smooth map derivative along Euler vector field

Question

let $p$ be a non zero integer. Let $U = \mathbb{R}^3 \setminus\{0\}$ and $f$ be a smooth map $f: U\rightarrow \mathbb{R}: \forall (x, y, z) ∈ U , \forall t>0:$

$f (tx, ty, tz) = t^p f (x, y, z)$

Let $E$ be Euler's field on $U: E=x\partial_x+y\partial_y+z\partial_z$

Find $E(f)$.

I don't know where to begin. Thank you

Answer 1

Hint: notice that $E(f)$ is the same as the directional derivative of $f$ along the vector $(x,y,z)$. More precisely, $$ E(f)(x,y,z) = \lim_{t\to 0} \frac{f(x+tx,y+ty,z+tz)-f(x,y,z)}{t}.$$

Answer 2

The flow of $E$ is $\phi_t(x,y,z)=(e^tx,e^ty,e^tz)$, $f(\phi_t(x,y,z))=f(e^tx,e^ty,e^tz)=e^{pt}f(x,y,z)$ implies that=t $E(f)={d\over{dt}}_{t=0}f(\phi_t(x,y,z))={d\over{dt}}_{t=0}e^{pt}f(x,y,z)=pf(x,y,z).$