The example 1.7 in "An introduction to Theoretical Fluid Mechanics" by Stephen Childress, he calculated the material derivative of the Jacobian determinant $Det \ \textbf{J}$ of the Lagrangian map
\begin{align*} X:\Omega_0 \times [0,+\infty) \to \Omega_t \subset \mathbb{R}^N, \ \ \ (\alpha,t) \mapsto X(\alpha,t)=\textbf{x}. \end{align*}
I think that the definition of $\textbf{J}$ is: \begin{align*} \textbf{J}=\begin{pmatrix} \frac{\partial X_1}{\partial \alpha_1} & \frac{\partial X_1}{\partial \alpha_2} & \cdots & \frac{\partial X_1}{\partial \alpha_N} \\ \frac{\partial X_2}{\partial \alpha_1} & \frac{\partial X_2}{\partial \alpha_2} & \cdots & \frac{\partial X_2}{\partial \alpha_N}\\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial X_N}{\partial \alpha_1} & \frac{\partial X_N}{\partial \alpha_2} & \cdots & \frac{\partial X_N}{\partial \alpha_N} \end{pmatrix} \end{align*}.
Then, he concluded that \begin{align*} \frac{D}{Dt}(Det \ \textbf{J})= div(\text{u})(Det \ \textbf{J}), \end{align*} where \begin{align*} \frac{D}{Dt}=\frac{\partial}{\partial t}+ \textbf{u}\cdot \nabla, \end{align*} is the material derivative.
On the other hand, in the book "A Mathematical Introduction to Fluid Mechanics" Third Edition by Chorin and Marsden, in the lemma of page 8, they asserts that (in the Childress notation) \begin{align*} \frac{\partial}{\partial t} Det \ \textbf{J}=(Det \ \textbf{J})div(\textbf{u}). \end{align*}
So, like \begin{align*} \frac{D}{Dt} Det \ \textbf{J}=\frac{\partial}{\partial t} (Det \ \textbf{J})+\textbf{u}\cdot \nabla(Det \ \textbf{J}), \end{align*} for the above, we can conclude that \begin{align*} \frac{\partial}{\partial t} (Det \ \textbf{J})+\textbf{u}\cdot \nabla(Det \ \textbf{J})=div(\textbf{u})(Det \ \textbf{J}), \end{align*} then \begin{align*} div(\textbf{u})(Det \ \textbf{J})+\textbf{u}\cdot \nabla(Det \ \textbf{J})=div(\textbf{u})(Det \ \textbf{J}), \end{align*} therefore \begin{align*} \textbf{u}\cdot\nabla (Det \ \textbf{J})=0. \end{align*}
It is right? Or am I misunderstanding the concept of material derivative?