My Calculus Project
This page is still not properly constructed. You can read my Calculus writeup.
It is a work in progress, take a look at the tables of content, the first one reflects
the material that I usually teach in my Elementary Calculus class for high school
students at MIT under HSSP initiative, the second is a tentative and more
elaborate, it is for a book that I hope to finish some day. I have not decided on the
name of the book, my latest favorite is "Caclulus: the Final Solution" with the subtitle:
"caution: reading this book may be harmful to your mental health." In accordance
with the modern tradition of intellectual freedom, it will include the source (LaTeX
and .fig, all the ones done so far are here now, the instructions are in README.txt)
and will be copylefted, see FDL for details.
I will appreciate any comments. Please, e-mail them to firstname.lastname@example.org,
email@example.com or firstname.lastname@example.org will work as well.
The idea to treat differentiation as division of functions came to me in the fall of 1997
when I worked as a part time recitation instructor at Suffolk University and was
disgusted by the textbook by Anton. I learned later that this idea had been used
by Caratheodory in his 1950 book on complex variables, and in fact (in a slightly
different form) by Weierstrass in his 1860-s lectures (see Ch.3 of Analysis by Its
History, which is one of the books related to Calculus that I recommend).
This insight was very useful, and the students liked this algebraic approach
that allows to deal efficiently with rational and elementary algebraic functions
without using limits (Susan Bassein's book from my list is a good example of
this ideology in action). It looked natural to me to treat differentiation as division
of continuous functions to develop the analytic aspects of Calculus. At this stage
I benefited a lot from my conversations with Mark Bridger who suggested using
uniform continuity (see his article in AMM ), some interest and incouragement
from Eric Towne who is a Calculus instructor at Harvard Extension School,
from David Mumford who was back then my close neighbor, from Victor Guillemin
and Richard Melrose who are still my close neighbors and from Richard Palais
who was my advisor at Brandeis in the late 80-s. The numerous discussions on
calc-reform and mathedu internet forums were also very helpful.
But the problem of explaining continuity (it was clear to me that continuity
should be dealt with before limits and limits should be explained in terms of continuity
as was done by E.Chech, according to Jerry Uhl ) still remained a daunting obstacle
to making Calculus elementary. The breakthrough came in May of 2001, when I met
with Hermann Karcher who was visiting Richard Palais in Weston, MA.
He told me about his brilliant idea to use uniform Lipschitz estimates
instead of uniform continuity, see the English summary of his approach
and his German lecture notes (see also Karl Dovermann's Applied Calculus based
on non-uniform Lipschitz estimates, that lead to a more cumbersome treatment,
as another attempt to avoid premature introduction of limits and continuity).
This proved to be the silver bullet that I needed to make Calculus truly
elementary without sacrificing much of its power. Unfortunately it looks
like popularizing these ideas among Calculus teachers may be an uphill battle because
of the conservative and entrenched nature of the establishment (I fondly call it
The Church of Limitology). The audience at Mathfest gave me a cold shoulder
in the August of 2001 when I tried to outline my plan for simlifying Calculus (to be fair,
I must admit that my talk was rather lousy and too short to make a point).
I got a warmer reception in September of 2001 at Williams College when Frank Morgan
invited me to give a colloquium talk. So I remain optimistic, most of all becuase I see
that the students who want to learn Calculus really like the approach that I take, and
to me it is the most important thing. In any case, here I have one more way to explain
Calculus, and the more ways we have - the better.
"Calculus without limits" T-shirt is available from my e-shop, take a look.
Slides (13 pages) for my talk on Friday the 13th in August 2004 at MathFest.
Slides (20 pages) for my talk at Wayne State on 10/11/2010.
My recent and still not quite finished article "You could simplify calculus" is on arxiv
and its slightly updated version is available too.