There
isn't any more. That was all of it. That field is dead, it's been
mined out, nobody's studying it any more."

I took a semester of point set topology, and I thought it was just the
coolest thing ever. The textbook (Munkres) was more fun to read than
a dime store novel, and the "mental pictures" it all produced were
fabulous. Talka about "visualizeable" -- it practically
defined that word! At the end of the course, I asked about the
next level of point set topology -- what was the follow-on
course? For surely, I wanted to take it! ... and they told me,
"There isn't any more. That was all of it. That field is dead, it's
been mined out, nobody's studying it any more." The "followon" to
point set topology was algebraic topology, which bears about as much
resemblance to point set topology as Mandarin Chinese bears to
Fortran.

Now, geometry.... Let me put it this way. I majored in math at a
good school, and in all my college years, I never took a geometry
course, in the sense that you mean it; what's more I'm not sure any
were offered at my school (aside from the differential/algebraic
sort). In fact, the only geometry class I've ever taken in my life
was a high school class called "coordinate geometry", and it didn't
bear much resemblance to the plane or solid geometry classes of
yesteryear. I started to read a bit about plane and solid geometry
last year, just for fun, from a book (published by Dover) in which the
author talks about the stuff "they never taught you in school"
because, for the most part ... geometry isn't even taught any more.
Calculus has almost entirely superseded it.

On the bright side, there are a lot of other fields in math which are
just as visual, and just as pleasing. The downside is that, for the
most part, you must work harder to visualize things when you're not
talking about the actual geometry of the universe we (appear to) live
in.

Quote Quote by Ulagatin View Post
Take note that these courses are generally taught by people who have a
really solid mathematics grounding (ie a degree in it, most commonly),
as opposed to what I've heard about the American AP Calculus courses
(I have heard rumours that they are often taught by incompetent
teachers. Is this true?).
I don't know; I only attended one high school, after all. The math
faculty in my high school was variable, some better than others;
however, the factulty were generally a whole lot better at college.

Quote Quote by Ulagatin View Post
Also, a few questions regarding analysis, before I finish writing this
post. How intuitive is this field?
Very, at least to start with.

Quote Quote by Ulagatin View Post
Does it throw you in the deep end?
I don't know what that means.

Since you've probably never encountered much in the way of serious
proofs before, and you've almost certainly never been called upon to
prove much of anything, it's likely to be difficult to start with, but
an introductory analysis class is geared to people who are all going
through the same struggle, so it's not really a problem.

The first analysis test I recall taking had ten questions, and each
one was just a statement, no questionmark, no "please do...", no
nothing. Just a statement. And for each one, we were expected to
either prove it or find a counterexample.

Quote Quote by Ulagatin View Post
And how elegant is the field?
That's a rather poorly defined term, isn't it?

Quote Quote by Ulagatin View Post
The "Topics in Advanced Mathematics" course covers various topics such
as geometry, set theory, number theory, history of mathematics and a
few others. If I'm not mistaken, you are able to choose the area of
the list of topics provided that you wish to study. The course itself
only requires that you have studied a second-year mathematics
unit. Perhaps if I can choose, I wold be wise to choose to study
algebraic and differential geometry?
It would be wise to choose something based on what you've learned at
the time you pick. It's too early to try to make a decision like
that; wait until you've gotten through calculus and algebra!

Quote Quote by Ulagatin View Post
Differential equations and linear algebra
are covered in the same course
Eh??

Quelle bizarre!