Time Derivative of Fluid Map Jacobian (Page 10 of 'A Mathematical Introduction to Fluid Mechanics' by Chorin and Marsden) Context and Notation
I'm self-studying from the book mentioned in the post title and I'm struggling to follow a calculation performed by the authors on pages 8 and 9. Please see the pictures attached for the authors' full calculation. The authors are calculating the partial time derivative of the fluid flow map's Jacobian. They use $\phi  = (\xi, \eta, \zeta)$ for the fluid flow map, and $(x,y,z)$ for the initial coordinates of a fluid particle at time $t=0$. The authors use u for the fluid velocity vector and its components are $(u,v,w)$.
Source of Confusion
I have attempted the calculation myself and in my calculations the divergence in the final result is calculated with respect to the fluid map coordinates. That is, the derivatives are taken with respect to $\xi$, $\eta$, and $\zeta$, respectively.  However the authors' calculations state the final result using the divergence with respect to the initial position coordinates $(x,y,z)$. I have been unable to see why the author's calculation is correct.
Request to Community
I was hoping someone may be able to point out to me why the author is correct, in particular how he uses the chain rule at the bottom of page 9 to make the conclusion.
Please note that a highly similar question was posted previously by another user at the link below. The main response to this question links to an article that calculates the divergence using  derivatives with respect to the fluid map coordinates. Thus (in my view) the response contradicts the author's result.
https://math.stackexchange.com/q/3166941/


 A: If you're saying that at the end of the calculation you got $\left(\frac{\partial u}{\partial \xi}+\frac{\partial v}{\partial \eta}+\frac{\partial w}{\partial \zeta}\right) J$, then that's fine. In any case, this is much less confusing if you write out everything precisely and indicate the points of evaluation. In precise notation, the final step should read
\begin{align}
\left[(\partial_1u)(\phi(\mathbf{x},t), t)+ (\partial_2v)(\phi(\mathbf{x},t), t)+ (\partial_3w)(\phi(\mathbf{x},t), t) \right]\cdot J(\mathbf{x},t)=(\text{div }\mathbf{u})(\phi(\mathbf{x},t))\cdot J(\mathbf{x},t).
\end{align}
Here, the subscript $\partial_if$ means differentiate the function $f:\Bbb{R}^n\to\Bbb{R}$ with respect to its $i^{th}$ variable, and $(\partial_if)(p)$ means the evaluation of this function on the point $p\in\Bbb{R}^n$. So, explicitly, $(\partial_if)(p)=\lim\limits_{h\to 0}\frac{f(p+ he_i)-f(p)}{h}$, where $e_i\in\Bbb{R}^3$ is the vector with $1$ in the $i^{th}$ spot, and $0$ elsewhere. This way, you do not have to introduce extraneous letters $x,y,z,\xi,\eta,\zeta$ (resulting in confusions like yours).
