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Math Help - Hessian

  1. #1
    Senior Member slevvio's Avatar
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    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 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

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

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

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

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

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

    BUT!

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

    |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
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  2. #2
    A Plied Mathematician
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    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.
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