1. ## Hessian

Hello there I am investigating catastrophe theory although I have discovered something in a book I am reading that I dont understand

we consider a smooth function of two variables with $\displaystyle f(0,0) = f_x(0,0) = f_y(0,0) = 0$, i.e. there is a critical point at the origin, so the Taylor series is

$\displaystyle f(x,y) = ax^2 + 2h x y + b y^2)$+ higher order terms

where $\displaystyle a = f_{xx}, b = f_{yy}, h = f_{xy}$

We consider the case when the hessian$\displaystyle ab - h^2 = 0$, and that not all the derivatives are zero

So the book says that this means $\displaystyle |ax^2 + 2h x y + b y^2 |$ is a perfect square and allows us to write

$\displaystyle |f(x,y)| = \frac{1}{2}(\sqrt{|a|}x + \sqrt{|b|}y)^2$ + higher order terms

BUT!

if we do $\displaystyle f(x,y) = \frac{1}{2}(2x^2 - 4 x y + 2 y^2)$, all the conditions are satisfied but

$\displaystyle |f(x,y)| \not=\frac{1}{2}((\sqrt{|a|}x + \sqrt{|b|}y)^2 = \frac{1}{2}(2x^2 + 4 x y + 2 y^2)$

could anyone explain? thanks very much

2. Just replace y with -y. Wouldn't that fix the problem? It seems to me that the issue is whether you allow negative y's in the picture. Why wouldn't you allow negative y's? It doesn't disprove what the author is saying.