Petersen's proof of Gauss Lemma

Question

I don't quite understand a step in Petersen's proof of Gauss Lemma (lemma 5.5.5 here). After introducing exponential normal coordinates, he defines a vector field $\partial_r$ on $U=\exp_p(B_\varepsilon)$ in coordinates by $\partial_r\vert_{\exp_px}=\frac{1}{r}x^i\partial_i\vert_{\exp_px}$. He then claims that the integral curves of this vector field are unit-speed geodesics and I don't understand why this is obvious.

Answer 1

That vector field is a normalized radial vector field (it is $\frac{x}{\Vert x \Vert}$). By construction of the exponential map, its integral curves are the unit-speed geodesics.