Comments on: Reforming the Calculus Class, Permanently

By: Walt

Walt — Sun, 23 Mar 2008 04:12:04 +0000

I posted a little bit about smooth infinitesimal analysis once before. (Almost two years ago; I didn’t realize until just now that I’d been doing this blog for that long.)

By: Todd Trimble

Todd Trimble — Fri, 21 Mar 2008 23:02:36 +0000

Yes, I’m a big fan of SIA too. I haven’t looked at Bell’s book, but of course there are a number of other texts on the subject. I expect that the book I’ve looked at the most, Models for Smooth Infinitesimal Analysis by Moerdijk and Reyes, is a bit more technical. A good online intro has been given by Mike Shulman.

By: Mark Hanlon

Mark Hanlon — Fri, 21 Mar 2008 20:03:27 +0000

I think this requires checking out smooth infinitesimal analysis (SIA); a version of analysis based on nilsquare infinitesimals and the principle of microstraightness. It is far superior to limit theory or non-standard analysis. The best book on the subject is A Primer of Infinitesimal Analysis by J L Bell.

By: Do, a derivative, a d/dx derivative… « Mathiness

Do, a derivative, a d/dx derivative… « Mathiness — Tue, 26 Feb 2008 02:24:45 +0000

[...] 25, 2008 in Uncategorized Tags: amusing, calculus Reforming the Calculus Class, Permanently [...]

By: crank

crank — Sat, 23 Feb 2008 21:46:38 +0000

… less or equal to K|x-a|^2

By: crank

crank — Sat, 23 Feb 2008 21:45:09 +0000

Sorry, &le |x-a|^2 should be

By: crank

crank — Sat, 23 Feb 2008 21:42:36 +0000

There is an article by Jeff Suzuki called “The Lost Calculus (637-1670): Tangency and Optimization Without Limits” in Mathematicas Magazine, Vol. 78, No.5, December 2005,
that won MAA Carl B. Allendoerfer Award for expository writing in 2006 that you may find of interest. As for many varables, the (locally) uniform estimate |g(x)-g(a)-g’(a)(x-a)| &le |x-a|^2 that obviously holds for polynomials can be taken as a definition for uniform Lipschitz differentiability, as I already mentioned in my comment 8.

By: Jacob Freeze

Jacob Freeze — Sat, 23 Feb 2008 19:45:44 +0000

$latex ( \partial f/\partial y)(a,b) $

I can’t see how this peculiar notation of Reformed Calculus is much of an improvement over Leibniz, although the introduction of dollar signs may draw a few more financially motivated students into the field.

By: Walt

Walt — Sat, 23 Feb 2008 17:17:46 +0000

I thought you were claiming you can (for nice-enough functions) always assign a meaning for 0/0. I was only pointing this is strictly a one-variable phenomenon.

By: crank

crank — Sat, 23 Feb 2008 13:33:24 +0000

And by the way, Walt, in your comment
comment 120 you overlook the fact that all the partial derivatives are the dirivatives in one variable. For example, if $latex f(x,y)$ is a function of 2 variables, then $latex ( \partial f/\partial y)(a,b) $ is the derivatife with respect to $latex y$ of the restriction of $latex f$ to the line $latex x=a $, calculated at $latex y=b$. In particular, it’s clear $latex (\partial x/\partial y)(0,0)=0$, and to calculate it you have do restrict $latex x$ to the line $latex x=0$ and then differentiate with respect to $latex y$, so your criticism misses the point.