On page 250 of Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, by Alfred Gray https://books.google.com/books?id=-LRumtTimYgC&lpg=PP1&pg=PA250#v=onepage&q&f=false
"Lemma 11.3. (The Chain Rule) Let $g_{1},\dots,g_{k}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ and $F:\mathbb{R}^{k}\rightarrow\mathbb{R}$ be differentiable functions and let $\boldsymbol{v}_{\boldsymbol{p}}\in\mathbb{R}_{\boldsymbol{p}}^{n}$ where $\boldsymbol{p}\in\mathbb{R}^{n}$. Write $f=F\left(g_{1},\dots,g_{k}\right)$. Then
$\boldsymbol{v}_{\boldsymbol{p}}\left[f\right]=\underset{j=1}{\overset{k}{\sum}}\frac{\partial F}{\partial u_{j}}\left(g_{1}\left(\boldsymbol{p}\right),\dots,g_{k}\left(\boldsymbol{p}\right)\right)\boldsymbol{v}_{\boldsymbol{p}}\left[g_{j}\right]$."
I contend that the partial differentiation should be with respect to $g_{j}$ instead of $u_{j}$. That is
$\boldsymbol{v}_{\boldsymbol{p}}\left[f\right]=\underset{j=1}{\overset{k}{\sum}}\frac{\partial F}{\partial g_{j}}\left(g_{1}\left(\boldsymbol{p}\right),\dots,g_{k}\left(\boldsymbol{p}\right)\right)\boldsymbol{v}_{\boldsymbol{p}}\left[g_{j}\right]$.
Am I correct?
The lemma is stated a bit differently in the third edition, but the expression in question is of the same form. See page 267, Lemma 9.6.
Gray is correct. But my alternative formulation is also correct.
$\frac{\partial F}{\partial u_{j}}$ is the rate of change of F with respect to the coordinate value $u_{j}$.
$\frac{\partial F}{\partial g_{j}}$ is the rate of change of F with respect to the coordinate value which is given as a function $g_{j}$.
The two denote the same value with different connotations.