Can Homogeneous Functions Help Solve Partial Derivative Relationships?

[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.

 

[jvicens]

Thank you for sharing your problem with us. I understand that you have been working on this for a few months now and are having trouble showing the relationship between the partial derivatives of f and g. First, I would like to clarify that a zero order homogeneous function means that the function does not change when all of its variables are scaled by a constant. In other words, multiplying all the variables by a constant does not change the value of the function. However, this does not necessarily mean that the function is not homogeneous at all.

To tackle this problem, I suggest starting with the definition of a homogeneous function. As you have mentioned, a first order homogeneous function f(x_1, x_2, x_3) satisfies the property \lambda f(x_1, x_2, x_3) = f(\lambda x_1, \lambda x_2, \lambda x_3). From this definition, we can rearrange the terms and rewrite it as f(\lambda x_1, \lambda x_2, \lambda x_3) = \lambda f(x_1, x_2, x_3). This means that f is also homogeneous with respect to the variables \lambda x_2 and \lambda x_3.

Now, let's look at g = \frac{1}{x_1}f. Since f is homogeneous with respect to \lambda x_2 and \lambda x_3, g must also be homogeneous with respect to these variables. This means that we can write g(\lambda m_2, \lambda m_3) = \frac{1}{\lambda x_1}f(\lambda x_1, \lambda x_2, \lambda x_3) = \frac{1}{x_1}f(x_1, x_2, x_3) = g(m_2, m_3). Therefore, g is also a first order homogeneous function.

Now, let's look at the relationship between the partial derivatives. We can write the partial derivative of f with respect to x_2 as \left( \frac{\partial f}{\partial x_2} \right)_{x_1, x_3} = \left( \frac{\partial f}{\partial m_2} \right)_{m_3}\frac{\partial m_2}{\partial x_2}. Since g =