We have that $$\frac{\partial}{\partial{t}}J=\begin{vmatrix} \frac{\partial}{\partial{t}}\frac{\partial{\xi}}{\partial{x}}& \frac{\partial{\eta}}{\partial{x}} & \frac{\partial{\zeta}}{\partial{x}} \\ \frac{\partial}{\partial{t}}\frac{\partial{\xi}}{\partial{y}} & \frac{\partial{\eta}}{\partial{y}} & \frac{\partial{\zeta}}{\partial{y}} \\ \frac{\partial}{\partial{t}}\frac{\partial{\xi}}{\partial{z}} & \frac{\partial{\eta}}{\partial{z}} & \frac{\partial{\zeta}}{\partial{z}} \end{vmatrix}+\begin{vmatrix} \frac{\partial{\xi}}{\partial{x}}& \frac{\partial}{\partial{t}}\frac{\partial{\eta}}{\partial{x}} & \frac{\partial{\zeta}}{\partial{x}} \\ \frac{\partial{\xi}}{\partial{y}} & \frac{\partial}{\partial{t}}\frac{\partial{\eta}}{\partial{y}} & \frac{\partial{\zeta}}{\partial{y}} \\ \frac{\partial{\xi}}{\partial{z}} & \frac{\partial}{\partial{t}}\frac{\partial{\eta}}{\partial{z}} & \frac{\partial{\zeta}}{\partial{z}} \end{vmatrix}+\begin{vmatrix} \frac{\partial{\xi}}{\partial{x}}& \frac{\partial{\eta}}{\partial{x}} & \frac{\partial}{\partial{t}}\frac{\partial{\zeta}}{\partial{x}} \\ \frac{\partial{\xi}}{\partial{y}} & \frac{\partial{\eta}}{\partial{y}} & \frac{\partial}{\partial{t}}\frac{\partial{\zeta}}{\partial{y}} \\ \frac{\partial{\xi}}{\partial{z}} & \frac{\partial{\eta}}{\partial{z}} & \frac{\partial}{\partial{t}}\frac{\partial{\zeta}}{\partial{z}} \end{vmatrix}$$
$$\phi(\overrightarrow{x}, t)=(\xi(\overrightarrow{x}, t),\eta (\overrightarrow{x}, t), \zeta (\overrightarrow{x}, t)) \\ \overrightarrow{u}=(u, v, w)$$
We have for example $$\frac{\partial}{\partial{t}}\frac{\partial{\xi}}{\partial{x}}(\overrightarrow{x}, t)=\frac{\partial}{\partial{x}}\frac{\partial{\xi}}{\partial{t}}(\overrightarrow{x}, t)=\frac{\partial}{\partial{x}}[u(\phi(\overrightarrow{x}, t), t)]=\frac{\partial}{\partial{x}}[u(\xi(\overrightarrow{x}, t),\eta (\overrightarrow{x}, t), \zeta (\overrightarrow{x}, t);t)]\\ =\frac{\partial{u}}{\partial{x}}\frac{\partial{\xi}}{\partial{x}}+\frac{\partial{u}}{\partial{y}}\frac{\partial{\eta}}{\partial{x}}+\frac{\partial{u}}{\partial{z}}\frac{\partial{\zeta}}{\partial{x}}$$ and similar the other ones...
How do we get then $$\frac{\partial}{\partial{t}}J=\left (\frac{\partial{u}}{\partial{x}}+\frac{\partial{v}}{\partial{y}}+\frac{\partial{w}}{\partial{z}}\right )J=\left (div \overrightarrow{u}\right )J$$ ??