Infinitesimal Calculus
by James M. Henle, Eugene M. Kleinberg · first published 2003
57% recommend43% neutral
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…modern non-standard analysis, minus rigor. It is also the way Newton and Leibniz thought about calculus, and the way most physicists intuitively think about problems. For a more mature audience, I like Infinitesimal Calculus by Henle & Kleinberg.
on “The Calculus Trap (2005)” ↗…mostly nonstandard proofs of many of the results in the textbook. I like it a lot, and anyone who's interested in this stuff should grab the PDF. Another book that is short and moderately elementary is "Infinitesimal Calculus" by James Henle and Eugene Kleinberg, also published by Dover [which also publishes a paper version of Keisler's text, and is generally a great publishing company]. When Keisler's book came out, someone…
on “Elementary Calculus: An Infinitesimal Approach (2000)” ↗One more book I found to be useful, at least for its introduction to the hyperreal number system: http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mat... It's short and focused.
on “Hyperreal numbers: infinities and infinitesimals” ↗While I haven't examined it closely, there's a well thought of, concise, no-fluff book on it: http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mat...
on “Ask HN: Books with a high signal to noise ratio?” ↗…Robinson's "Non-standard Analysis" and Jerome Keisler's (now free) calculus book someone else mentioned (and the supplement which provides a lot of the theory), people who are interested in this can check out "Infinitesimal Calculus" by James Henle and Eugene Kleinberg (https://store.doverpublications.com/0486428869.html - a short introduction) or "Applied Nonstandard Analysis" by Martin Davis (https://store.doverpublications.com/0486442292.html…
on “Lessons I wish I had learned before I started teaching differential equations [pdf] (1997)” ↗…Probability Theory (Hoel) * The Little Schemer (Friedman) The proceed to more advanced texts like: * Naive Set Theory (Halmos) * Linear Algebra Done Right (Axler) * Geometry Revisited (Coxeter) * Infinitesimal Calculus (Keisler) * Concrete Mathematics (Graham) * Information Theory, Inference and Learning (MacKay) * SICP (Abelson)
on “How I faced my fears and learned to be good at math” ↗Depends what you're looking for. Full Frontal Calculus[0], Intuitive Infinitesimal Calculus[1], and Elementary Calculus[2] are all textbooks on the calculus sequence using an infinitesimal pov. The basic approach is to extend the Real numbers to include infinitesimals (greater than zero but…
on “Simplifying and Refactoring Introductory Calculus” ↗Looks similar to this book, which I bought after a recommendation here: https://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Ma...
on “Big Picture of Calculus (2010) [video]” ↗This is one book on infinitesimal calculus: https://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Ma... . However, and this is very amusing to me, it turns out that the process of automatic differentiation (see https://en.wikipedia.org/wiki/Automatic_differentiation,…
on “Notation as a Tool of Thought (1979)” ↗I second this thinking. You may be interested in reading about Geometric Algebra[1] and Infinitesimal Calculus[2], as two other alternative frameworks for the symbolic tooling used in the physical sciences. Together with symbolic manipulation and calculation software, I also think alternative, simpler symbolic…
on “Identities in Higher Calculus and Special Functions” ↗…isomorphic to the reals, so the dual space [where dx lives] is too.) (Calculus via infinitesimals is pretty cool; a good resource for this is H. Jerome Keisler's "Elementary Calculus" and "Foundations of Infinitesimal Calculus", both available for free: https://people.math.wisc.edu/~hkeisler/) I second the recommendation for Barrett O'Neill's book - I used it in my differential geometry class at MIT.
on “On Leibniz Notation” ↗