If the cartesian coordinates are given as functions of general curvilinear coordinates $a_1,a_2,a_3$, i.e.

$x=x(a_1,a_2,a_3),$

$y=y(a_1,a_2,a_3),$

$z=z(a_1,a_2,a_3),$

(G.1)

it follows for the infinitesimal volume element in curvilinear coordinates

$dV=\left|J\right|d{a}_{1}d{a}_{2}d{a}_3.$

(G.2)

Thereby the Jacobian determinant $J$ is defined as

(G.3)

With $v_i=\frac{d{x}_i}{dt}$ it follows for the time derivative of $J$

	${\displaystyle \frac{dJ}{dt}}$	$={\displaystyle \frac{d}{dt}}{\displaystyle \sum _{i,j,k}}{ϵ}_{ijk}{\displaystyle \frac{\partial x}{\partial a_i}}{\displaystyle \frac{\partial y}{\partial a_j}}{\displaystyle \frac{\partial z}{\partial a_k}}$		(G.4)
		$={\displaystyle \sum _{i,j,k}}{ϵ}_{ijk}\left({\displaystyle \frac{\partial v_x}{\partial a_i}}{\displaystyle \frac{\partial y}{\partial a_j}}{\displaystyle \frac{\partial z}{\partial a_k}}+{\displaystyle \frac{\partial x}{\partial a_i}}{\displaystyle \frac{\partial v_y}{\partial a_j}}{\displaystyle \frac{\partial z}{\partial a_k}}+{\displaystyle \frac{\partial x}{\partial a_i}}{\displaystyle \frac{\partial y}{\partial a_j}}{\displaystyle \frac{\partial v_z}{\partial a_k}}\right).$		(G.5)

From $\frac{\partial v_k}{\partial a_i}={\sum }_l\frac{\partial v_k}{\partial x_l}\frac{\partial x_l}{\partial a_i}$ $(k\in x,y,z)$ we get:

${\displaystyle \frac{dJ}{dt}}={\displaystyle \sum _{i,j,k,l}}{ϵ}_{ijk}\left({\displaystyle \frac{\partial v_x}{\partial x_l}}{\displaystyle \frac{\partial x_l}{\partial a_i}}{\displaystyle \frac{\partial y}{\partial a_j}}{\displaystyle \frac{\partial z}{\partial a_k}}+{\displaystyle \frac{\partial x}{\partial a_i}}{\displaystyle \frac{\partial v_y}{\partial x_l}}{\displaystyle \frac{\partial x_l}{\partial a_j}}{\displaystyle \frac{\partial z}{\partial a_k}}+{\displaystyle \frac{\partial x}{\partial a_i}}{\displaystyle \frac{\partial y}{\partial a_j}}{\displaystyle \frac{\partial v_z}{\partial x_l}}{\displaystyle \frac{\partial x_l}{\partial a_k}}\right).$

(G.6)

Analysis of the first term in the bracket for

	$l$	$=1:{\displaystyle \frac{\partial v_x}{\partial x}}{\displaystyle \frac{\partial x}{\partial a_i}}{\displaystyle \frac{\partial y}{\partial a_j}}{\displaystyle \frac{\partial z}{\partial a_k}},$
	$l$	$=2:{\displaystyle \frac{\partial v_x}{\partial y}}\underset{\text{symmetric}}{\underbrace{{\displaystyle \frac{\partial y}{\partial a_i}}{\displaystyle \frac{\partial y}{\partial a_j}}}}{\displaystyle \frac{\partial z}{\partial a_k}}\text{ is (multiplied with }{ϵ}_{ijk}\text{) 0 when summed up},$
	$l$	$=3:{\displaystyle \frac{\partial v_x}{\partial z}}\overbrace{{\displaystyle \frac{\partial z}{\partial a_i}}}{\displaystyle \frac{\partial y}{\partial a_j}}\overbrace{{\displaystyle \frac{\partial z}{\partial a_k}}}\text{ is (multiplied with }{ϵ}_{ijk}\text{) 0 when summed up}.$

So for symmetry reasons most of the contributions are zero when summed up. In we end we see: For the first term there is only a non-vanishing contribution for $l=1$, for the second term there is only one for $l=2$ and for the third term there is only one for $l=3$. So we get

${\displaystyle \frac{dJ}{dt}}={\displaystyle \sum _{i,j,k}}{ϵ}_{ijk}\left({\displaystyle \frac{\partial v_x}{\partial x}}+{\displaystyle \frac{\partial v_y}{\partial y}}+{\displaystyle \frac{\partial v_z}{\partial z}}\right){\displaystyle \frac{\partial x}{\partial a_i}}{\displaystyle \frac{\partial y}{\partial a_j}}{\displaystyle \frac{\partial z}{\partial a_k}}=(\nabla \cdot 𝒗)J.$

(G.7)