ODE for the Jacobian determinant of the flow map I'm trying to follow the steps starting at the last section of page 2 here.
I'm asked to show that if $|J(x,y,z,t)|$ is the (spatial) Jacobian determinant of the flow map $\boldsymbol{\varphi}(x,y,z,t)$, then
$$\frac{\partial |J|}{\partial t}=|J| \operatorname{div}(\mathbf{v}) $$
where $\mathbf{v}=\partial \boldsymbol{\varphi}/\partial t $ is the velocity field of the flow.
I couldn't prove it for the 3D case, so I tried the 1D analogue, which boils down to the equation
$$\frac{\partial^2 \varphi}{\partial t \partial x}= \frac{\partial \varphi}{\partial x} \frac{\partial^2 \varphi}{\partial x \partial t}.$$
I can't see why this simpler equation must hold either. I'd like some help with both of this cases please. Thank you. 
 A: Changing notation from $(x,y,z)$ to $(x_1,x_2,x_3)$, the flow map $\boldsymbol{\varphi}$ maps the coordinates $(x_1,x_2,x_3)$ of a fluid particle at time $t= 0$ onto the coordinates $(\varphi_1, \varphi_2,\varphi_3)$  at time $t$.
The Jacobian determinant $|J|$ can be expressed as
$$|J| = \sum_{(j_1,j_2,j_3)}s(j_1,j_2,j_3) \frac{\partial\varphi_1}{\partial x_{j_1}}\frac{\partial\varphi_2}{\partial x_{j_2}}\frac{\partial\varphi_3}{\partial x_{j_3}},$$
where the sum is taken over all permutations $(j_1,j_2,j_3)$ of $(1,2,3)$ and $s(j_1,j_2,j_3)$ is the sign of the permutation.
Since 
$$\frac{\partial}{\partial t}\frac{\partial \varphi_k}{\partial x_{j_k}} = \frac{\partial}{\partial x_{j_k}}\frac{\partial \varphi_k}{\partial t} = \frac{\partial v_k}{\partial x_{j_k}}$$
we have
$$\tag{*}\begin{align}\frac{\partial}{\partial t} |J| &= \sum_{(j_1,j_2,j_3)}s(j_1,j_2,j_3) \frac{\partial v_1}{\partial x_{j_1}}\frac{\partial\varphi_2}{\partial x_{j_2}}\frac{\partial\varphi_3}{\partial x_{j_3}} \\ &+ \sum_{(j_1,j_2,j_3)}s(j_1,j_2,j_3) \frac{\partial \varphi_1}{\partial x_{j_1}}\frac{\partial v_2}{\partial x_{j_2}}\frac{\partial\varphi_3}{\partial x_{j_3}} \\ &+ \sum_{(j_1,j_2,j_3)}s(j_1,j_2,j_3) \frac{\partial \varphi_1}{\partial x_{j_1}}\frac{\partial\varphi_2}{\partial x_{j_2}}\frac{\partial v_3}{\partial x_{j_3}}   \end{align}.$$
By the chain rule,
$$\frac{\partial v_1}{\partial x_{j_1}} = \frac{\partial v_1}{\partial \varphi_1}\frac{\partial \varphi_1}{\partial x_{j_1}} + \frac{\partial v_1}{\partial \varphi_2}\frac{\partial \varphi_2}{\partial x_{j_1}} + \frac{\partial v_3}{\partial \varphi_1}\frac{\partial \varphi_3}{\partial x_{j_1}}. $$
Applying this to (*) the first sum on the RHS becomes
$$\begin{align}\sum_{(j_1,j_2,j_3)}s(j_1,j_2,j_3) \frac{\partial v_1}{\partial x_{j_1}}\frac{\partial\varphi_2}{\partial x_{j_2}}\frac{\partial\varphi_3}{\partial x_{j_3}} &= \frac{\partial v_1}{\partial \varphi_1}\sum_{(j_1,j_2,j_3)}s(j_1,j_2,j_3) \frac{\partial \varphi_1}{\partial x_{j_1}}\frac{\partial\varphi_2}{\partial x_{j_2}}\frac{\partial\varphi_3}{\partial x_{j_3}} \\ &+ \frac{\partial v_2}{\partial \varphi_1}\sum_{(j_1,j_2,j_3)}s(j_1,j_2,j_3) \frac{\partial \varphi_2}{\partial x_{j_1}}\frac{\partial\varphi_2}{\partial x_{j_2}}\frac{\partial\varphi_3}{\partial x_{j_3}}  \\ &+  \frac{\partial v_3}{\partial \varphi_1}\sum_{(j_1,j_2,j_3)}s(j_1,j_2,j_3) \frac{\partial \varphi_3}{\partial x_{j_1}}\frac{\partial\varphi_2}{\partial x_{j_2}}\frac{\partial\varphi_3}{\partial x_{j_3}} \\ \end{align} $$
The second and third sums on the RHS of (**) are determinants with identical rows and must vanish.
Hence,
$$\sum_{(j_1,j_2,j_3)}s(j_1,j_2,j_3) \frac{\partial v_1}{\partial x_{j_1}}\frac{\partial\varphi_2}{\partial x_{j_2}}\frac{\partial\varphi_3}{\partial x_{j_3}} =   \frac{\partial v_1}{\partial \varphi_1}\sum_{(j_1,j_2,j_3)}s(j_1,j_2,j_3) \frac{\partial \varphi_1}{\partial x_{j_1}}\frac{\partial\varphi_2}{\partial x_{j_2}}\frac{\partial\varphi_3}{\partial x_{j_3}} =   \frac{\partial v_1}{\partial \varphi_1} |J|$$
Applying the chain rule, similarly, to the second and third sums on the RHS of (*) and adding we obtain
$$\frac{\partial}{\partial t} |J|= \left(\frac{\partial v_1}{\partial \varphi_1} + \frac{\partial v_2}{\partial \varphi_2} + \frac{\partial v_3}{\partial \varphi_3} \right)|J|  = \operatorname{div}(\mathbf{v}) |J|$$
A: This answer is adapted from About the derivative of the Jaobian in fluid dynamics with new variables naming.
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}
@time $t=0$ the fluid particle is at position $\mathbf{x}$ and at a later time, it is at position $\mathbf{\varphi}(\mathbf{x},t)$ (which is a 3-D vector).
The velocity of the particle (in the moving frame) is denoted $\mathbf{v}$ with tht $i$-th component given by
$v_i = \frac{\partial \varphi_i}{\partial t}$.
The key observation is
$$
(\dot{\mathbf{J}})_{ij}
=
\frac{\partial}{\partial t}
\left(
\frac{\partial \varphi_i}{\partial x_j}
\right)
=
\frac{\partial}{\partial x_j}
\left(
\frac{\partial \varphi_i}{\partial t}
\right)
=
\frac{\partial v_i}{\partial x_j}
=
\sum_k
\frac{\partial v_i}{\partial \varphi_k}
\frac{\partial \varphi_k}{\partial x_j}
=
\sum_k
\frac{\partial v_i}{\partial \varphi_k}
J_{kj}
$$
In matrix form, this writes
$$
\dot{\mathbf{J}}
=
\begin{pmatrix}
\frac{\partial v_1}{\partial \varphi_1}
&
\frac{\partial v_1}{\partial \varphi_2}
&
\frac{\partial v_1}{\partial \varphi_3} \\
\frac{\partial v_2}{\partial \varphi_1}
&
\frac{\partial v_2}{\partial \varphi_2}
&
\frac{\partial v_2}{\partial \varphi_3} \\
\frac{\partial v_3}{\partial \varphi_1}
&
\frac{\partial v_3}{\partial \varphi_2}
&
\frac{\partial v_3}{\partial \varphi_3}
\end{pmatrix}
\mathbf{J}
=\mathbf{A}\mathbf{J}
$$
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}
and thus
$$
\frac{\partial |\mathbf{J}|}{\partial t}
=
|\mathbf{J}|
\operatorname{div}(\mathbf{v})
$$
