Proving Euler's identity (fluid mechanics) by differentiating the determinant $J=\partial(x,y,z)/\partial(X,Y,Z)$ I am looking at some fluid mechanics lecture notes. In this context, $X,Y,Z$ are Lagrangian variables (at time $t = 0$) and $x,y,z$ are Eulerian variables (at time $t &gt; 0$). Euler's identity says that $$\frac{\mathrm{D}J}{\mathrm{D}t}=J\mathbf{\nabla}\cdot \mathbf{u}$$ where $\mathbf{u}=\mathbf{u}(x(t),y(t),z(t),t)$ is the velocity of the fluid, $J=\frac{\partial (x,y,z)}{\partial(X,Y,Z)}$ is the Jacobian determinant and $\frac{\mathrm{D}}{\mathrm{D}t}=\frac{\partial }{\partial t}+\mathbf{u \cdot \nabla}$ is the convective differential operator.

The notes say this:



Whilst this seems to look quite straightforward, I am not sure what property has been used here or why it is permissible for us to be able to differentiate each row of the determinant separately and add up the resulting determinants. Could somebody please offer an input or key result that is being used?
Consider the case $n=2$. Observe
\begin{align}
\frac{d}{dt}
\begin{vmatrix}
a_{11}(t) &amp; a_{12}(t)\\
a_{21}(t) &amp; a_{22}(t)
\end{vmatrix}
=&amp;\ \frac{d}{dt} \sum_{\sigma \in S_2}(-1)^{\operatorname{sgn}\sigma} a_{1\sigma(1)}(t) a_{2\sigma(2)}(t)\\
=&amp;\ \sum_{\sigma \in S_2}(-1)^{\operatorname{sgn}\sigma} [a_{1\sigma(1)}'(t) a_{2\sigma(2)}(t)+a_{1\sigma(1)}(t) a_{2\sigma(2)}'(t)]\\
=&amp;\ \sum_{\sigma \in S_2}(-1)^{\operatorname{sgn}\sigma} a_{1\sigma(1)}'(t) a_{2\sigma(2)}(t)+\sum_{\sigma \in S_2}(-1)^{\operatorname{sgn}\sigma} a_{1\sigma(1)}(t) a_{2\sigma(2)}'(t)\\
=&amp;\ 
\begin{vmatrix}
a_{11}'(t) &amp; a_{12}'(t)\\
a_{21}(t) &amp; a_{22}(t)
\end{vmatrix}
+
\begin{vmatrix}
a_{11}(t) &amp; a_{12}(t)\\
a_{21}'(t) &amp; a_{22}'(t)
\end{vmatrix}.
\end{align}
Note I have used the definition: If $A=(a_{ij})$ is an $n\times n$ matrix then
\begin{align}
\det A = \sum_{\sigma \in S_n} (-1)^{\operatorname{sgn}\sigma} a_{1 \sigma(1)}\cdots a_{n\sigma(n)}
\end{align}
where $S_n$ is the permutation group of $n$ objects.
An additional resource which I found was useful for this problem are those slides: 

http://www3.dicca.unige.it/rrepetto/linked-files/fluid-dynamics-lecture-notes.pdf 

The appendix (Slide 154 - 156) it explains the topic as well. 

The approach there is as follows: 

$$J=\operatorname{det}\left(\begin{array}{ccc}\frac{\partial x_{1}}{\partial \xi_{1}} &amp; \frac{\partial x_{1}}{\partial \xi_{2}} &amp; \frac{\partial x_{1}}{\partial \xi_{3}} \\ \frac{\partial x_{2}}{\partial \xi_{1}} &amp; \frac{\partial x_{2}}{\partial \xi_{2}} &amp; \frac{\partial x_{2}}{\partial \xi_{3}} \\ \frac{\partial x_{3}}{\partial \xi_{1}} &amp; \frac{\partial x_{3}}{\partial \xi_{2}} &amp; \frac{\partial x_{3}}{\partial \xi_{3}}\end{array}\right)$$

Definition of the material derivative:

$$\frac{D}{D t}\left(\frac{\partial x_{i}}{\partial \xi_{j}}\right)=\frac{\partial}{\partial \xi_{j}} \frac{D x_{j}}{D t}=\frac{\partial u_{i}}{\partial \xi_{j}}$$

Apply the chain rule:

$$\frac{\partial u_{i}}{\partial \xi_{j}}=\frac{\partial u_{i}}{\partial x_{1}} \frac{\partial x_{1}}{\partial \xi_{j}}+\frac{\partial u_{i}}{\partial x_{2}} \frac{\partial x_{2}}{\partial \xi_{j}}+\frac{\partial u_{i}}{\partial x_{3}} \frac{\partial x_{3}}{\partial \xi_{j}}=\frac{\partial u_{i}}{\partial x_{k}} \frac{\partial x_{k}}{\partial \xi_{j}}$$

As an example for the first row, insert the previous results:

$$ \operatorname{det}\left(\begin{array}{ccc}
\frac{\partial u_{1}}{\partial x_{k}} \frac{\partial x_{k}}{\partial \xi_{1}} &amp; \frac{\partial u_{1}}{\partial x_{k}} \frac{\partial x_{k}}{\partial \xi_{2}} &amp; \frac{\partial u_{1}}{\partial x_{k}} \frac{\partial x_{k}}{\partial \xi_{3}} \\
\frac{\partial x_{2}}{\partial \xi_{1}} &amp; \frac{\partial x_{2}}{\partial \xi_{2}} &amp; \frac{\partial x_{2}}{\partial \xi_{3}} \\
\frac{\partial x_{3}}{\partial \xi_{1}} &amp; \frac{\partial x_{3}}{\partial \xi_{2}} &amp; \frac{\partial x_{3}}{\partial \xi_{3}}
\end{array}\right)
$$

Simplify the result:

$$ \operatorname{det} = \left(\begin{array}{ccc}\frac{\partial u_{1}}{\partial x_{1}} \frac{\partial x_{1}}{\partial \xi_{1}} &amp; \frac{\partial u_{1}}{\partial x_{1}} \frac{\partial x_{1}}{\partial \xi_{2}} &amp; \frac{\partial u_{1}}{\partial x_{1}} \frac{\partial x_{1}}{\partial \xi_{3}} \\ \frac{\partial x_{2}}{\partial \xi_{1}} &amp; \frac{\partial x_{2}}{\partial \xi_{2}} &amp; \frac{\partial x_{2}}{\partial \xi_{3}} \\ \frac{\partial x_{3}}{\partial \xi_{1}} &amp; \frac{\partial x_{3}}{\partial \xi_{2}} &amp; \frac{\partial x_{3}}{\partial \xi_{3}}\end{array}\right)=\frac{\partial u_{1}}{\partial x_{1}}J$$