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Tom Apostol RIP


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This man had a big impact on my view of calculus teaching.  His book Calculus is a wonderful resource for any high level introduction to the topic, and has been so used for decades at many places.  Ironically, the advance of popularity of accelerated high school presentation of math ("AP" programs), including calculus for most students has largely killed off the outstanding college programs using his book.  The problem is that high school students are usually not offered the subject at so high a level, but then when they enter college they do not want to begin over at the beginning.  So for example at Stanford and Harvard, students are not offered the chance to be introduced to this high level of beginning calculus, they must have alteady had it, and there are few chances to get it in high school.  I.e. when my son went there Stanford offered only volume 2 of Apostol, not volume 1.  University of Georgia kept the tradition going until very recently of offering a comparable course from Spivak's similarly high level Calculus, but I am told it tooi was recently abandoned for lack of an appropriate incoming audience.  This makes almost the only remaining audience for his fine book the home school crowd.  When I retired I donated my copies of his books to the undergraduate student math library at UGA, as well as copies of Spivak, and Courant.  I still miss them from time to time.  For those still seeking such courses, they existed until recently still at University of Chicago (math 16000, Spivak based) and MIT ("calculus with theory") from Apostol).

 

A few years ago when teaching Euclid and Archimedes to brilliant 10 year old, I discovered that (by using "Cavalieri's" principle) Archimedes had found the volume not only of a sphere but also of a "bicylinder" , the region between two perpendicular intersecting cylinders of the same radius, and realized how to generalize the calculation he had made from the sphere to that case, a much easier argument than the one using calculus that we teach in college. Then I found a current article by Apostol generalizing these two solids to a whole range of similar ones. I have forgotten it now but it involved solids of a certain degree of convexity for which the calculation of surface area followed directly from that of volume.  E.g. for a sphere the solid is viewed as a cone with vertex at the center, hence the volume is 1/3 the product of the surface area and the radius (the "height" of the "cone").  It was very pleasing to learn that such a man as Apostol also found interest in relatively elementary geometry deriving from that of the great Greeks, and even added to it.  I didn't know then he was of Greek descent.  I recommend anything he has written for seriously mathy types.

 

http://mamikon.com/USArticles/CircumSolids.pdf

 

http://www.maa.org/press/books/new-horizons-in-geometry

Edited by mathwonk
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