# Homogeneous functions

1. Dec 1, 2009

### Ionophore

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 $$f(x_1, x_2, x_3)$$. In other words, f has the property that: $$\lambda f(x_1, x_2, x_3) = f(\lambda x_1, \lambda x_2, \lambda x_3)$$.

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

I'm trying to show that:
$$\left( \frac{\partial f}{\partial x_2} \right)_{x_1, x_3} = \left( \frac{\partial g}{\partial m_2} \right)_{m_3}$$

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.