1.1.1 The Euler equations in differential form (vectors)

The Euler equations in differential conservation form in vector notation are

$${ \renewedcommand{arraystretch}{2.0} \begin{array}{cclcc} ... ...dmath$\scriptscriptstyle v$}}} \right) & = & 0 \end{array}}$$ (1)

They describe the inviscid flow of density $\rho$, momentum $\rho\ensuremath{\mathchoice{\mbox{\boldmath$\displayst... ...ox{\boldmath$\scriptstyle v$}} {\mbox{\boldmath$\scriptscriptstyle v$}}}$, and total energy $\rho e_{\rm ik}$, with

$\ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle v$... ...ox{\boldmath$\scriptstyle v$}} {\mbox{\boldmath$\scriptscriptstyle v$}}}$	velocity vector
$e_{\rm ik}$	total (internal + kinetic) energy per mass unit
$P$	pressure
$\bar{\bar{\ensuremath{\mathsf{I}}}}$	unity tensor  .