Leonhard Euler: Everything, Everywhere, All at Once

Course Description

Summary

Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." The thesis of this class is that this is still true today. Leonhard Euler's (1707–1783) collected works run to 81 volumes and over 35,000 pages, the publication only having been (mostly) completed in 2022. Most of Euler's works have never been translated out of Latin. 

Euler comes at a time when the tools of calculus are relatively new. Euler, together with Newton, Leibniz, the Bernoullis, LaGrange and others, applies these tools to a dizzying array of old and new mathematical problems. Euler is particularly striking for his joy, in how he applies this new technology in novel, creative, astounding ways. In this way, he is reminiscent of his near-contemporary, Johann Sebastian Bach. Euler is also striking in how readable he is; Georg Polya says, "among old mathematicians, I was most influenced by Euler and mostly because Euler did something that no other great mathematician of his stature did. He explained how he found his results and I was deeply interested in that. It has to do with my interest in problem solving." 

This class is not primarily a history class; the main focus is the mathematics, embedded in the historical narrative. We follow the original motivations, and the story of the mathematics as it was developed, in particular reading original papers. However, we will simplify, and jump forward in time as needed, in order to make the mathematics more clear. The mathematics is still very contemporary; it is only the approach that is historically oriented.

The main topic of the class could be roughly called "advanced calculus", especially infinite series. However, Euler's work, (like contemporary creative mathematical work), does not obey any artificial boundaries. Euler often applies calculus ideas in surprising ways to problems in combinatorics, or the theory of whole numbers. Students in the class will develop many calculus techniques, but starting from problems, not from a list of topics.

Students may take this class at different levels of mathematical intensity: it can either be an intermediate or advanced class, depending on how much of the mathematical detail you get into. What level an individual student completes will be reflected in the narrative evaluation.  The final evaluation will be relative to the student goals. For example, a student can get an "A" if they choose to take this as an intermediate class, doing only the core ideas, and doing a good job with those; the evaluation would identify that the student took it as an intermediate class. 

The prerequisite for the class is a familiarity with fundamental tools of calculus, at the level of MAT 4288 Calculus: A Classical Approach. 

The overall scope of the course is similar to William Dunham, Euler: The Master of Us All. This course counts as a one-semester class in Advanced Calculus.