Math 331, Fall 2004: Problems 8-9

NOTE 1: Each exercise is worth 10 points and can be turned in at any time before its ``expiration date''. At the end of the semester, I will expect you to have turned in at least half of the exercises assigned. If you do more, I will pick your best grades. If you do less, the missing grades will be counted as zeros. Altogether, these will count the same as one project.

NOTE 2: Maple is not necessary to the next two exercises, although you can use is as a word processor.

11.

(expires 10/5)    Following Section 4 of the notes, prove that if we describe the circle of center $(a,b)$ and radius $r$ using the parameters $(a,b,k)$, with $k = a^2 + b^2 - r^2$, rather than the more natural parameters $(a,b,r)$, then the error function $H(a,b,k) = E(a,b,\sqrt{a^2 + b^2 -k})$ is quadratic in $a,b$ and $k$. What does this imply about the number of critical points?

12.

(expires 10/5)    With reference to Problem #11, show that, for $r > 0$, the transformation $(a,b,r) \mapsto (a,b,k)$ is a valid change of variables, that is, it is one-to-one. This should help you prove that $E(a,b,r)$ has only one ``physical'' critical point, which is a minimum, and is mapped, through the transformation, into the unique critical point of $H(a,b,k)$.