Continuity equation in Lagrangian coordinates

[c0der]

From the Eulerian form of the continuity equation, where x is the Eulerian coordinate:

\frac {\partial \rho}{ \partial t } + u \frac {\partial \rho}{\partial x} + \rho \frac { \partial u}{\partial x} = 0

The incremental change in mass is, where m is the Lagrangian coordinate:

dm = \rho dx

The specific volume is:

V = \frac{1}{\rho}

How does one get the final form of the continuity equation in Lagrangian coordinates as follows:

\frac{\partial V}{\partial t} = \frac{\partial u}{\partial m}

 

[wrobel]

$$\rho (t,\xi)\mathrm{det}\,\Big(\frac{\partial \xi}{\partial\hat \xi}\Big)=\hat\rho(\hat\xi),$$ here $\hat \xi$ are the Lagrangian coordinates in the initial moment $t=0$ and $\xi=\xi(t,\hat\xi)$ are the current Lagrangian coordinates; $\hat\rho$ is the initial desity and $\rho (t,\xi)=\hat\rho(\xi(t, \hat\xi))$ is the current density