One advantage of the original book by Thomas is its inclusion of techniques for using centers of mass to make some calculations easier, such as work. He does not give exactly the following one, but it illustrates the power of the technique and is similar to his explanation of how to calculate work and potential energy ( see below). Problem is to find the 4 dimensional volume of the unit 4-ball, i.e. the interior of the set with equation x^2+y^2+z^2+t^2 = 1, in 4 space.
By integral calculus, and the usual method of slicing taught in most of these books, one is led to integrate the 3 diml volume of the slice at height t, namely (4π/3)(1-t^2)^3/2 dt, from t=-1 to t=1, not so easy for the average student.
But using centers of mass, and the ancient technique named after Pappus, one can compute the volume by considering the 4-ball as obtained by revolving half a 3 -ball around a 2-diml axis in 4 space, (just as one can represent a 3 -ball by revolving a half 2-disc around a line in 3 space).
It seems we need to know the center of mass of the half 3 - ball, to compute how far it travels under the rotation, but we can finesse that since by Archimedes, we know the volume of half a 3-ball equals the difference between the volume of a cylinder and that of a cone, and even Archimedes knew their centers of mass. I.e. the center of mass of the cylinder of height 1 and base radius 1, is at height 1/2, and the center of mass of the (inverted) cone of height 1 and base radius 1, is at height 3/4.
So we just revolve a cylinder of volume π around a circular path of radius 1/2, getting as product: volume times length of path = π^2, and then subtract the result of revolving a cone of volume π/3 around a circular path of radius 3/4 getting a product of π^2/2. The difference is the volume of the unit 4 - ball, namely π^2/2.
The point is that by teaching students the wisdom of the ancients, calculations are sometimes made far easier than forcing them to use more difficult methods in circumstances where they are not the most appropriate. Thomas' original book was written with a sensitivity to doing things more naturally, using appropriate methods, and I noticed that some of this is lost in the rewrites of his book.
You may ask why a physicist wants to know about 4 dimensional volume, since only a mathematician does such exotic examples! But if you write down the integral for the work done by raising a half 3-ball through a certain distance, e.g. the work done by pumping out the water from a swimming pool shaped like half a 3-ball, you will see that the integral is the same as that for computing the volume generated by revolving a half 3-ball around a 2 diml axis in 4 space, using the method of cylindrical shells, except for a factor of 2π. I.e. work computations done when moving 3 dimensional objects, or potential energy calculations on solid objects situated at a certain height, are exactly the same, mathematically, as computing 4 dimensional volumes!
Now just as the example above shows how to simplify such a volume computation using centers of mass, Thomas explains, in his original book, how to simplify computing work using centers of mass. Now I may well be wrong, since I no longer have access to the later books, but my memory is that the later books teach work computations first, using more difficult methods, and then teach centers of mass only later, and then do not explain how to use them to simplify work computations. This criticism is at least consistent with the fact that in the table of contents I found online, work occurs earlier than centers of mass in the 14th edition of this book. Maybe someone who has access to a copy, can check whether after teaching centers of mass, the 14th edition then uses that concept to simplify work calculations, or for mathematics students, whether they teach Pappus’ theorem to do volumes of solids of revolution. Indeed I hope so. Ok, end of rant. peace. and “to be fair” one can learn something even from later editions, just not as much, and at a much higher price. (I seem to recall now the first version of these Thomas rewrites that was forced on me, and inspired this rant, was called "University Calculus", by Hass, Weir, and Thomas (and published the year he died. It is now described on amazon as a "streamlined" version of Thomas, at only 960 pages.)
what frustrates me as a teacher, is why we deny students knowledge of concepts that were known centuries ago, and instead teach them to use modern ideas that actually make some computations harder. hello? isn't the point of education to make solving problems easier?
