Vector field invariant under certain diffeomorphisms This probably should only require elementary differential geometry but somehow I can't seem to prove it.
How can I show the vector field $\displaystyle \sum_{i=1}^n x_i \frac{\partial}{\partial x_i}$ on $\mathbb{R}^n$ is invariant under the action of $\mathrm{SO}(n)$?

This is called the "Euler vector field" and it is actually invariant under all of $\mathrm{GL}(n,\mathbb{R})$. A nice way to see this is to look at its flow which is the scaling action
$$ \phi_t(x) = e^t x.$$
It is quite clear that this flow is invariant, because linear maps commute with scalar multiplication, and this is equivalent to invariance of the generating vector field.
