In the title of your question you mention the Euler vector field, so I suspect that your question was about the following fact:
for any vector bundle $\pi:E\to M$, the Euler vector field $X$ on $E$, i.e. the infinitesimal generator of the homotheties of $E$, has the expression $X=u\frac{\partial}{\partial u}$ w.r.t. standard coordinates $(x,u)$.
If I have understood your question then below there is my answer.
If $\pi:E\to M$ is a vector bundle then its Euler vector field $X$ is defined as the vector field on $E$ which is the infinitesimal generator of the action $\Phi$ of $\mathbb{R}$ on $E$ given by the fiberwise scalar multiplication.
$\Phi:(t,e)\in\mathbb{R}\times E\to\Phi_t(e):=\exp(t).e\in E.$ $X_e:=\left.\frac{d}{dt}\right|_{t=0}\ \Phi_t(e),\ \forall e\in E.$
Let $x:U\to\mathbb{R}^n$ be coordinates on $M$ and $(x,u):\pi^{-1}(U)\to\mathbb{R}^n\times\mathbb{R}^n$ the associate standard coordinates on $E$. In such coordinates we find the following explicit expressions: $\Phi(t,(x,u))=(x,\exp(t).u),$ $X_{(x,u)}=\left.\frac{d}{dt}\right|_{t=0}\ (x,\exp(t).u)=\left.u\frac{\partial}{\partial u}\right|_{(x,u)}.$