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Euler homogeneous formula and applications


See the following pdf file: EulerHomogeneity (source: http://www.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf; )

In the previous article and in this link you can see the definitions of homogeneous functions of certain degree and their property, that if a nth degree homogeneous function f: \Bbb{R}^p \to \Bbb{R} is differentiable then the following relation holds:

x \cdot \nabla f(x)=nf(x),\ \forall x \in \Bbb{R}^p (where the \cdot product is the usual inner product).

As I was preparing some students for their exams, I encountered numerous problems which dealt with proving that a differentiable function satisfies some partial differential equation, and the theorem mentioned above is quite useful when dealing with this.

For example take g: \Bbb{R}^2 \to \Bbb{R}, of class \mathcal{C}^2 and define f(x,y)=g(\displaystyle\frac{x}{y},\ln x-\ln y). Calculate the following expression \displaystyle x^2\frac{\partial^2 f}{\partial x^2}+ y^2\frac{\partial^2 f}{\partial y^2}.

A straight forward calculation would lead us to the right result, but this problem has another approach, by the theorem mentioned in the beginning of the article. It is enough to see that f is homogeneous of degree 0, and by the given theorem x\displaystyle \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=0. Differentiate this relation with respect to x, multiply the resylt with x,  do the same thing with y, add the resulting relations and obtain that the expression to be calculated is equal to zero.

Short easy solution if you know the right results. 🙂

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