Full Frontal Calculus: An Infinitesimal Approach

by Seth Braver

Written by Beckenbach-Prize-winning author Seth Braver, this textbook is designed for a standard single-variable calculus class. It is distinguished by its concision, its use of infinitesimals (though limits are here, too), and the high quality of its exposition. The emphasis throughout is on deep intuition and computational facility.

Genres: Mathematics

192 pages, Paperback

Published July 28, 2019

Reviews

[Gavin · Rating 4 out of 5]

For you [the beginner], the importance of Robinson’s work lies not in its formidable logical details, but rather in the retrospective blessing it bestows upon the centuries-old tradition of infinitesimal thinking, a tradition that will help you understand how to think about calculus – how to recognize when calculus is an appropriate tool for a problem, how to formulate such problems in the language of calculus, how to understand why calculus’s computational tricks work as they do. All of this becomes considerably easier when we allow ourselves the luxury of working with infinitesimals. We need no longer, as in the 1950’s, blush to say “the i-word”. And so... infinitesimals shall parade proudly through its pages, naked and unashamed.

St. Peter places an hourglass before you. “Mark ye how the sands fall into the lower chamber,” he instructs, “Yea, even as souls fall into perdition. The heap of fallen grains is conical, and lo, its height always equaleth the radius of its base. Observe the steady rate at which the sands fall: five cubic cubits every minute.” Frightened, but fascinated, you look, and verily, it is so. “Ere these sands run out,” Peter resumes, with a grave look, “tell me, when the sandy cone is 2 cubits high, how rapidly doth its height increase?

Thanks to Arthur Schopenhauer, Philip Larkin, Hermann Melville, and Franz Schubert for many hours of stimulation and consolation. Thanks to Olympia’s great blue herons, wood ducks, mallards, mourning doves, and bald eagles.

we use radians precisely because they simplify sine’s derivative; if calculus didn’t exist, neither would radians. Similarly, we use 𝑒 as a base for exponential functions and logarithms because it simplifies their derivatives. If calculus did not exist, no one would bother with 𝑒. Calculus forces strange things up from the depths.

One of my weaknesses is that I need stuff like that ^ to read maths. Austere transcendental timelessness doesn't do it for me, I just don't care. This makes me vulnerable to eloquent cranks - and so of course I read a Nonstandard Analysis textbook instead of Rudin.

Nah but really: it's a great alternative if you bounce off the limit approach. This is clear, historically-motivated, passionate, and short. The typesetting isn't beautiful (Google Doc Chic) but not terrible. There are hundreds of exercises, all placed immediately after each concept is introduced. (This has a handholding quality but I didn't mind.)

It's not rigorous ("model theory", which first gave us back infinitesimals, appears nowhere) and doesn't pretend to be. Here's a lovely exercise: experimental mathematics using the Wolfram microscope:

To better appreciate my claim that elementary functions often lack elementary antiderivatives, try composing some basic precalculus functions, and then let Wolfram Alpha (or the computer program of your choice) try to integrate them. While playing around in this way, try to find some examples of elementary functions whose antiderivatives involve exotic functions such as the log-integral function discussed above, or whose antiderivatives cannot be expressed even in terms of these.

Maybe the synthesis is: use nonstandard analysis for the intro, then do standard limits stuff, so that you can talk to other users (and to computers).

Stage: just post-precalc
Rigour: 2/10
Friendliness: 10/10.
Motivation: 8/10
Typesetting: 4/10.
Worked examples: Yes.
Exercises + solutions: 9/10.

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