# Euler's relations and positively homogeneous functions

Let $U \subset \mathbb{R}^{m}$ be an open set such that $x \in U, t>0 \Rightarrow tx \in U$. If $f:U \longrightarrow \mathbb{R}$ is differentiable then $f$ is positively homogeneous of degree $k$ if, only if, satisfies the Euler's relations $\displaystyle \sum \frac{\partial f}{\partial x_{i}}(x)x_{i} = kf(x)$.

I have no ideia how to start this question. I still cannot use Rademacher's Theorem or grad. Any hints? I don't need the complete solution, a suggestion would be great

• Start by worrying about the "only if." You need to start by writing down the definition of homogeneity of degree $k$. You have $f(tx) = t^kf(x)$ for all $t>0$. What you're trying to prove makes it look like you want to take a derivative. – Ted Shifrin Apr 13 '18 at 6:11
• Possibly you want to have a look at math.stackexchange.com/questions/2694590/… – Jens Schwaiger Apr 13 '18 at 7:44

## 1 Answer

Let $x\in U$. Define $I=(0,\infty)$ and $g:I\to\Bbb R$ by $g(t)=f(tx)$. Euler's relations induce a differential equation for $g$. Solve it.