Math 331, Fall 2002: Problems 11-12

NOTE: 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 2/5 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.

11.
(expires 10/14)    Following Section 4 of the notes, prove that if we describe the circle of center and radius using the parameters , with , rather than the more natural parameters , then the error function is quadratic in and . What does this imply about the number of critical points?

12.
(expires 10/14)    With reference to Problem #11, show that, for , the transformation is a valid change of variables, that is, it is one-to-one. This should help you prove that has only one physical'' critical point, which is a minimum, and is mapped, through the transformation, into the unique critical point of .

MAT 331 2002-09-25