Tuesday, January 28, 2020

Thoughts on Strogatz's Infinite Powers: Great intuition on calculus, from a master teacher

I enjoyed Steven Strogatz's new work, Infinite Powers: How Calculus Reveals the Secrets of the Universe. The book gives an excellent overview of calculus, which permeates all branches of mathematics and so much of life. I should say at the outset that I had the great opportunity of being taught calculus in college by Dr. Strogatz. After reading this book, I feel even more fortunate for this experience because he's such a gifted communicator.

What I liked mainly about the book was the intuitive way Strogatz describes differentials and the development of calculus from Newton and Leibniz onwards. He introduces these concepts in several ways. My favorite was the way he demonstrated simply cubing the number 2 and contrasting it with cubing 2.01, where the latter can be expressed as the cube of a sum (2 + .01) and then expanded out with Pascal's triangle. From merely looking at the multiplication of these numbers, one can immediately get a sense of which terms can be neglected in this specific sum and in the whole process of differentiation.

Strogatz also clearly explains many classic equations in mathematics and physics, such as the heat and the wave equations. I particularly liked the way he described the development of the Fourier series and how this series converts differentiation of sine and cosine into a simple multiplication by minus one, making it easy to deal with. I also liked how he explained how one can easily express even very angular shapes such as a triangular waveform in terms of Fourier series.

I enjoyed many of the practical examples of how we can see calculus in everyday life, ranging from the oscillations of HIV in people, as tracked by Alan Perelson and David Ho, to the development of CT scans by Hounsfield and Cormack. Strogatz gives an especially hands-on understanding of the fundamental theorem of calculus by describing it in terms of a well-known paint roller analogy and how it can link together the disparate ideas of the slope of a function and the area under a curve.

Finally, I enjoyed the discussion of many of the personalities in mathematics, such as Descartes and Fermat. I hadn't appreciated the famous feud between these two until I read the book.

Overall, a great read. I'd highly recommend it, especially for anyone studying or using calculus.

Infinite Powers: How Calculus Reveals the Secrets of the Universe
by Steven Strogatz