Sunday, September 16, 2007

Engineering Calculus

I saw a comment over on Learning Curves about having to sub for an "engineering calculus" class, and decided it was worth posting a comment on the genre.

As a math BS and physics PhD with a father who is an engineer plus my own love of old textbooks, I have a unique perspective. I have actually seen a true "engineering calculus" text book, and suspect there is a place for something like it today.

My dad's calculus book looks bizarre to someone raised on Thomas (3rd edition) who has seen nothing but its clones and copycats over the years ... both back when I taught calculus recitations and today when I poach some parts of the subject by looking over the shoulders of my physics students when they are doing math homework before class.

The "engineering calculus" book dates to the post-war period, definitely before Thomas appeared (along with all the other post-Sputnik books?) in the 60s. It appears to have been designed for a quarter system and optimized to fit in with a calculus-based physics class that had its (very easy) first section as a pre-requisite. Also, from my current perspective, it might have accidentally managed to deal with the lack of abstract thinking skills so many kids bring to calculus.

It starts with the calculus as arithmetic, literally about doing a specific kind of calculation. It does not ignore the limit (like the infinitesimal approaches that were tried) but it does minimize it, thereby eliminating the several weeks of abstraction related to functional analysis in the typical text. Its goal for the first 10 weeks is to teach you to solve derivatives, integrals, and differential equations involving polynomials. (The only thing you really need for the first bits of physics, until you get to simple harmonic motion and need more.) I think they also did the log integral so they could do first-order linear diff.eqs which are also needed in physics.

The book then goes from concrete to abstract, revisiting the main topics to put in enough functional analysis to do derivations of derivatives of trig and inverse functions, chain rule, etc, and integrals of those functions. That is enough for simple harmonic motion by the time you get to it in physics (even if taken at the same time as this 2nd 10 weeks class). By the third 10 weeks, you are doing all the methods of integration and on to multi-variable calculus, just as in a conventional book, except the co-requisite requirements might ensure that the calc students knew some physics by this time, so the applications could be taken more seriously (assuming the math prof also knew that physics).

The rationale for this could be a simple as the fact that Newton had no need of functional analysis to do what he needed to do in the Principia. That approach to teaching calculus came much later, and mostly serves to puzzle the freshman taking calc I who wonders when they will ever get to calculus. (Answer: Sometimes it is after the first exam!)

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