Leibnizian calculus textbook - MathOverflow

Leibnizian calculus textbook

2011-03-11T19:32:00Z

Where can I find a calculus textbook that emphasizes differentials? Is there such a book that I could realistically require my calculus students to use?

I want a textbook that supports me when I tell my students something like:

$\Delta((x^2+1)^5)\approx5(x^2+1)^4\Delta(x^2+1)\approx5(x^2+1)^4(2x\Delta x)$

$d((x^2+1)^5)=5(x^2+1)^4d(x^2+1)=5(x^2+1)^4(2x\ dx)$

Or:

$\Sigma_{k=1}^n 3x_k^2\Delta x_k\approx\Sigma_{k=1}^n\Delta(x_k^3)=x_n^3-x_1^3$

$\int_{x=0}^{x=4}3x^2\ dx=\int_{x=0}^{x=4}d(x^3)=4^3-0^3=64$

Perhaps I could write this book someday, but it'd be a lot easier for me if my students and I could just buy and/or download a book that takes this approach without neglecting to provide a cornucopia of exercises, examples, and applications similar to what's available in today's most popular calculus textbooks.

Answer by Dick Palais for Leibnizian calculus textbook

2011-03-11T19:59:19Z

There is a marvelous old book (19th Century if I recall correctly) where I learned Calculus the first time, called "Calculus Made Easy" by Sylvanus P. Thompson, and subtitled "What one fool can do another can". He explains that dx means a "little bit of x" and shows a square with sides x and x + dx and you can see why you can "ignore dx^2". Of course it isn't rigorous in any sense, but it uses differentials to get all the essential ideas of both differential and integral Calculus across quickly and smoothly. Needless to say, once I had absorbed all these essential ideas I went on to read more rigorous books where limits were introduced and used to make precise what I already understood well from this intuitive introduction. If I recall correctly Calculus Made Easy was republished some years back (Dover?) and was quite popular. I would suggest that you recommend it to your students, with appropriate caveats.

(Added later) I checked online and indeed there is a recent reprinting (available from Amazon and the other usual places). Moreover it has three new chapters written by the late great Martin Gardner aimed at the modern reader. I'm going to buy myself a copy!

Answer by Toby Bartels for Leibnizian calculus textbook

2011-04-03T23:57:11Z

This approach is suggested by Tevian Dray and Corinne Manogue in their program of Bridging the Vector Calculus Gap. They focus on multivariable calculus and differential forms, but they discuss single-variable calculus (pdf) once. Unfortunately, they don't seem to have a textbook for that.

Answer by lhf for Leibnizian calculus textbook

2011-04-05T11:50:30Z

http://en.wikipedia.org/wiki/Non-standard_calculus#References links to two texts available online.

Answer by jim-hefferon for Leibnizian calculus textbook

2012-08-23T12:21:19Z

Henle and Kleinberg's Infinitesimal Calculus is available in paperback for less than ten dollars and the reviews at Amazon are very strong, at least for use as a supplemental text.

Answer by katz for Leibnizian calculus textbook

2013-04-18T09:24:05Z

Does Vakil's book "real analysis through modern infinitesimals" http://www.google.co.il/books?id=hyFjtJ3Wq24C&source=gbs_navlinks_s contain the additional material you are looking for?

Answer by katz for Leibnizian calculus textbook

2013-05-10T07:31:37Z

If what you are looking for is "multivariable calculus with infinitesimals", you might be interested in Stroyan's book at http://library.wolfram.com/infocenter/Books/6877/ The table of contents is here: http://homepage.math.uiowa.edu/~stroyan/MultiCalc/iMultiCalcTOC/iMultiCalcIntro.html