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 michaelliv@gmail.com,

michaell@world.std.com or michaell@theworld.com 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.