1.2.4 Changes of the conserved quantities

Without source terms the only change between time $\bgroup\color{DEFcolor}$t_1$\egroup$ and $\bgroup\color{DEFcolor}$t_0$\egroup$ to the amount of a conserved variable inside a volume $\bgroup\color{DEFcolor}$V$\egroup$ comes from the flux through its surface $\bgroup\color{DEFcolor}$\partial V$\egroup$ .

Integrating Eqs. (8) to (10) over that surface and time thus gives

$\begin{displaymath} \int_{V} \rho \left( \ensuremath{\mathchoice{\mbox{\boldmat... ... n$}} {\mbox{\boldmath$\scriptscriptstyle n$}}}} \; d\!A dt \end{displaymath}$

(11)

$\begin{displaymath} \int_{V} \rho\ensuremath{\mathchoice{\mbox{\boldmath$\displ... ... n$}} {\mbox{\boldmath$\scriptscriptstyle n$}}}} \; d\!A dt \end{displaymath}$

(12)

$\begin{displaymath} \int_{V} \rho e_{\rm ik}\left( \ensuremath{\mathchoice{\mbo... ... n$}} {\mbox{\boldmath$\scriptscriptstyle n$}}}} \; d\!A dt \end{displaymath}$

(13)

The unity tensor $\bgroup\color{DEFcolor}$\bar{\bar{\ensuremath{\mathsf{I}}}}$\egroup$ has been squeezed in,

$\begin{displaymath} P\hat{\ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle... ...yle n$}} {\mbox{\boldmath$\scriptscriptstyle n$}}}} \enspace . \end{displaymath}$

(14)