MAT331 homework problems

NOTE: Neither of these problems involve Maple, except as a word processor to write your solution. If you like, you are welcome to turn in a printed or handwritten version, if you are more comfortable with that.

11.

(expires 2/25)    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 2/25)    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 $H(a,b,k)$.