Part III

Rudin has a nice chapter on Fourier
series, that is, representing a periodic
function f(t) with a sum of e^(iwt) for
overtones at frequencies w and where
each frequency w is a whole number
multiple of a fundamental frequency.
The linear algebra and geometry here are
that the e^(iwt) are perpendicular
projections of the f(t). So, the e^(iwt)
are orthogonal axes.

The line integral generalizes to the
exterior calculus of differential forms
(keep track of signs from the direction do
the integrations, i.e., as in line
integrals) and the fundamental theorem of
calculus generalizes to Stokes theorem
crucial in Maxwell's equations, fluid
flow, and partial differential equations.
Now we are close to differential geometry,
and Rudin also does the inverse and
implicit function theorems important in
differential geometry and parts of
nonlinear optimization, e.g., Lagrange
multipliers.

I warmly suggest that take a really fun,
one weekend, pass through Stokes theorem
parts of

Tom M. Apostol, Mathematical Analysis: A
Modern Approach to Advanced Calculus,
Addison-Wesley, Reading, Massachusetts,
1957.

Get it used, and pay whatever you have to.
This way get to see the 2-3 dimensional
versions and, really, mostly enough for
what physics and engineering do with
vector analysis and Stokes theorem.

In H. Royden, Real Analysis and the
first half of W. Rudin, Real and Complex
Analysis, can see the Lebesgue integral,
due to H. Lebesgue near 1900. In
essentially all cases where the Riemann
integral is defined, the Lebesgue integral
is also defined and gives the same
numerical value. But the Lebesgue
integral has more powerful theorems and is
defined in more general situations.

As in a 1933 paper of A. Kolmogorov, the
Lebesgue integral gives a solid foundation
to probability, statistics, and stochastic
processes.

In W. Rudin, Functional Analysis get a
treatment of distributions that cleans
up what physics tries to do with the Dirac
delta function and also covers spectral
theory.

From there, can go for a Ph.D. For that
will need mostly (A) pass the qualifying
exams and (B) do some original research.
The standards are commonly something like
"an original contribution to knowledge
worthy of publication" and for
publication, "new, correct, and
significant".

For the qualifying exams, first pick a
department, pure/applied math or some math
area in engineering, optimization,
probability, economics, computer science,
etc., get their description of their
qualifying exam topics and references,
study, and take and pass the exams. For
the research, do some.

One suggestion: For the research, pick a
real problem and use some math, at least
in part new, to get a good solution. Can
get "significant" from the importance of
the real problem. Can get much of "new" if
have the first or better attack on the
real problem. Can get "correct" if do the
math with careful theorems and proofs.
Can publish if pick an appropriate
journal. Might get to regard the math as
significant based on what it does for
the real problem and not just its
contribution to pure math.

Even the best US grad schools are hungry
for good students. With a good ugrad math
major and good work on independent study,
should get a good reception at grad
schools and at least a tuition
scholarship. In that way I got accepted
to grad school at math departments at
Cornell, Brown, Princeton, and more.

In a sense, independent study is
recommended: (A) That is basically what
research profs have to do for all their
careers. (B) At least at one time the
math department at Princeton stated that
no courses were offered for preparation
for the qualifying exams, students were
expected to prepare for the exams on their
own, courses were introductions to
research by experts in their fields, and
students should have some research
underway in their first year.

For a little on how to do the original
parts of math research, there is the A.
Wiles comment:

"Perhaps I could best describe my
experience of doing mathematics in terms
of entering a dark mansion. You go into
the first room and it's dark, completely
dark. You stumble around, bumping into
the furniture. Gradually, you learn where
each piece of furniture is. And finally,
after six months or so, you find the light
switch and turn it on. Suddenly it's all
illuminated and you can see exactly where
you were. Then you go into the next dark
room ...."

That's some of how to learn some math,
largely with independent study, and maybe
to get a Ph.D. But you might want more:
You might want your Ph.D. work to get you
a good start on a good career in
pure/applied math research/applications.
For that, look around, pick up what you
can in seminars and conferences, and get
what your profs can explain to you.

Broad point: Long the best opportunities
in applied math, even with some advanced
pure math prerequisites, have been in US
national security, especially within 100
miles of the Washington Monument.