About the derivative of the Jacobian in fluid dynamics I was studying a book on the mathematics of fluid dynamics in which there was a lemma on how to find the derivative of the Jacobian. The explanation is as follows (Sorry if it's too long):
There is a region $D$ in Euclidean space where there is a fluid whose velocity at any point $\mathbf {\vec x}\in D$ at any time $t$ is given by the vector field $\mathbf{\vec u}(\mathbf{\vec x}, t)$. Let us write $\mathbf{\vec \varphi}(\mathbf{\vec x}, t)$ for the trajectory followed by the particle that is at
point $\mathbf{\vec x}$ at time $t = 0$. We will assume $\varphi$ is smooth enough so the following
manipulations are legitimate and for fixed $t$, $\mathbf{\vec \varphi}$ is an invertible mapping.
Let $\varphi_t$ denote the map $\mathbf{\vec x} \rightarrow \mathbf{\vec \varphi}(\mathbf{\vec x}, t)$; that is, with fixed $t$, this map advances
each fluid particle from its position at time $t = 0$ to its position at time $t$.
Here, of course, the subscript does not denote differentiation. We call $\mathbf{\vec \varphi}$ the
fluid flow map. If $W$ is a region in $D$, then $\varphi_t(W) = W_t$ is the volume
$W$ moving with the fluid as shown in figure.

Now let's say that $\mathbf{\vec x}=x\hat i+y\hat j+z\hat k$; $\mathbf{\vec \varphi}=\epsilon \hat i+\eta\hat j+\zeta \hat k$ and $\mathbf{\vec u}=u\hat i+v\hat j+w\hat k$. Since $\mathbf{\vec \varphi}$ is the displacement field and $\mathbf{\vec u}$ is the velocity field so we have
$$\frac{\delta}{\delta t}(\mathbf{\vec \varphi}(\mathbf{\vec x}, t))=\mathbf{\vec u}(\mathbf{\vec \varphi}(\mathbf{\vec x}, t), t)$$
So
$$\frac{\delta\epsilon}{\delta t}=u;\ \frac{\delta\eta}{\delta t}=v;\ \frac{\delta\zeta}{\delta t}=w\ \ \ \ \ \cdots(i)$$
The Jacobian of $\mathbf{\vec x}$ w.r.t $\mathbf{\vec \varphi}(\mathbf{\vec x}, t)$ is given as:
$$J(\mathbf {\vec x},t)=\left[\begin{matrix}
\displaystyle{\frac{\delta\epsilon}{\delta x}}&\displaystyle\frac{\delta\eta}{\delta x}&\displaystyle\frac{\delta\zeta}{\delta x}\\
\displaystyle{\frac{\delta\epsilon}{\delta y}}&\displaystyle\frac{\delta\eta}{\delta y}&\displaystyle\frac{\delta\zeta}{\delta y}\\
\displaystyle{\frac{\delta\epsilon}{\delta z}}&\displaystyle\frac{\delta\eta}{\delta z}&\displaystyle\frac{\delta\zeta}{\delta z}
\end{matrix}\right]$$
Now the task is to prove that

$$\displaystyle\frac{\delta}{\delta t}J(\mathbf{\vec x}, t) = J(\mathbf{\vec x}, t)\left[\text{div }\mathbf{\vec u}(\mathbf{\vec \varphi}(\mathbf{\vec x}, t), t)\right]$$

What I did:
$$\begin{align}
\displaystyle\frac{\delta}{\delta t}J&=\left[\begin{matrix}
\displaystyle{\frac{\delta}{\delta t}\frac{\delta\epsilon}{\delta x}}&\displaystyle\frac{\delta\eta}{\delta x}&\displaystyle\frac{\delta\zeta}{\delta x}\\
\displaystyle{\frac{\delta}{\delta t}\frac{\delta\epsilon}{\delta y}}&\displaystyle\frac{\delta\eta}{\delta y}&\displaystyle\frac{\delta\zeta}{\delta y}\\
\displaystyle{\frac{\delta}{\delta t}\frac{\delta\epsilon}{\delta z}}&\displaystyle\frac{\delta\eta}{\delta z}&\displaystyle\frac{\delta\zeta}{\delta z}
\end{matrix}\right]+\left[\begin{matrix}
\displaystyle{\frac{\delta\epsilon}{\delta x}}&\displaystyle{\frac{\delta}{\delta t}\frac{\delta\eta}{\delta x}}&\displaystyle\frac{\delta\zeta}{\delta x}\\
\displaystyle{\frac{\delta\epsilon}{\delta y}}&\displaystyle{\frac{\delta}{\delta t}\frac{\delta\eta}{\delta y}}&\displaystyle\frac{\delta\zeta}{\delta y}\\
\displaystyle{\frac{\delta\epsilon}{\delta z}}&\displaystyle{\frac{\delta}{\delta t}\frac{\delta\eta}{\delta z}}&\displaystyle\frac{\delta\zeta}{\delta z}
\end{matrix}\right]+\left[\begin{matrix}
\displaystyle{\frac{\delta\epsilon}{\delta x}}&\displaystyle\frac{\delta\eta}{\delta x}&\displaystyle{\frac{\delta}{\delta t}\frac{\delta\zeta}{\delta x}}\\
\displaystyle{\frac{\delta\epsilon}{\delta y}}&\displaystyle\frac{\delta\eta}{\delta y}&\displaystyle{\frac{\delta}{\delta t}\frac{\delta\zeta}{\delta y}}\\
\displaystyle{\frac{\delta\epsilon}{\delta z}}&\displaystyle\frac{\delta\eta}{\delta z}&\displaystyle{\frac{\delta}{\delta t}\frac{\delta\zeta}{\delta z}}
\end{matrix}\right]\\
&=\left[\begin{matrix}
\displaystyle{\frac{\delta u}{\delta x}}&\displaystyle\frac{\delta\eta}{\delta x}&\displaystyle\frac{\delta\zeta}{\delta x}\\
\displaystyle{\frac{\delta u}{\delta y}}&\displaystyle\frac{\delta\eta}{\delta y}&\displaystyle\frac{\delta\zeta}{\delta y}\\
\displaystyle{\frac{\delta u}{\delta z}}&\displaystyle\frac{\delta\eta}{\delta z}&\displaystyle\frac{\delta\zeta}{\delta z}
\end{matrix}\right]+\left[\begin{matrix}
\displaystyle{\frac{\delta\epsilon}{\delta x}}&\displaystyle\frac{\delta v}{\delta x}&\displaystyle\frac{\delta\zeta}{\delta x}\\
\displaystyle{\frac{\delta\epsilon}{\delta y}}&\displaystyle\frac{\delta v}{\delta y}&\displaystyle\frac{\delta\zeta}{\delta y}\\
\displaystyle{\frac{\delta\epsilon}{\delta z}}&\displaystyle\frac{\delta v}{\delta z}&\displaystyle\frac{\delta\zeta}{\delta z}
\end{matrix}\right]+\left[\begin{matrix}
\displaystyle{\frac{\delta\epsilon}{\delta x}}&\displaystyle\frac{\delta\eta}{\delta x}&\displaystyle\frac{\delta z}{\delta x}\\
\displaystyle{\frac{\delta\epsilon}{\delta y}}&\displaystyle\frac{\delta\eta}{\delta y}&\displaystyle\frac{\delta z}{\delta y}\\
\displaystyle{\frac{\delta\epsilon}{\delta z}}&\displaystyle\frac{\delta\eta}{\delta z}&\displaystyle\frac{\delta z}{\delta z}
\end{matrix}\right]\ \ \ \ [\text{By }(i)]
\end{align}$$
At this point the book says to use the fact that
$$\frac{\delta u}{\delta x}=\frac{\delta u}{\delta \epsilon}\frac{\delta \epsilon}{\delta x}+\frac{\delta u}{\delta \eta}\frac{\delta \eta}{\delta x}+\frac{\delta u}{\delta \zeta}\frac{\delta \zeta}{\delta x}$$
$$\frac{\delta u}{\delta y}=\frac{\delta u}{\delta \epsilon}\frac{\delta \epsilon}{\delta y}+\frac{\delta u}{\delta \eta}\frac{\delta \eta}{\delta y}+\frac{\delta u}{\delta \zeta}\frac{\delta \zeta}{\delta y}$$
$$\vdots$$
$$\frac{\delta w}{\delta z}=\frac{\delta w}{\delta \epsilon}\frac{\delta \epsilon}{\delta z}+\frac{\delta w}{\delta \eta}\frac{\delta \eta}{\delta z}+\frac{\delta w}{\delta \zeta}\frac{\delta \zeta}{\delta z}$$
However I can't quite understand how to use these equations to continue the determinant. Please help me in this problem.
Thanks for the attention.
 A: A straightforward derivation is to use the differential of a determinant that can be written as
\begin{eqnarray}
d|\mathbf{J}|
&=&
|\mathbf{J}| 
\mathrm{tr}
\left( \mathbf{J}^{-1} d\mathbf{J} \right)
=
|\mathbf{J}| 
\mathrm{tr}
\left( \mathbf{J}^{-1} \dot{\mathbf{J}} \right)
dt
\tag{1}
\end{eqnarray}
The key observation is
$$
(\dot{\mathbf{J}})_{ij}
=
\frac{\partial}{\partial t}
\left(
\frac{\partial x_i}{\partial \xi_j}
\right)
=
\frac{\partial}{\partial \xi_j}
\left(
\frac{\partial x_i}{\partial t}
\right)
=
\frac{\partial v_i}{\partial \xi_j}
=
\sum_k
\frac{\partial v_i}{\partial x_k}
\frac{\partial x_k}{\partial \xi_j}
=
\sum_k
\frac{\partial v_i}{\partial x_k}
J_{kj}
$$
In matrix form, this writes
$$
\dot{\mathbf{J}}
=
\begin{pmatrix}
\frac{\partial v_1}{\partial x_1}
&
\frac{\partial v_1}{\partial x_2}
&
\frac{\partial v_1}{\partial x_3} \\
\frac{\partial v_2}{\partial x_1}
&
\frac{\partial v_2}{\partial x_2}
&
\frac{\partial v_2}{\partial x_3} \\
\frac{\partial v_3}{\partial x_1}
&
\frac{\partial v_3}{\partial x_2}
&
\frac{\partial v_3}{\partial x_3}
\end{pmatrix}
\mathbf{J}
=\mathbf{A}\mathbf{J}
$$
The relation (1) writes
\begin{eqnarray}
d|\mathbf{J}|
&=&
|\mathbf{J}| 
\mathrm{tr}
\left( \mathbf{J}^{-1} \mathbf{A}\mathbf{J} \right)
dt
=
|\mathbf{J}| 
\mathrm{tr}
\left( \mathbf{A} \right)
dt
=
|\mathbf{J}|
\operatorname{div}(\mathbf{v})
dt
\end{eqnarray}
A: https://www.owlnet.rice.edu/~ceng501/Chap4.pdf
This link might be helpful to answer your question.
There are cancellations when you expand the determinant.
