College Math Teaching

May 4, 2015

Hitting the bat with the ball….the vector calculus integral theorems….

Filed under: calculus, editorial, vector calculus — Tags: , — collegemathteaching @ 4:43 pm

When I was a small kid, my dad would play baseball with me. He’d pitch the ball and try to hit my bat with the ball so I could think I was actually hitting the ball.

Well, fast forward 50 years to my vector calculus final exam; we are covering the “big integral” theorems.

Yeah, I know; it is \int_{\partial \Omega} \sigma = \int_{\Omega} d \sigma but, let’s just say that we aren’t up to differential forms as yet. 🙂

And so I am giving them classical Green’s Theorem, Stokes’ Theorem and Divergence Theorem problems….and everything in sight basically boils down to integrating a constant over a rectangle, box, sphere, ball or disk.

I am hitting their bats with the ball; I wonder how many will notice. 🙂

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June 12, 2013

Just a 3 page paper…

Filed under: academia, advanced mathematics, editorial, research, topology — Tags: — collegemathteaching @ 9:17 pm

I just sent off the final (I think) revisions of a 3 page paper that has been accepted for publication.
Now, as I get ready to start writing another paper (different area entirely), I picked up an old notebook: 65+ pages of notes of work are related to this paper!
I submitted it; had a rejection (due to writing form), re did it, resubmitted it, had to do more revisions, etc. I talked about this in a seminar, had a colleague show me that a similar result had appeared (but mine was different enough to warrant publication)

Of course, I did things a different way and I created a “new to me” technique for “smoothing” a piecewise linear half-line in such a way that the resulting curve is C^1 (has a continuous first derivative) and remains convex.

And the result: 3 pages in a journal. In terms of time, that is one page per year!

So what was in all of these notes?

Some of these notes were about the idea itself (some elementary point set topology of the plane was involved); the idea: if one has a collection of points in the plane that has a limit point, can one run a locally piecewise linear arc though a selected convergent subset of points to the limit point, while keeping the resulting arc convex? (Yes, you can)

Now is there a way to “smooth” this arc (round off the corners) so as to pass though an infinite subset of this convergent sub-sequence of points so as to produce a C^1 curve? (Yes, there is; that is where the spline construction came in).

Can this curve be made into a C^2 curve? NO!!! The counter example is rather indirect.

Anyway, the details of the above is what fills up 65+ pages of my little notebook.

My point: expect research to take a while; there are starts, false starts, dead ends, revisions and revisions to the revisions, BEFORE you send off the first draft to the journal for consideration!

What comes next
I have one result ready to be proofed and another that I am writing up; hopefully these will be sent to a journal in a month’s time. But I won’t be surprised if these papers also take quite a bit of time.

So, for my academic year: what do I do as far as research?

1. Work on a specific problem?
2. Learn something “new to me” but related to my research?
3. Explore new topics (new to me) at a shallow level?
4. Work on a lower division book?

We shall see.

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