Radial Vector Field in Normal Neighborhood

Theorem
Let $\struct {M, g}$ be an $n$-dimensional Riemannian manifold.

Let $U_p$ be the normal neighborhood of $p \in M$.

Let $\partial_r$ be the radial vector field on $U_p \setminus \set p$, where $\setminus$ denotes the set difference.

Then $\partial_r$ is well-defined in $U_p \setminus \set p$ independently of the choice of normal coordinates.

Furthermore, $\partial_r$ is smooth on $U_p \setminus \set p$.