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.
"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.)
In my experience, almost nobody who has not actually learned tensor
calculus has a clue what a tensor is
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
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
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.
Mathwonk can be seriously obscure when he puts his mind to it, IMHO.
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.
The term "level" is undefined in this context, so I can't really
No. You are thinking of Laplace transforms; they are not at all the
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.
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
that word! At the end of the course, I asked about the
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
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
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.
Very, at least to start with.
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.
That's a rather poorly defined term, isn't it?
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!