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.
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 , i.e. there is a critical point at the origin, so the Taylor series is
+ higher order terms
where
We consider the case when the hessian , and that not all the derivatives are zero
So the book says that this means is a perfect square and allows us to write
+ higher order terms
BUT!
if we do , all the conditions are satisfied but
could anyone explain? thanks very much