Picture below is from 242th page of
Huisken, Gerhard, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom. 20, 237-266 (1984). ZBL0556.53001.
$H$ is the mean curvature. I don't know how to get the red line under this frame. In my view, $$ \nabla H =(\frac{\partial H}{\partial x_1},...,\frac{\partial H}{\partial x_n}) $$ and $$ \frac{\partial }{\partial x_1}=e_1 =\frac{\nabla H}{|\nabla H|} $$
"A frame" is not a global coordinate system, but rather a collection of bases for tangent spaces, say near a point as in the above text. If one introduces a frame where $e_1(p)=\frac{\nabla H(p)}{|\nabla H(p)|}$ then when using the basis that is given by the frame the vector $\nabla H(p)$ is written as a linear combination of basis vectors as $$\nabla H(p)= |\nabla H(p)| e_1(p) +0 e_2(p)+\ldots +0 e_n(p)$$ Thus, since the frame is orthonormal at each point, $\nabla_i H(p)= \nabla H(p) \cdot e_i(p)$ is as in the "red line".