Section 0: Problem 1 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

(a) Show that for fixed values of $(y,z)$ , $x$ is a repeated root of $x^{3}+xy+z=0$ if and only if $x=-3z/2y$ and $4y^{3}+27z^{2}=0$ ;

(b) there are $3$ distinct roots if and only if $4y^{3}+27z^{2}<0$ ;

(d) now open up any book or article on catastrophe theory and compare.

(a) $x^{3}+xy+z=$ $(x-a)^{2}(x-b)=$ $x^{3}-(2a+b)x^{2}+(a^{2}+2ab)x-a^{2}b$ iff $0=-2a-b$ , $y=a^{2}+2ab$ , and $z=-a^{2}b$ iff $b=-2a$ , $y=-3a^{2}$ , and $z=2a^{3}$ . And we have, $a=-3z/2y$ (in fact, there is also the case $x=y=z=0$ ) and $4y^{3}+27z^{2}=0$ .

(b) From (a) we have inequality. Further, by taking the first derivative, we determine critical points: $3x^{2}+y=0$ , $x^{2}=-y/3$ . There are three distinct roots iff there are two critical points with opposite sign values iff $y<0$ , $-\tfrac{2}{3}y\sqrt{-\tfrac{y}{3}}+z>0>\tfrac{2}{3}y\sqrt{-\tfrac{y}{3}}+z$ iff $y<0$ and $|z|<-\tfrac{2}{3}y\sqrt{-\tfrac{y}{3}}$ iff $z^{2}<-\tfrac{4}{27}y^{3}$ iff

(c), (d) https://www.wolframalpha.com/input/?i=plot+x^3%2Bxy%2Bz%3D0.

https://en.wikipedia.org/wiki/Catastrophe_theory.

Subpages

Section 0: Problem 1 Solution

Section 0: Problem 1 Solution