Reforming the Calculus Class, Permanently
February 19th, 2008 by WaltCalculus has been the subject of immense amounts of educational material, ranging from textbooks to blog posts. Unfortunately, that is now all obsolete. The definitive presentation of calculus is here: Calculus, the Musical.
February 22nd, 2008 at 4:35 am
IT’S “ABOUT TIME” (?)!
February 23rd, 2008 at 5:45 am
Nah, they still sing about limits and infinitesmals, we are still in the Dark Ages dominated by the Church of Limitology and mystified by the Cult of Infinedesimalogy. It will take at least another century to free our minds and get it straight.
February 23rd, 2008 at 6:33 am
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.
February 23rd, 2008 at 10:17 am
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.
February 23rd, 2008 at 12:45 pm
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.
February 23rd, 2008 at 2:42 pm
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.
February 23rd, 2008 at 2:45 pm
Sorry, &le |x-a|^2 should be
February 23rd, 2008 at 2:46 pm
… less or equal to K|x-a|^2
February 25th, 2008 at 7:24 pm
[...] 25, 2008 in Uncategorized Tags: amusing, calculus Reforming the Calculus Class, Permanently [...]
March 21st, 2008 at 1:03 pm
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.
March 21st, 2008 at 4:02 pm
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.
March 22nd, 2008 at 9:12 pm
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.)