Posted by
Frank Wimberly on
Oct 09, 2005; 6:11pm
URL: http://friam.383.s1.nabble.com/Getting-Math-Chops-Back-Up-tp520638p520640.html
I should just say, "What she (Cheryl) said..."
I had sent the following to Owen "offline" but I was encouraged to send it
to the list:
"Owen,
I have a lot to say about this. In particular, there is this difference
between applied (or applicable) mathematics and "pure" mathematics.
Mathematics departments tend to emphasize the latter. After my sophomore
year at Berkeley the only thing we did in math courses was to learn to prove
theorems. Our former classmates in Calculus 1-4 who were physics majors,
for example, took "advanced calculus for science and engineering" while we
math majors took "introductory real analysis". They learned differential
equations and we learned the Heine-Borel theorem. They learned about
matrices and linear transformations; we learned about groups, rings and
fields. When we asked, "When are we going to learn about the stuff the
physicists are learning?" we were told, "If you learn this stuff you can
always learn that stuff."
Maybe this was just characteristic of that time and place (Berkeley, 1960's)
but I doubt it.
Frank"
In my graduate course in complex analysis we worked through Ahlfors in great
detail. There were several problems the professor couldn't do (he was a
specialist in the area).
---
Frank C. Wimberly 140 Calle Ojo Feliz Santa Fe, NM 87505
(505) 995-8715 or (505) 670-9918 (cell)
wimberly3 at earthlink.net or wimberly at andrew.cmu.edu or
wimberly at cal.berkeley.edu
-----Original Message-----
From:
[hidden email] [mailto:
[hidden email]] On Behalf
Of Cheryl Fillekes
Sent: Sunday, October 09, 2005 12:43 PM
To: Friam at redfish.com
Subject: Re: [FRIAM] Getting Math Chops Back Up
Interesting story --
For some of the mechanics of working problems in grad school, I
still would often go back to my original notes from Richard
Rand's classes at Cornell. He is now known as an absolutely
legendary educator now, but back then he was just another professor
-- whose lectures just happened to be gripping, fascinating...
almost unforgettable. I eventually got rid of my record collection,
but I sure hung on to those course notes, even after ditching the
course textbooks.
When I started graduate school at Chicago in Geophysics, I'd
come out of a full 3 years of engineering math from Rand,
including a graduate level courses that were weeding-out
courses for physics grad students (I got a B) as an engineering
undergraduate -- so in my first year at Chicago, I decided to
sign up for what looked like the equivalent graduate level
courses in differential equations...for review.
My grad advisor in our "get to know your faculty advisor and
choose your courses" meeting thought this was overly ambitious
and suggested I take "at least a course in *calculus.*"
I pointed out that I'd done that in uh, high school. He countered
(definitively revealing that he'd not even read my transcript)
"well that was a long time ago." When I pointed out that I'd
taken half a dozen engineering mathematics and physics courses
that required calculus as a prerequisite, and used calculus
almost continually, he stiffened and answered that "Mathematics is
Different Here at Chicago." (!!!) So I was like, "Oh what, you
mean like 2+2=5 here?"
He suggested a compromise whereby I'd sign up for sophomore-level
complex analysis first quarter and ODE's second. I figured this
one wasn't worth fighting, and besides I could use the easy "A"
if I didn't get too bored in the mean time.
In practice, I was pleasantly surprised. Whereas my
engineering math courses had focussed primarily on technique,
the mechanics of solving specific problems, and I could do
Schwartz-Christoffel Transforms in my sleep already --these
courses at Chicago focussed almost exclusively on proving a
variety of properties of functions in the complex plane, i.e.
analytic functions vs piecewise continuous functions, contour
integration and so forth. In other words, it was complex *analysis*
based on Ahlfors' text, not Complex Functions based on, say,
Church. What had previously seemed to be a chore with some
incomprehensible beauty behind it, was now was something truly
beautiful I was getting the tools to actually take apart and
put back together, lectures from people with some real insight
and understanding.
ODEs and PDEs were even better in that regard, the ODEs course
being based on Birkhoff and Gian-Carlo Rota's text, which is
so beautifully written, it reads more like an exciting novel
in places, *particularly* the proofs. I'd been through Green's
functions at least 3 times in different courses, for example,
and again, could blow through the problem sets -- but it was
just symbol manipulation. It never even occured to me to even
ask *why* Green's functions gave you the particular solution.
It was just the technique you applied when you had a forcing
function, and it worked.
So one night, I'm studying for the midterm, and get sidetracked
reading Gian-Carlo's one-page proof on Green's functions. He
actually drew me in to the story, when I "should have been
studying" in the only way I knew how back then: working problems
(in this case correcting some of the mistakes in Birkhoff and Rota).
I thought for sure I was going to blow the exam, but this proof was
cool and so interesting and so clearly written -- that I was
able to reproduce the proof on the exam the next day...and I
was the only one in the class able to do that. So what I thought
was "being sidetracked" -- actually taking an interest in the
material for its own sake rather than chugging through that
odious chore called math homework -- turned out to be a more
effective study technique as well as a whole lot more fun.
I had amost the same experience with a proof of the uniqueness
and completeness of Fourier Series in PDEs, which Chicago taught
from Weinberger's text.
That first year at Chicago, math went from being a Beautiful BFJ
to something even more beautiful and engaging -- like great art.
When I was working through Guckenheimer and Holmes on my own
(there wasn't a course at Chicago that used it) I used Hirsch and
Smale as my ODEs reference rather than Birkhoff and Rota, because
Hirsch and Smale uses the same notation and way of expressing
things (Guckenheimer was Smale's student, after all).
The nice thing about these classics is that you can go back to them and
re-read them like a good novel. They're incredibly enjoyable as well
as merely useful.
Some of these are really expensive these days, but I think most of
them are on the bookshelf at SFI:
Complex Analysis:
Ahlfors
http://www.amazon.com/exec/obidos/tg/detail/-/0070006571/qid=1128869678/sr=8-1/ref=pd_bbs_1/102-6245318-2684139?v=glance&s=books&n=507846
ODEs:
Birkhoff and Rota
http://www.amazon.com/exec/obidos/tg/detail/-/0471860034/qid=1128869773/sr=1-1/ref=sr_1_1/102-6245318-2684139?v=glance&s=books
Hirsch and Smale
http://www.amazon.com/exec/obidos/tg/detail/-/0123495504/qid=1128871551/sr=1-3/ref=sr_1_3/102-6245318-2684139?v=glance&s=books
PDEs:
Weinberger
http://www.amazon.com/exec/obidos/tg/detail/-/048668640X/qid=1128869893/sr=1-1/ref=sr_1_1/102-6245318-2684139?v=glance&s=books
Nonlinear Dynamics:
Guckenheimer and Holmes
http://www.amazon.com/exec/obidos/tg/detail/-/0387908196/qid=1128869955/sr=2-1/ref=pd_bbs_b_2_1/102-6245318-2684139?v=glance&s=books
Note that none of these really drag you into Courant and Hilbert territory.
Cheryl
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