Friday, March 29, 2019

AP Calculus exam: Liouville's Theorem (there are two of them), and why we exist at all (and why I feel like a substitute teacher again today)



OK, you AP calculus students, here is your test this Friday.   Or it is a free-response question for college credit.  This was meaningful twelve years ago when I worked as a substitute teacher (high school), which is a saga in itself.  You have to prove two controversial theorems in complex variables, and one in statistics.

Ritvikmath, in this video, proves two of the least intuitive theorems in mathematics (complex variables), the Morera Theorem and the LiouvilleTheorem (that a bounded everywhere-differentiable function remains a constant everywhere, like Facebook).   It probably does not apply to quaternions.  There is a related theorem in statistics of significance in fluid mechanics.  (The spelling of the mathematician's name is different from the Kentucky city's.) 


The math professor’s technique is simple: write the steps of the proof on simple sheets of paper and use colored felt ink.  You don’t need fancy cameras to make the videos.

The Liouville (an odd spelling of a French mathematician’s name) was a question in my Master’s Orals at the University of Kansas, I believe on January 18, 1968, on an unusually mild winter day in Lawrence. I stumbled through this theorem (its in the Chapter 7 of my DADT-III book) and probably barely passed the orals.  (Yup, “You passed”.  But “He’s going into the service”.) 

On campus, we used to talk about “all this useless math” that gave people deferments from going into Vietnam combat.  No matter, quaternions probably explain the elementary particles of physics, and the entire Universe follows the laws of mathematics and statistics to a tee. We wouldn’t have Facebook’s algorithms or today’s Internet without it.  We all exist because statistics says we must at some point.

I gave a much better technical talk, on my Master’s Thesis (Minimax Rational Function Approximation) at one of my first job interviews in Princeton NJ in December 1969 just before getting out of the Army.
  
What “professionalism” means has changed so much since then.

Also, for a video about a physics exam, try this

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