Due at the beginning of class, Tuesday Sep 21.

1. Holmes, Exercise 1.5.1(e), p23. (You are expected to read Holmes section 1.5! :-) )

2. Fowkes & Mahony, Ex 4.1, p108. (No calc. of var. required!)
Hint: Some integral equations (equations where the unknown function appears inside an integral, can be converted into a differential equation by differentiating the whole equation.

3. Brachistocrone: Use Maple to plot the family of brachistocrones (cycloids) all starting at (x_A,y_A)=(0,0). Make your plot range x in [0,1], y in [0,3]. Choose 10 or more values for a (the radius of the rolling wheel), to give a sense of the shape of the fastest path for x_B=1, and y_B ranging throughout [0,3]. Describe how the path looks for large and small vertical drops ( y_B-y_A ).

Maple suggestions:
Use the parametric form of plot: plot([x,y,theta=0..2*Pi]);
Use the following useful version of seq: seq( foo(a), a=[0.1,0.5,20.2]);
Use "scaling=constrained" in displaying your plot, because it matters that the horizontal and vertical scales are the same.

4. Byron & Fuller, Problem 3, p80 (geodesics on a sphere). Additional notes: (1) I had difficulty going from Euler-Lagrange to the desired answer until I worked backwards from the desired answer and "met in the middle". (2) For the second part, you may choose your zero of longitude (phi) so that "alpha" is zero.