From 2012 Petrova, "Finite Volume Methods- Powerful Means of Engineering Design"

$\bar\phi = \frac{1}{\Delta t} \int_{t_0}^{t_0 + \Delta t} \phi dt = $ Reynolds averaging over time.

$\tilde \phi = \frac{1}{\bar{\rho}} \frac{1}{\Delta t} \int_{t_0}^{t_0 + \Delta t} (\rho \phi) dt = $ Favre averaging.


variables:
${ n} ^i$ = flux surface normal, in units of $[1]$
$\bar{\rho}$ = Reynolds-averaged density, in units of $\frac{kg}{{m}^{3}}$
${ \tilde{v}} ^i$ = Favre-averaged velocity, in units of ${\frac{1}{s}} {m}$
${{ \bar{m}} ^i} = {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}$ $= \overline{\rho v^i} =$ Reynolds-averaged momentum, in units of $\frac{kg}{{{{m}^{2}}} {{s}}}$
$k$ = turbulent kinetic energy, in units of $\frac{{m}^{2}}{{s}^{2}}$
$\omega$ = specific turbulent dissipation rate, in units of ???
$\tilde{T}$ = Favre-averaged temperature, in units of $K$
${C_v}$ = constant-volume heat capacity, in units of $\frac{{{K}} {{{s}^{2}}}}{{m}^{2}}$
${C_p}$ = constant-pressure heat capacity, in units of $\frac{{{K}} {{{s}^{2}}}}{{m}^{2}}$
$R$ = specific heat constant, in units of $\frac{{m}^{2}}{{{K}} {{{s}^{2}}}}$
${R} = {{{C_p}}{-{{C_v}}}}$
$\gamma$ = heat capacity ratio, unitless
${\gamma} = {{\frac{1}{{C_v}}} {{C_p}}}$
${\bar{P}} = {{{\bar{\rho}}} \cdot {{R}} {{\tilde{T}}}}$ = Reynolds-averaged pressure, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${P^*}$ = ${\bar{P}} + {{{\frac{2}{3}}} {{\bar{\rho}}} \cdot {{k}}}$ = ${\frac{1}{3}} {{{\bar{\rho}}} \cdot {{\left({{{{2}} {{k}}} + {{{3}} {{R}} {{\tilde{T}}}}}\right)}}}$ = static pressure, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{c_s}} = {{{\frac{1}{\sqrt{{C_v}}}}} {{\frac{1}{\sqrt{\bar{\rho}}}}} {{\sqrt{{{{{C_v}}} \cdot {{{P^*}}} \cdot {{{|n|}^{2}}}} + {{{{P^*}}} \cdot {{R}} {{{|n|}^{2}}}}{-{{{R}} {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^c}} {{{ \tilde{v}} ^d}} {{{ n} _c}} {{{ n} _d}}}}}}}}$ = speed of sound in units of ${\frac{1}{s}} {m}$
${{ g} _i} _j$ = metric tensor, in units of $[1]$
${{(\tilde{v})^2}} = {{{{ \tilde{v}} ^k}} {{{ \tilde{v}} ^l}} {{{{ g} _k} _l}}}$ = Favre-averaged velocity, norm squared, in units of $\frac{{m}^{2}}{{s}^{2}}$

${{\tilde{e}_{kin}}} = {{{\frac{1}{2}}} {{{ \tilde{v}} ^k}} {{{ \tilde{v}} ^l}} {{{{ g} _k} _l}}}$ = Favre-averaged specific kinetic energy, in units of $\frac{{m}^{2}}{{s}^{2}}$

${\tilde{e}_{int}}$ = ${{{C_v}}} \cdot {{\tilde{T}}}$ = $\frac{{{{C_v}}} \cdot {{\left({{{{3}} {{{P^*}}}}{-{{{2}} {{\bar{\rho}}} \cdot {{k}}}}}\right)}}}{{{3}} {{R}} {{\bar{\rho}}}}$ = Favre-averaged specific internal energy, in units of $\frac{{m}^{2}}{{s}^{2}}$

${{\tilde{e}_{total}}} = {{{\tilde{e}_{int}}} + {{\tilde{e}_{kin}}}}$ = Favre-averaged densitized total energy, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{\tilde{e}_{total}}} = {{{{{C_v}}} \cdot {{{P^*}}} \cdot {{\frac{1}{R}}} {{\frac{1}{\bar{\rho}}}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{\frac{1}{R}}}}}$

Conservative and primitive variables:
${{ W} ^I} = {\left[\begin{array}{c} \bar{\rho}\\ { \tilde{v}} ^i\\ {P^*}\\ k\\ \omega\end{array}\right]}$
${{ U} ^I} = {\left[\begin{array}{c} \bar{\rho}\\ {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}\\ {{\bar{\rho}}} \cdot {{{\tilde{e}_{total}}}}\\ {{\bar{\rho}}} \cdot {{k}}\\ {{\bar{\rho}}} \cdot {{\omega}}\end{array}\right]}$
Partial of conservative quantities wrt primitives:
${\frac{\partial { U} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccccc} \frac{\partial \bar{\rho}}{\partial \bar{\rho}}& \frac{\partial \bar{\rho}}{\partial { \tilde{v}} ^j}& \frac{\partial \bar{\rho}}{\partial {P^*}}& \frac{\partial \bar{\rho}}{\partial k}& \frac{\partial \bar{\rho}}{\partial \omega}\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}\right)\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{{\tilde{e}_{total}}}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{{\tilde{e}_{total}}}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{{\tilde{e}_{total}}}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{{\tilde{e}_{total}}}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{{\tilde{e}_{total}}}}}\right)\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{k}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{k}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{k}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{k}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{k}}}\right)\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{\omega}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{\omega}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{\omega}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{\omega}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{\omega}}}\right)\end{array}\right]}$
${\frac{\partial { U} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccccc} \frac{\partial \bar{\rho}}{\partial \bar{\rho}}& \frac{\partial \bar{\rho}}{\partial { \tilde{v}} ^j}& \frac{\partial \bar{\rho}}{\partial {P^*}}& \frac{\partial \bar{\rho}}{\partial k}& \frac{\partial \bar{\rho}}{\partial \omega}\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}}}\right)\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{\left({{{{\frac{1}{2}}} {{{ \tilde{v}} ^k}} {{{ \tilde{v}} ^l}} {{{{ g} _k} _l}}} + {\frac{{{{C_v}}} \cdot {{\left({{{{3}} {{{P^*}}}}{-{{{2}} {{\bar{\rho}}} \cdot {{k}}}}}\right)}}}{{{3}} {{R}} {{\bar{\rho}}}}}}\right)}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{\left({{{{\frac{1}{2}}} {{{ \tilde{v}} ^k}} {{{ \tilde{v}} ^l}} {{{{ g} _k} _l}}} + {\frac{{{{C_v}}} \cdot {{\left({{{{3}} {{{P^*}}}}{-{{{2}} {{\bar{\rho}}} \cdot {{k}}}}}\right)}}}{{{3}} {{R}} {{\bar{\rho}}}}}}\right)}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{\left({{{{\frac{1}{2}}} {{{ \tilde{v}} ^k}} {{{ \tilde{v}} ^l}} {{{{ g} _k} _l}}} + {\frac{{{{C_v}}} \cdot {{\left({{{{3}} {{{P^*}}}}{-{{{2}} {{\bar{\rho}}} \cdot {{k}}}}}\right)}}}{{{3}} {{R}} {{\bar{\rho}}}}}}\right)}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{\left({{{{\frac{1}{2}}} {{{ \tilde{v}} ^k}} {{{ \tilde{v}} ^l}} {{{{ g} _k} _l}}} + {\frac{{{{C_v}}} \cdot {{\left({{{{3}} {{{P^*}}}}{-{{{2}} {{\bar{\rho}}} \cdot {{k}}}}}\right)}}}{{{3}} {{R}} {{\bar{\rho}}}}}}\right)}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{\left({{{{\frac{1}{2}}} {{{ \tilde{v}} ^k}} {{{ \tilde{v}} ^l}} {{{{ g} _k} _l}}} + {\frac{{{{C_v}}} \cdot {{\left({{{{3}} {{{P^*}}}}{-{{{2}} {{\bar{\rho}}} \cdot {{k}}}}}\right)}}}{{{3}} {{R}} {{\bar{\rho}}}}}}\right)}}}\right)\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{k}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{k}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{k}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{k}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{k}}}\right)\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{\omega}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{\omega}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{\omega}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{\omega}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{\omega}}}\right)\end{array}\right]}$
${\frac{\partial { U} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ { \tilde{v}} ^i& {{\bar{\rho}}} \cdot {{{{ δ} ^i} _j}}& 0& 0& 0\\ {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{\frac{1}{R}}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}}}& {{\bar{\rho}}} \cdot {{{ \tilde{v}} _j}}& {{{C_v}}} \cdot {{\frac{1}{R}}}& {{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{\frac{1}{R}}}& 0\\ k& 0& 0& \bar{\rho}& 0\\ \omega& 0& 0& 0& \bar{\rho}\end{array}\right]}$
Expanded:
${\frac{\partial { U} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccccccc} 1& 0& 0& 0& 0& 0& 0\\ { \tilde{v}} ^x& \bar{\rho}& 0& 0& 0& 0& 0\\ { \tilde{v}} ^y& 0& \bar{\rho}& 0& 0& 0& 0\\ { \tilde{v}} ^z& 0& 0& \bar{\rho}& 0& 0& 0\\ \frac{{{{3}} {{{(\tilde{v})^2}}} \cdot {{R}}}{-{{{4}} {{{C_v}}} \cdot {{k}}}}}{{{6}} {{R}}}& {{\bar{\rho}}} \cdot {{{ \tilde{v}} _x}}& {{\bar{\rho}}} \cdot {{{ \tilde{v}} _y}}& {{\bar{\rho}}} \cdot {{{ \tilde{v}} _z}}& {\frac{1}{R}} {{C_v}}& -{\frac{{{2}} {{{C_v}}} \cdot {{\bar{\rho}}}}{{{3}} {{R}}}}& 0\\ k& 0& 0& 0& 0& \bar{\rho}& 0\\ \omega& 0& 0& 0& 0& 0& \bar{\rho}\end{array}\right]}$
${{(\tilde{v})^2}} = {{{{{ \tilde{v}} ^x}} {{{ \tilde{v}} _x}}} + {{{{ \tilde{v}} ^y}} {{{ \tilde{v}} _y}}} + {{{{ \tilde{v}} ^z}} {{{ \tilde{v}} _z}}}}$
${\frac{\partial { W} ^I}{\partial { U} ^J}} = {\left[\begin{array}{ccccccc} 1& 0& 0& 0& 0& 0& 0\\ -{{\frac{1}{\bar{\rho}}} {{ \tilde{v}} ^x}}& \frac{1}{\bar{\rho}}& 0& 0& 0& 0& 0\\ -{{\frac{1}{\bar{\rho}}} {{ \tilde{v}} ^y}}& 0& \frac{1}{\bar{\rho}}& 0& 0& 0& 0\\ -{{\frac{1}{\bar{\rho}}} {{ \tilde{v}} ^z}}& 0& 0& \frac{1}{\bar{\rho}}& 0& 0& 0\\ \frac{{{R}} {{{(\tilde{v})^2}}}}{{{2}} {{{C_v}}}}& -{{\frac{1}{{C_v}}} {{{R}} {{{ \tilde{v}} _x}}}}& -{{\frac{1}{{C_v}}} {{{R}} {{{ \tilde{v}} _y}}}}& -{{\frac{1}{{C_v}}} {{{R}} {{{ \tilde{v}} _z}}}}& {\frac{1}{{C_v}}} {R}& \frac{2}{3}& 0\\ -{{\frac{1}{\bar{\rho}}} {k}}& 0& 0& 0& 0& \frac{1}{\bar{\rho}}& 0\\ -{{\frac{1}{\bar{\rho}}} {\omega}}& 0& 0& 0& 0& 0& \frac{1}{\bar{\rho}}\end{array}\right]}$
${\frac{\partial { W} ^I}{\partial { U} ^J}} = {\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ {\frac{1}{\bar{\rho}}}{\left({-{{ \tilde{v}} ^i}}\right)}& {\frac{1}{\bar{\rho}}} {{{ δ} ^i} _j}& 0& 0& 0\\ {{\frac{1}{2}}} {{{\frac{1}{{C_v}}} {R}}} {{{(\tilde{v})^2}}}& {-{{\frac{1}{{C_v}}} {R}}} {{{ \tilde{v}} _j}}& {\frac{1}{{C_v}}} {R}& \frac{2}{3}& 0\\ {\frac{1}{\bar{\rho}}}{\left({-{k}}\right)}& 0& 0& \frac{1}{\bar{\rho}}& 0\\ {\frac{1}{\bar{\rho}}}{\left({-{\omega}}\right)}& 0& 0& 0& \frac{1}{\bar{\rho}}\end{array}\right]}$

Flux:
${{ F} ^I} = {\left[\begin{array}{c} {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^j}} {{{ n} _j}}\\ {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ \tilde{v}} ^j}} {{{ n} _j}}} + {{{{ n} ^i}} {{{P^*}}}}\\ {{{ \tilde{v}} ^j}} {{{ n} _j}} {{\left({{{{\bar{\rho}}} \cdot {{{\tilde{e}_{total}}}}} + {{P^*}}}\right)}}\\ {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^j}} {{{ n} _j}} {{k}}\\ {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^j}} {{{ n} _j}} {{\omega}}\end{array}\right]}$
${{ F} ^I} = {\left[\begin{array}{c} {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^j}} {{{ n} _j}}\\ {{{{P^*}}} \cdot {{{ n} ^i}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ \tilde{v}} ^j}} {{{ n} _j}}}\\ {{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ \tilde{v}} ^j}} {{{ n} _j}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ \tilde{v}} ^j}} {{{ n} _j}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^j}} {{{ n} _j}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^j}} {{{ n} _j}} {{\frac{1}{R}}}}\\ {{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^j}} {{{ n} _j}}\\ {{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ \tilde{v}} ^j}} {{{ n} _j}}\end{array}\right]}$

Flux derivative wrt primitive variables:
${\frac{\partial { F} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccccc} {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{{{P^*}}} \cdot {{{ n} ^i}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{{{P^*}}} \cdot {{{ n} ^i}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{{{P^*}}} \cdot {{{ n} ^i}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}}\right)& {\frac{\partial}{\partial k}}\left({{{{{P^*}}} \cdot {{{ n} ^i}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{{{P^*}}} \cdot {{{ n} ^i}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}}\right)\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}}}\right)& {\frac{\partial}{\partial k}}\left({{{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}}}\right)\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)\\ {\frac{\partial}{\partial \bar{\rho}}}\left({{{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial { \tilde{v}} ^j}}\left({{{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial {P^*}}}\left({{{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial k}}\left({{{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)& {\frac{\partial}{\partial \omega}}\left({{{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}}\right)\end{array}\right]}$
${\frac{\partial { F} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccccc} {{{ \tilde{v}} ^k}} {{{ n} _k}}& {{\bar{\rho}}} \cdot {{{ n} _j}}& 0& 0& 0\\ {{{ \tilde{v}} ^i}} {{{ \tilde{v}} ^k}} {{{ n} _k}}& {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ n} _j}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}} {{{{ δ} ^i} _j}}}& { n} ^i& 0& 0\\ {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}}& {{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _j}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ n} _j}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{\bar{\rho}}} \cdot {{{ n} _j}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ n} _j}} {{\frac{1}{R}}}}& {{{{C_v}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}} + {{{{ \tilde{v}} ^k}} {{{ n} _k}}}& {{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}& 0\\ {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}}& {{\bar{\rho}}} \cdot {{k}} {{{ n} _j}}& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}& 0\\ {{\omega}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}& {{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ n} _j}}& 0& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}\end{array}\right]}$
${\frac{\partial { F} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccccc} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{{ n} _j}}& 0& 0& 0\\ {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^i}} {{{ n} _a}}& {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ n} _j}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{{{ δ} ^i} _j}}}& { n} ^i& 0& 0\\ {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}}& {{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _j}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ n} _j}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{\bar{\rho}}} \cdot {{{ n} _j}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ n} _j}} {{\frac{1}{R}}}}& {{{{C_v}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}} + {{{{ \tilde{v}} ^a}} {{{ n} _a}}}& {{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}& 0\\ {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{k}} {{{ n} _j}}& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& 0\\ {{\omega}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ n} _j}}& 0& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}\end{array}\right]}$
${\frac{\partial { F} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccccc} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{{ n} _j}}& 0& 0& 0\\ {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^i}} {{{ n} _a}}& {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ n} _j}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{{{ δ} ^i} _j}}}& { n} ^i& 0& 0\\ {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}}& {{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _j}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ n} _j}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{\bar{\rho}}} \cdot {{{ n} _j}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ n} _j}} {{\frac{1}{R}}}}& {{{{C_v}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}} + {{{{ \tilde{v}} ^a}} {{{ n} _a}}}& {{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}& 0\\ {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{k}} {{{ n} _j}}& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& 0\\ {{\omega}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ n} _j}}& 0& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}\end{array}\right]}$
${\frac{\partial { F} ^I}{\partial { W} ^J}} = {\left[\begin{array}{ccccc} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{{ n} _j}}& 0& 0& 0\\ {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^i}} {{{ n} _a}}& {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ n} _j}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{{{ δ} ^i} _j}}}& { n} ^i& 0& 0\\ {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}} + {{{\frac{1}{2}}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^b}} {{{ \tilde{v}} _b}} {{{ n} _a}}}& {{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _j}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ n} _j}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ n} _j}} {{\frac{1}{R}}}} + {{{\frac{1}{2}}} {{\bar{\rho}}} \cdot {{{ \tilde{v}} _a}} {{{ \tilde{v}} ^a}} {{{ n} _j}}}& {{{{C_v}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}} + {{{{ \tilde{v}} ^a}} {{{ n} _a}}}& {{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}& 0\\ {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{k}} {{{ n} _j}}& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& 0\\ {{\omega}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ n} _j}}& 0& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}\end{array}\right]}$

Flux derivative wrt conserved variables:
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\frac{\partial { F} ^I}{\partial { W} ^L}}} {{\frac{\partial { W} ^L}{\partial { U} ^J}}}}$
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\left[\begin{array}{ccccc} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{{ n} _k}}& 0& 0& 0\\ {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^i}} {{{ n} _a}}& {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^i}} {{{ n} _k}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{{{ δ} ^i} _k}}}& { n} ^i& 0& 0\\ {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}} + {{{\frac{1}{2}}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^b}} {{{ \tilde{v}} _b}} {{{ n} _a}}}& {{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _k}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ n} _k}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ n} _k}} {{\frac{1}{R}}}} + {{{\frac{1}{2}}} {{\bar{\rho}}} \cdot {{{ \tilde{v}} _a}} {{{ \tilde{v}} ^a}} {{{ n} _k}}}& {{{{C_v}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}} + {{{{ \tilde{v}} ^a}} {{{ n} _a}}}& {{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}& 0\\ {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{k}} {{{ n} _k}}& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& 0\\ {{\omega}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ n} _k}}& 0& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}\end{array}\right]}} {{\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ {\frac{1}{\bar{\rho}}}{\left({-{{ \tilde{v}} ^k}}\right)}& {\frac{1}{\bar{\rho}}} {{{ δ} ^k} _j}& 0& 0& 0\\ {{\frac{1}{2}}} {{{\frac{1}{{C_v}}} {R}}} {{{(\tilde{v})^2}}}& {-{{\frac{1}{{C_v}}} {R}}} {{{ \tilde{v}} _j}}& {\frac{1}{{C_v}}} {R}& \frac{2}{3}& 0\\ {\frac{1}{\bar{\rho}}}{\left({-{k}}\right)}& 0& 0& \frac{1}{\bar{\rho}}& 0\\ {\frac{1}{\bar{\rho}}}{\left({-{\omega}}\right)}& 0& 0& 0& \frac{1}{\bar{\rho}}\end{array}\right]}}}$
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{ccccc} {{{{ \tilde{v}} ^a}} {{{ n} _a}}} + {{{-1}} {{{ \tilde{v}} ^k}} {{{ n} _k}}}& {{{ n} _k}} {{{{ δ} ^k} _j}}& 0& 0& 0\\ {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{R}} {{{ n} ^i}} {{\frac{1}{{C_v}}}}} + {{{-1}} {{{ \tilde{v}} ^i}} {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^i}} {{{ n} _a}}} + {{{-1}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^k}} {{{ n} _a}} {{{{ δ} ^i} _k}}}& {{{-1}} {{R}} {{{ \tilde{v}} _j}} {{{ n} ^i}} {{\frac{1}{{C_v}}}}} + {{{{ \tilde{v}} ^i}} {{{ n} _k}} {{{{ δ} ^k} _j}}} + {{{{ \tilde{v}} ^a}} {{{ n} _a}} {{{{ δ} ^i} _k}} {{{{ δ} ^k} _j}}}& {{R}} {{{ n} ^i}} {{\frac{1}{{C_v}}}}& {{2}} \cdot {{\frac{1}{3}}} {{{ n} ^i}}& 0\\ {{{-1}} {{{C_v}}} \cdot {{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}} {{\frac{1}{\bar{\rho}}}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{R}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{{C_v}}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{R}}}} + {{{-1}} {{{P^*}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}} {{\frac{1}{\bar{\rho}}}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} _a}} {{{ \tilde{v}} ^k}} {{{ n} _k}}} + {{{\frac{1}{2}}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^b}} {{{ \tilde{v}} _b}} {{{ n} _a}}}& {{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _k}} {{{{ δ} ^k} _j}} {{\frac{1}{R}}} {{\frac{1}{\bar{\rho}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{{ n} _k}} {{{{ δ} ^k} _j}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ n} _k}} {{{{ δ} ^k} _j}} {{\frac{1}{\bar{\rho}}}}} + {{{-1}} {{R}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} _j}} {{{ n} _a}} {{\frac{1}{{C_v}}}}} + {{{-1}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} _j}} {{{ n} _a}}} + {{{\frac{1}{2}}} {{{ \tilde{v}} _a}} {{{ \tilde{v}} ^a}} {{{ n} _k}} {{{{ δ} ^k} _j}}}& {{{{ \tilde{v}} ^a}} {{{ n} _a}}} + {{{R}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{{C_v}}}}}& {{2}} \cdot {{\frac{1}{3}}} {{{ \tilde{v}} ^a}} {{{ n} _a}}& 0\\ {{-1}} {{k}} {{{ \tilde{v}} ^k}} {{{ n} _k}}& {{k}} {{{ n} _k}} {{{{ δ} ^k} _j}}& 0& {{{ \tilde{v}} ^a}} {{{ n} _a}}& 0\\ {{-1}} {{\omega}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _k}}& {{\omega}} \cdot {{{ n} _k}} {{{{ δ} ^k} _j}}& 0& 0& {{{ \tilde{v}} ^a}} {{{ n} _a}}\end{array}\right]}$
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{ccccc} 0& { n} _j& 0& 0& 0\\ {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{R}} {{{ n} ^i}} {{\frac{1}{{C_v}}}}} + {{{-1}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^i}} {{{ n} _a}}}& {{{{ \tilde{v}} ^i}} {{{ n} _j}}} + {{{-1}} {{R}} {{{ \tilde{v}} _j}} {{{ n} ^i}} {{\frac{1}{{C_v}}}}} + {{{{ \tilde{v}} ^a}} {{{ n} _a}} {{{{ δ} ^i} _j}}}& {{R}} {{{ n} ^i}} {{\frac{1}{{C_v}}}}& {{2}} \cdot {{\frac{1}{3}}} {{{ n} ^i}}& 0\\ {{{-1}} {{{C_v}}} \cdot {{{P^*}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}} {{\frac{1}{\bar{\rho}}}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{R}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{{C_v}}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}} + {{{-1}} {{{P^*}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{\bar{\rho}}}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \tilde{v}} _a}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^b}} {{{ n} _b}}} + {{{\frac{1}{2}}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^b}} {{{ \tilde{v}} _b}} {{{ n} _a}}}& {{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _j}} {{\frac{1}{R}}} {{\frac{1}{\bar{\rho}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{{ n} _j}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ n} _j}} {{\frac{1}{\bar{\rho}}}}} + {{{-1}} {{R}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} _j}} {{{ n} _a}} {{\frac{1}{{C_v}}}}} + {{{\frac{1}{2}}} {{{ \tilde{v}} _a}} {{{ \tilde{v}} ^a}} {{{ n} _j}}} + {{{-1}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} _j}} {{{ n} _a}}}& {{{{ \tilde{v}} ^a}} {{{ n} _a}}} + {{{R}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{{C_v}}}}}& {{2}} \cdot {{\frac{1}{3}}} {{{ \tilde{v}} ^a}} {{{ n} _a}}& 0\\ {{-1}} {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{k}} {{{ n} _j}}& 0& {{{ \tilde{v}} ^a}} {{{ n} _a}}& 0\\ {{-1}} {{\omega}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\omega}} \cdot {{{ n} _j}}& 0& 0& {{{ \tilde{v}} ^a}} {{{ n} _a}}\end{array}\right]}$

Acoustic matrix:
${{{{ A} ^I} _J} + {{{{\tilde{v}_n}}} \cdot {{{{ δ} ^I} _J}}}} = {{{\frac{\partial { W} ^I}{\partial { U} ^K}}} {{\frac{\partial { F} ^K}{\partial { W} ^J}}}}$
${{{{ A} ^I} _J} + {{{{\tilde{v}_n}}} \cdot {{{{ δ} ^I} _J}}}} = {{{\left[\begin{array}{ccccc} 1& 0& 0& 0& 0\\ {\frac{1}{\bar{\rho}}}{\left({-{{ \tilde{v}} ^i}}\right)}& {\frac{1}{\bar{\rho}}} {{{ δ} ^i} _k}& 0& 0& 0\\ {{\frac{1}{2}}} {{{\frac{1}{{C_v}}} {R}}} {{{(\tilde{v})^2}}}& {-{{\frac{1}{{C_v}}} {R}}} {{{ \tilde{v}} _k}}& {\frac{1}{{C_v}}} {R}& \frac{2}{3}& 0\\ {\frac{1}{\bar{\rho}}}{\left({-{k}}\right)}& 0& 0& \frac{1}{\bar{\rho}}& 0\\ {\frac{1}{\bar{\rho}}}{\left({-{\omega}}\right)}& 0& 0& 0& \frac{1}{\bar{\rho}}\end{array}\right]}} {{\left[\begin{array}{ccccc} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{{ n} _j}}& 0& 0& 0\\ {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^k}} {{{ n} _a}}& {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^k}} {{{ n} _j}}} + {{{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{{{ δ} ^k} _j}}}& { n} ^k& 0& 0\\ {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}} + {{{\frac{1}{2}}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^b}} {{{ \tilde{v}} _b}} {{{ n} _a}}}& {{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _j}} {{\frac{1}{R}}}} + {{{{P^*}}} \cdot {{{ n} _j}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{k}} {{{ n} _j}} {{\frac{1}{R}}}} + {{{\frac{1}{2}}} {{\bar{\rho}}} \cdot {{{ \tilde{v}} _a}} {{{ \tilde{v}} ^a}} {{{ n} _j}}}& {{{{C_v}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}} + {{{{ \tilde{v}} ^a}} {{{ n} _a}}}& {{-2}} \cdot {{\frac{1}{3}}} {{{C_v}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{R}}}& 0\\ {{k}} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{k}} {{{ n} _j}}& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& 0\\ {{\omega}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{\omega}} \cdot {{{ n} _j}}& 0& 0& {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ n} _a}}\end{array}\right]}}}$
${{{{ A} ^I} _J} + {{{{\tilde{v}_n}}} \cdot {{{{ δ} ^I} _J}}}} = {\left[\begin{array}{ccccc} {{{ \tilde{v}} ^a}} {{{ n} _a}}& {{\bar{\rho}}} \cdot {{{ n} _j}}& 0& 0& 0\\ {{{-1}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^i}} {{{ n} _a}} {{\frac{1}{\bar{\rho}}}}} + {{{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^k}} {{{ n} _a}} {{{{ δ} ^i} _k}} {{\frac{1}{\bar{\rho}}}}}& {{{-1}} {{{ \tilde{v}} ^i}} {{{ n} _j}}} + {{{{ \tilde{v}} ^k}} {{{ n} _j}} {{{{ δ} ^i} _k}}} + {{{{ \tilde{v}} ^a}} {{{ n} _a}} {{{{ δ} ^i} _k}} {{{{ δ} ^k} _j}}}& {{{ n} ^k}} {{{{ δ} ^i} _k}} {{\frac{1}{\bar{\rho}}}}& 0& 0\\ {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{R}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{{C_v}}}}} + {{{\frac{1}{2}}} {{R}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^b}} {{{ \tilde{v}} _b}} {{{ n} _a}} {{\frac{1}{{C_v}}}}} + {{{-1}} {{R}} {{{ \tilde{v}} ^a}} {{{ \tilde{v}} ^k}} {{{ \tilde{v}} _k}} {{{ n} _a}} {{\frac{1}{{C_v}}}}}& {{{{P^*}}} \cdot {{{ n} _j}}} + {{{{P^*}}} \cdot {{R}} {{{ n} _j}} {{\frac{1}{{C_v}}}}} + {{{\frac{1}{2}}} {{{(\tilde{v})^2}}} \cdot {{R}} {{\bar{\rho}}} \cdot {{{ n} _j}} {{\frac{1}{{C_v}}}}} + {{{\frac{1}{2}}} {{R}} {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ \tilde{v}} _a}} {{{ n} _j}} {{\frac{1}{{C_v}}}}} + {{{-1}} {{R}} {{\bar{\rho}}} \cdot {{{ \tilde{v}} _k}} {{{ \tilde{v}} ^k}} {{{ n} _j}} {{\frac{1}{{C_v}}}}} + {{{-1}} {{R}} {{\bar{\rho}}} \cdot {{{ \tilde{v}} ^a}} {{{ \tilde{v}} _k}} {{{ n} _a}} {{{{ δ} ^k} _j}} {{\frac{1}{{C_v}}}}}& {{{{ \tilde{v}} ^a}} {{{ n} _a}}} + {{{R}} {{{ \tilde{v}} ^a}} {{{ n} _a}} {{\frac{1}{{C_v}}}}} + {{{-1}} {{R}} {{{ \tilde{v}} _k}} {{{ n} ^k}} {{\frac{1}{{C_v}}}}}& 0& 0\\ 0& 0& 0& {{{ \tilde{v}} ^a}} {{{ n} _a}}& 0\\ 0& 0& 0& 0& {{{ \tilde{v}} ^a}} {{{ n} _a}}\end{array}\right]}$
${{{{ A} ^I} _J} + {{{{\tilde{v}_n}}} \cdot {{{{ δ} ^I} _J}}}} = {\left[\begin{array}{ccccc} {\tilde{v}_n}& {{\bar{\rho}}} \cdot {{{ n} _j}}& 0& 0& 0\\ 0& {{{\tilde{v}_n}}} \cdot {{{{ δ} ^i} _j}}& {{{ n} ^i}} {{\frac{1}{\bar{\rho}}}}& 0& 0\\ 0& {{{{P^*}}} \cdot {{{ n} _j}}} + {{{{P^*}}} \cdot {{R}} {{{ n} _j}} {{\frac{1}{{C_v}}}}} + {{{-1}} {{R}} {{\bar{\rho}}} \cdot {{{\tilde{v}_n}}} \cdot {{{ \tilde{v}} _j}} {{\frac{1}{{C_v}}}}}& {\tilde{v}_n}& 0& 0\\ 0& 0& 0& {\tilde{v}_n}& 0\\ 0& 0& 0& 0& {\tilde{v}_n}\end{array}\right]}$
${{{ A} ^I} _J} = {\left[\begin{array}{ccccc} 0& {{\bar{\rho}}} \cdot {{{ n} _j}}& 0& 0& 0\\ 0& 0& {\frac{1}{\bar{\rho}}} {{ n} ^i}& 0& 0\\ 0& {\frac{1}{{C_v}}}{\left({{{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _j}}} + {{{{P^*}}} \cdot {{R}} {{{ n} _j}}}{-{{{R}} {{\bar{\rho}}} \cdot {{{\tilde{v}_n}}} \cdot {{{ \tilde{v}} _j}}}}}\right)}& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]}$

${{c_s}} = {\frac{\sqrt{{{{{P^*}}} \cdot {{\gamma}} \cdot {{{|n|}^{2}}}}{-{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{{\tilde{v}_n}}^{2}}}}}}}{\sqrt{\bar{\rho}}}}$
${{P^*}} = {\frac{{{\bar{\rho}}} \cdot {{\left({{{{c_s}}^{2}} + {{{{\gamma_1}}} \cdot {{{{\tilde{v}_n}}^{2}}}}}\right)}}}{{{\gamma}} \cdot {{{|n|}^{2}}}}}$
Acoustic matrix, expanded:
${{{ A} ^I} _J} = {\left[\begin{array}{ccccccc} 0& {{\bar{\rho}}} \cdot {{{ n} _x}}& {{\bar{\rho}}} \cdot {{{ n} _y}}& {{\bar{\rho}}} \cdot {{{ n} _z}}& 0& 0& 0\\ 0& 0& 0& 0& {\frac{1}{\bar{\rho}}} {{ n} ^x}& 0& 0\\ 0& 0& 0& 0& {\frac{1}{\bar{\rho}}} {{ n} ^y}& 0& 0\\ 0& 0& 0& 0& {\frac{1}{\bar{\rho}}} {{ n} ^z}& 0& 0\\ 0& {\frac{1}{{C_v}}}{\left({{{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _x}}} + {{{{P^*}}} \cdot {{R}} {{{ n} _x}}}{-{{{R}} {{\bar{\rho}}} \cdot {{{\tilde{v}_n}}} \cdot {{{ \tilde{v}} _x}}}}}\right)}& {\frac{1}{{C_v}}}{\left({{{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _y}}} + {{{{P^*}}} \cdot {{R}} {{{ n} _y}}}{-{{{R}} {{\bar{\rho}}} \cdot {{{\tilde{v}_n}}} \cdot {{{ \tilde{v}} _y}}}}}\right)}& {\frac{1}{{C_v}}}{\left({{{{{C_v}}} \cdot {{{P^*}}} \cdot {{{ n} _z}}} + {{{{P^*}}} \cdot {{R}} {{{ n} _z}}}{-{{{R}} {{\bar{\rho}}} \cdot {{{\tilde{v}_n}}} \cdot {{{ \tilde{v}} _z}}}}}\right)}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]}$

...in just the x-axis (using ${{ n} _x} = {1}$, ${{ n} _y} = {0}$, ${{ n} _z} = {0}$ )
${{{ A} ^I} _J} = {\left[\begin{array}{ccccccc} 0& \bar{\rho}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {\frac{1}{\bar{\rho}}} {{ n} ^x}& 0& 0\\ 0& 0& 0& 0& {\frac{1}{\bar{\rho}}} {{ n} ^y}& 0& 0\\ 0& 0& 0& 0& {\frac{1}{\bar{\rho}}} {{ n} ^z}& 0& 0\\ 0& {\frac{1}{{C_v}}}{\left({{{{{C_v}}} \cdot {{{P^*}}}} + {{{{P^*}}} \cdot {{R}}}{-{{{R}} {{\bar{\rho}}} \cdot {{{\tilde{v}_n}}} \cdot {{{ \tilde{v}} _x}}}}}\right)}& -{{\frac{1}{{C_v}}} {{{R}} {{\bar{\rho}}} \cdot {{{\tilde{v}_n}}} \cdot {{{ \tilde{v}} _y}}}}& -{{\frac{1}{{C_v}}} {{{R}} {{\bar{\rho}}} \cdot {{{\tilde{v}_n}}} \cdot {{{ \tilde{v}} _z}}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]}$

...with a Cartesian metric (using ${{ n} ^x} = {1}$, ${{ n} ^y} = {0}$, ${{ n} ^z} = {0}$ )
${{{ A} ^I} _J} = {\left[\begin{array}{ccccccc} 0& \bar{\rho}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\bar{\rho}}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& {\frac{1}{{C_v}}}{\left({{{{{C_v}}} \cdot {{{P^*}}}} + {{{{P^*}}} \cdot {{R}}}{-{{{R}} {{\bar{\rho}}} \cdot {{{ \tilde{v}} _x}} {{{ \tilde{v}} _x}}}}}\right)}& -{{\frac{1}{{C_v}}} {{{R}} {{\bar{\rho}}} \cdot {{{ \tilde{v}} _y}} {{{ \tilde{v}} _x}}}}& -{{\frac{1}{{C_v}}} {{{R}} {{\bar{\rho}}} \cdot {{{ \tilde{v}} _z}} {{{ \tilde{v}} _x}}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]}$

speed of sound in Cartesian x-axis:
${{c_s}} = {\frac{\sqrt{{{P^*}} + {{{{P^*}}} \cdot {{{\gamma_1}}}}{-{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{{ \tilde{v}} _x}^{2}}}}}}}{\sqrt{\bar{\rho}}}}$
using ${R} = {{{C_p}}{-{{C_v}}}}$ , ${{C_p}} = {{{{C_v}}} \cdot {{\gamma}}}$ , ${{\gamma_1}} = {{\gamma}{-{1}}}$
${{{ A} ^I} _J} = {\left[\begin{array}{ccccccc} 0& \bar{\rho}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\bar{\rho}}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& {{{{P^*}}} \cdot {{\gamma}}}{-{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{{ \tilde{v}} _x}^{2}}}}}& -{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _x}} {{{ \tilde{v}} _y}}}& -{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _x}} {{{ \tilde{v}} _z}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]}$

Acoustic matrix eigen-decomposition:
${{{ A} ^I} _J} = {{{{{ {(R_A)}} ^I} _M}} {{{{ {(\Lambda_A)}} ^M} _N}} {{{{ {(L_A)}} ^N} _J}}}$
A charpoly: ${-{{\frac{1}{\bar{\rho}}} {{{{λ}^{5}}} {{\left({{-{{{{P^*}}} \cdot {{\gamma}}}} + {{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{{ \tilde{v}} _x}^{2}}}} + {{{\bar{\rho}}} \cdot {{{λ}^{2}}}}}\right)}}}}} = {0}$
${R} = {\left[\begin{array}{ccccccc} 0& 0& 0& 1& 0& \frac{1}{{{c_s}}^{2}}& \frac{1}{{{c_s}}^{2}}\\ 0& 0& 0& 0& \frac{1}{\bar{\rho}}& \frac{1}{{{{c_s}}} \cdot {{\bar{\rho}}}}& \frac{-1}{{{{c_s}}} \cdot {{\bar{\rho}}}}\\ 0& 0& \frac{-{{ \tilde{v}} _z}}{{ \tilde{v}} _x}& 0& \frac{{{c_s}}^{2}}{{{{\gamma_1}}} \cdot {{\bar{\rho}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}}}& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 1\\ 0& 1& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\end{array}\right]}$
${\Lambda} = {\left[\begin{array}{ccccccc} 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -{{c_s}}& 0\\ 0& 0& 0& 0& 0& 0& {c_s}\end{array}\right]}$
${L} = {\left[\begin{array}{ccccccc} 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 1& 0& 0& 0& -{\frac{1}{{{c_s}}^{2}}}& 0& 0\\ 0& 0& \frac{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}}}{{{c_s}}^{2}}& \frac{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}} {{{ \tilde{v}} _z}}}{{{{ \tilde{v}} _x}} {{{{c_s}}^{2}}}}& 0& 0& 0\\ 0& {\frac{1}{2}} {{{\bar{\rho}}} \cdot {{{c_s}}}}& -{\frac{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}}}{{{2}} {{{c_s}}}}}& -{\frac{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}} {{{ \tilde{v}} _z}}}{{{2}} {{{c_s}}} \cdot {{{ \tilde{v}} _x}}}}& \frac{1}{2}& 0& 0\\ 0& -{{\frac{1}{2}} {{{\bar{\rho}}} \cdot {{{c_s}}}}}& \frac{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}}}{{{2}} {{{c_s}}}}& \frac{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}} {{{ \tilde{v}} _z}}}{{{2}} {{{c_s}}} \cdot {{{ \tilde{v}} _x}}}& \frac{1}{2}& 0& 0\end{array}\right]}$
reconstructed:
${A} = {\left[\begin{array}{ccccccc} 0& -{\bar{\rho}}& \frac{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}}}{{{c_s}}^{2}}& \frac{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}} {{{ \tilde{v}} _z}}}{{{{ \tilde{v}} _x}} {{{{c_s}}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0& -{\frac{1}{\bar{\rho}}}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& -{{{\bar{\rho}}} \cdot {{{{c_s}}^{2}}}}& {{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}}& \frac{{{\bar{\rho}}} \cdot {{{\gamma_1}}} \cdot {{{ \tilde{v}} _u}} {{{ \tilde{v}} _v}} {{{ \tilde{v}} _z}}}{{ \tilde{v}} _x}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]}$