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Homogeneous functions

  1. Dec 1, 2009 #1
    Hi all,

    This is a problem i've been working on, off and on, for a few months now. It seems like it should be possible but I just can't figure it out.

    Suppose you have a first order homogeneous function [tex]f(x_1, x_2, x_3)[/tex]. In other words, f has the property that: [tex]\lambda f(x_1, x_2, x_3) = f(\lambda x_1, \lambda x_2, \lambda x_3)[/tex].

    I define a second function, g, as [tex]g = \frac{1}{x_1}f[/tex]. If we define [tex]m_2 = \frac{x_2}{x_1}[/tex] and [tex]m_3 = \frac{x_3}{x_1}[/tex], then I can write [tex]g(m_2, m_3)[/tex]. g is a zero order homogeneous function of [tex]x_1, x_2[/tex], and [tex]x_3[/tex], but i don't think is homogeneous at all (in general) expressed as a function of [tex]m_2[/tex] and [tex]m_3[/tex].

    I'm trying to show that:
    [tex]

    \left( \frac{\partial f}{\partial x_2} \right)_{x_1, x_3} = \left( \frac{\partial g}{\partial m_2} \right)_{m_3}

    [/tex]

    I can do if for specific cases of f, but i just can't figure out how to show it in general. Any help would be appreciated.
     
  2. jcsd
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