Math 629: History of Mathematics
Frank Sottile

• Opening Remarks:
      This week's reading is on mechanics, which is an important influence of Physics on Mathematics. As an algebraist, physics major, and now applied mathemetician, I appreciate these topics. This chapter is slightly out of place, as it predates and motivates the Calculus, but then it continues quite a bit past the time of calculus. Stillwell probably did not want to split this into two chapters, hence the slight mismatch. At any given time, mathematics is being developed along several fronts, and thus a subject-based historical survey of mathematics (like our book) will have to jump around in time a lot.
      This chapter on mechanics covers a lot of history, from the late medieval period until 1900. This begins with relations between velocity, acceleration, and distance, and ends with chaotic motion and fluid mechanics. I suspect that the compression of topics is due to Stilwell's interests, and probably also the need to keep the book from getting too long.
      The historical discussions in Sections 13.1 and 13.2 should be fascinating to all of us. We see how Archimedes was influenced by physical thinking in his work on areas and volumes (recall our earlier discussion of his Method), and then how the precursors to the calculus arose in thinking about motion. It was particularly intersting to see how Roberval in the 17th century grasped the idea of vector addition so early. (Vectors came into their own only in the late 19th century).
      Later sections describe quite a lot of high points of mechanics. The history of the special properties of the catenary and cycloid are fascinating, and are examples of results that could not have been obtained without methods developed in calculus. Do read in detail about the story of the Bernoulli's; they really had a family business, as well as a deep influence on the development of mathematics in the 18th century.
  
• Reading:
  • Chapter 13 in Stillwell's book. Pages 262–263 are a bit cumbersome, and Fulling has a succinct and easy summary of d'Alembert's derivation of the wave equation.
  • Watch the 3Blue1Brown video interviewing Steven Strogartz about the Brachistochrome.
  • Read the following: Here is a handout I created for a course in differential equations at the University of Toronto, when the students needed a proof that mixed partials derivatives commuted.
• Assignment: Due Monday, 18 March 2024. (HW 9)
  Here is a .pdf and a LaTeX source of the assignment.
  The group assignment is relatively light, I am asking that you verify the steps in Fulling's explanation of d'Alembert's derivation of the wave equation. Here is a pdf of the group assignment (which just repeats the previous text).
  To hand in: We are using Gradescope for homework submission.
  1. Exercises 13.5.1 and 13.5.2 from Stillwell.
  2. Do the three problems in the handout on Clairaut's Theorem about mixed partial derivatives.

Last modified: Mon Mar 18 08:18:34 PDT 2024 by sottile