Overview
the required book in in the file
•Chapters 20-22 ?These chapters are on more difficult mathematics. This in turn will make it difficult to fully engage with them. Go into the reading being aware of that. ?Read for a sense of what was done and who did what. A few small examples of the mathematics may be presented in class, and this computationally is what to focus on. Otherwise read the blue boxes, read the summary text and try to get a feel for what the chapter is about. ?Focus topics: ?Chapter 20: Euler on Topology ?Chapter 21: Gauss, Galois, Eigenspace, Groups and Fields. Very tough section. Do not try to read everything. ?Chapter 22: Tough section. Skip 22.3 and beyond and just read for big ideas before this. Homework Problems
1. [Graded] First explain Johann Bernoulli’s Solution (in brief) to the Brachistochrone (shortest time) problem. Not every detail, just what he did and his line of thinking. In class we studied the cyclotoid solution to the Brachistichrone Problem. Draw a circle of radius 1 centered at (0,1). Roll the circle along the x-axis for one full revolution to generate a cyclotoid. Give parametric equations x(theta)= y(theta)= For theta ranging from 0 to 2*pi, make a plot of the curve, and show the derivation principles in your solution. Each explain your work. 2. [graded ]Watch the 3 Blue 1 Brown video: “Bayes theorem, the geometry of changing beliefs”Solve Katz 18.32, using ideas from the video. 3. Deposit $50 in the beginning of each year for 5 years. How much is the account worth if it pays 6% interest once at the end of each year? How much would it be worth after 50 years? 4. Write a computer program, use Desmos, or by hand (with construction points plotted), draw an accurate Witch of Agnesi curve. 5. What does the De Moivre–Laplace theorem say? State the theorem. Explain it in simple language. Give an example of how to test the theorem by experiment.
6. Explain what is: “topology” and how it is different from “differential geometry.” Give a little history of the development of these two branches. Also give 2 modern examples of actual use cases / applications for each of these subjects.