In the section about the asymptotic expansion for the heat kernel in the "Heat and wave equations" chapter of John Roe's book, he includes the following in one of this proposition:
Suppose that $M$ is a Riemannian manifold with metric $g$ and that $S \to M$ is a Clifford bundle over $M$. Fix a point $q$ in $M$ and take a geodesic coordinate system ${x^i}$ with $q$ as the origin. Let $r^2=\sum (x^i)^2 = \sum g^{ij}x^{i}x^{j}$ so that $r$ is the geodesic distance from $q$, and define $h=(4\pi t)^{-n/2}e^{-r^2/4t}$. Then $\nabla h$ has the expression $-\frac{h}{2t}r\frac{\partial}{\partial r}$ in this coordinate system.
What does $\frac{\partial}{\partial r}$ mean? I've only seen this notation for a vector when that vector is the image of the tangent vector to the coordinate curve $x^i$ in $U \subset \mathbb{R}^n$ for some coordinate system $\{x^i\}$ on our manifold $M$. Is there some such coordinate neighborhood that's being implicitly used here? The notation is suggestive of polar coordinates. Should I first take polar coordinates on the tangent space at $q$ and then exponentiate to get this coordinate system?