Vector fields and plateau functions. Let $M$ a differentiable manifold and $X$ a vector field on $M$. Given $m \in M$, there exists $(W; x^1,...,x^n)$ a coordinate system around $m$. In $W$, the vector field $X$ can be written as $\sum a_i \frac{\partial}{\partial x^i}$ for some smooth functions $a_i$. I have read that if $\rho$ is a plateau function around $m$, then $$\rho^2 X=\sum \rho a_i \rho \frac{\partial}{\partial x^i}$$ My question is: why is true the last equality ?

Well, you are just regrouping the multiplication by $\rho^2$. But the point is that $$\sum (\rho a_i)\left(\rho \frac{\partial}{\partial x^i}\right)$$ is making a globally defined expression for $\rho^2 X$: Even though $a_i$ and $\partial/\partial x^i$ are only defined in the chart $W$, $\rho a_i$, for example, is globally defined, since $\rho$ is compactly supported in $W$.