$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

In a spatial (Eulerian) frame, the momentum balance equation for a continuum is $$\begin{equation} \rho\mathbf{a}=\divg\boldsymbol{\sigma}+\rho\mathbf{b}\,,\label{eq:momentum-balance} \end{equation}$$ where $\rho$ is the density, $\boldsymbol{\sigma}$ is the Cauchy stress, $\mathbf{b}$ is the body force per mass, and $\mathbf{a}$ is the acceleration, given by the material time derivative of the velocity $\mathbf{v}$ in the spatial frame, $$\begin{equation} \mathbf{a}=\dot{\mathbf{v}}=\frac{\partial\mathbf{v}}{\partial t}+\mathbf{L}\cdot\mathbf{v}\,,\label{eq:acceleration} \end{equation}$$ where $\mathbf{L}=\grad\mathbf{v}$ is the spatial velocity gradient. The mass balance equation is $$\begin{equation} \dot{\rho}+\rho\divg\mathbf{v}=0\,,\label{eq:mass-balance} \end{equation}$$ where the material time derivative of the density in the spatial frame is $$\begin{equation} \dot{\rho}=\frac{\partial\rho}{\partial t}+\grad\rho\cdot\mathbf{v}\,.\label{eq:density-material-derivative} \end{equation}$$ Let $\mathbf{F}$ denote the deformation gradient (the gradient of the motion with respect to the material coordinate). The material time derivative of $\mathbf{F}$ is related to $\mathbf{L}$ via $$\begin{equation} \dot{\mathbf{F}}=\mathbf{L}\cdot\mathbf{F}\,.\label{eq:F-dot} \end{equation}$$ Let $J=\det\mathbf{F}$ denote the Jacobian of the motion (the volume ratio, or ratio of current to referential volume, $J>0$ ); then, the dilatation (relative change in volume between current and reference configurations) is given by $e=J-1$ . Using the chain rule, $J$ 's material time derivative is $\dot{J}=J\mathbf{F}^{-T}:\dot{\mathbf{F}}$ which, when combined with eq.(2.11.1-5), produces a kinematic constraint between $\dot{J}$ and $\divg\mathbf{v}$ , $$\begin{equation} \dot{J}=J\,\divg\mathbf{v}\,.\label{eq:divv-kinematic-relation} \end{equation}$$ Substituting this relation into the mass balance, eq.(2.11.1-3), produces $\dot{\overline{\rho J}}=0$ , which may be integrated directly to yield $$\begin{equation} \rho=\rho_{r}/J\,,\label{eq:mass-balance-integrated} \end{equation}$$ where $\rho_{r}$ is the density in the reference configuration (when $J=1$ ). Since $\rho_{r}$ is obtained by integrating the above material time derivative of $\rho J$ , it is an intrinsic material property that must be invariant in time and space.