Riemannian geometry identity:
${\Gamma^i}_{ki} = \frac{1}{2 g} g_{,k} = ln(\sqrt{g})_{,k} = \frac{(\sqrt{g})_{,k}}{\sqrt{g}}$
Where $g_{ij}$ is the metric associated with ${\Gamma^i}_{jk}$
Euler Fluid Equations
variables:
$n^i =$ flux surface normal, units of [1].
$\rho =$ density, in units of $\frac{kg}{m^3}$
$v^i =$ velocity, in units of $\frac{m}{s}$
$m^i = \rho v^i = $ momentum, in units of $\frac{kg}{m^2 \cdot s}$
$\gamma =$ heat capacity ratio, in units of $[1]$
$\tilde\gamma = \gamma - 1$
$P = \tilde\gamma \rho e_{int} = \tilde\gamma E_{int} = \tilde\gamma (E_{total} - E_{kin}) = $ pressure, in units of $\frac{kg}{m \cdot s^2}$
$g_{ij} = $ metric tensor, in units of $[1]$.
$v^2 = v^i g_{ij} v^j$
$e_{kin} = \frac{1}{2} v^2 = $ specific kinetic energy, in units of $\frac{m^2}{s^2}$
$E_{kin} = \rho e_{kin} = \frac{1}{2} \rho v^2$ densitized kinetic energy, in units of $\frac{kg}{m \cdot s^2}$
$e_{int} = \frac{P}{\tilde\gamma \rho} =$ specific internal energy, in units of $\frac{m^2}{s^2}$
$E_{int} = \rho e_{int} = \frac{P}{\tilde\gamma} = $ densitized internal energy, in units of $\frac{kg}{m \cdot s^2}$
$e_{total} = e_{kin} + e_{int} =$ specific total energy, in units of $\frac{kg}{m \cdot s^2}$
$E_{total} = E_{kin} + E_{int} = $ densitized total energy, in units of $\frac{kg}{m \cdot s^2}$
$H_{total} = E_{total} + P = \frac{1}{2} \rho v^2 + \frac{\gamma}{\tilde\gamma} P$
$ = E_{total} + \tilde\gamma E_{total} + \tilde\gamma E_{kin} = \gamma E_{total} + \tilde\gamma E_{kin}$, total enthalpy, in units of $\frac{kg}{m \cdot s^2}$
Let capital indexes $IJK...$ span all conserved quantity variables $\{ \rho, v^i, E_{total} \}$
Conservation form:
$\int ( \partial_t U^I + \nabla_n F^I ) dV = 0$
Conservative and primitive variables:
$W^I = \left[\begin{matrix} \rho \\ v^i \\ P \end{matrix}\right]$
in units of $\left[\begin{matrix} \frac{kg}{m^3} \\ \frac{m}{s} \\ \frac{kg}{m \cdot s^2} \end{matrix}\right]$
$U^I = \left[\begin{matrix} \rho \\ \rho v^i \\ E_{total} \end{matrix}\right]$
in units of $\left[\begin{matrix} \frac{kg}{m^3} \\ \frac{kg}{m^2 \cdot s} \\ \frac{kg}{m \cdot s^2} \end{matrix}\right]$
Partial of conserved quantities wrt primitivies:
$\frac{\partial U^I}{\partial W^J} = \downarrow I \overset{\rightarrow J}{\left[\begin{matrix}
1 & 0 & 0 \\
v^i & \rho \delta^i_j & 0 \\
\frac{1}{2} v^2 & \rho v_j & \frac{1}{\tilde\gamma}
\end{matrix}\right]}$
$\frac{\partial W^I}{\partial U^J} = \downarrow I \overset{\rightarrow J}{\left[\begin{matrix}
1 & 0 & 0 \\
-\frac{1}{\rho} v^i & \frac{1}{\rho} \delta^i_j & 0 \\
\frac{1}{2} \tilde\gamma v^2 & -\tilde\gamma v_j & \tilde\gamma
\end{matrix}\right]}$
Flux:
$F^I (n)$
$ = \left[\begin{matrix} \rho v^l n_l \\ \rho v^i v^l n_l + n^i P \\ v^l n_l H_{total} \end{matrix}\right]$
$ = \left[\begin{matrix}
m^j n_j \\
\frac{1}{\rho} m^i m^j n_j + n^i \tilde\gamma (E_{total} - \frac{1}{2} \frac{1}{\rho} m^2) \\
\frac{1}{\rho} m^j n_j (\gamma E_{total} - \tilde\gamma \frac{1}{2} \frac{1}{\rho} m^2)
\end{matrix}\right]$
in units of $\left[\begin{matrix} \frac{kg}{m^2 \cdot s} \\ \frac{kg}{m \cdot s^2} \\ \frac{kg}{s^3} \end{matrix}\right]$
Flux derivative with respect to primitive variables:
$\frac{\partial F^I}{\partial W^J} = \downarrow I \overset{\rightarrow J}{
\left[\begin{matrix}
v^l n_l & \rho n_j & 0 \\
v^i v^l n_l & \rho v^l n_l \delta^i_j + \rho v^i n_j & n^i \\
\frac{1}{2} v^2 v^l n_l & n_j H_{total} + v^l n_l \rho v_j & v^l n_l \frac{\gamma}{\tilde\gamma}
\end{matrix}\right]
}$
$\frac{\partial F^I}{\partial U^J} = \frac{\partial F^I}{\partial W^K} \cdot \frac{\partial W^K}{\partial U^J}$:
$ =
\left[\begin{matrix}
v^l n_l & \rho n_k & 0 \\
v^i v^l n_l & \rho v^l n_l \delta^i_k + \rho v^i n_k & n^i \\
\frac{1}{2} v^2 v^l n_l & n_k H_{total} + v^l n_l \rho v_k & v^l n_l \frac{\gamma}{\tilde\gamma}
\end{matrix}\right]
\left[\begin{matrix}
1 & 0 & 0 \\
-\frac{1}{\rho} v^k & \frac{1}{\rho} \delta^k_j & 0 \\
\frac{1}{2} \tilde\gamma v^2 & -\tilde\gamma v_j & \tilde\gamma
\end{matrix}\right]
$
$ =
\left[\begin{matrix}
0 & n_j & 0 \\
- v^i v^k n_k + \frac{1}{2} \tilde\gamma v^2 n^i &
v^l n_l \delta^i_j + (2 - \gamma) n^i v_j &
\tilde\gamma n^i \\
-v^k n_k (h_{total} - \frac{1}{2} v^2 \tilde\gamma) &
n_j h_{total} - v^l n_l \tilde\gamma v_j &
v^l n_l \gamma
\end{matrix}\right]
$
TODO beneath this line replace F as a function of $g^{ij}$ with F as a function fo $n_i$
Flux partial derivative:
${F^{Ij}}_{,j}$
$= \left[\begin{matrix}
m^j \\
\frac{1}{\rho} m^i m^j + g^{ij} \tilde\gamma (E_{total} - \frac{1}{\rho} \frac{1}{2} m^2) \\
\frac{1}{\rho} m^j (\gamma E_{total} - \tilde\gamma \frac{1}{\rho} \frac{1}{2} m^2)
\end{matrix}\right]_{,j}$
$= \left[\begin{matrix}
(m^j)_{,j} \\
(\frac{1}{\rho} m^i m^j + g^{ij} \tilde\gamma (E_{total} - \frac{1}{\rho} \frac{1}{2} m^2))_{,j} \\
(\frac{1}{\rho} m^j (\gamma E_{total} - \tilde\gamma \frac{1}{\rho} \frac{1}{2} m^2))_{,j}
\end{matrix}\right]$
$ = \left[\begin{matrix}
{m^j}_{,j} \\
-\frac{1}{\rho^2} \rho_{,j} m^i m^j
+ \frac{1}{\rho} {m^i}_{,j} m^j
+ \frac{1}{\rho} m^i {m^j}_{,j}
+ {g^{ij}}_{,j} P
+ g^{ij} \tilde\gamma (
E_{total,j}
+ \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
- \frac{1}{\rho} {m^k}_{,j} m_k
- \frac{1}{2} \frac{1}{\rho} m^k m^l g_{kl,j}
)
\\
(-\frac{1}{\rho^2} \rho_{,j} m^j + \frac{1}{\rho} {m^j}_{,j}) H_{total}
+ \frac{1}{\rho} m^j (
\gamma E_{total,j}
+ \tilde\gamma \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
- \tilde\gamma \frac{1}{\rho} {m^k}_{,j} m_k
- \tilde\gamma \frac{1}{2} \frac{1}{\rho} m^k m^l g_{kl,j}
)
\end{matrix}\right]$
$ = \left[\begin{matrix}
0 & \delta^j_k & 0 \\
- \frac{1}{\rho^2} m^i m^j
+ g^{ij} \tilde\gamma \frac{1}{2} \frac{1}{\rho^2} m^2
&
\frac{1}{\rho} (
m^j \delta^i_k
+ m^i \delta^j_k
- \tilde\gamma m_k g^{ij}
)
&
g^{ij} \tilde\gamma
\\
\frac{1}{\rho^2} (
- H_{total} m^j
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^j m^2
)
&
\frac{1}{\rho} (
H_{total} \delta^j_k
- \tilde\gamma \frac{1}{\rho} m^j m_k
)
&
\frac{1}{\rho} \gamma m^j
\end{matrix}\right] \left[\begin{matrix}
\rho \\
m^k \\
E_{total}
\end{matrix}\right]_{,j} + \left[\begin{matrix}
0
\\
{g^{ij}}_{,j} P
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^k m^l g_{kl,j} g^{ij}
\\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l g_{kl,j}
\end{matrix}\right]$
Flux covariant derivative:
${F^{Ij}}_{;j}$
$ = \left[\begin{matrix}
{m^j}_{;j} \\
-\frac{1}{\rho^2} \rho_{,j} m^i m^j
+ \frac{1}{\rho} {m^i}_{;j} m^j
+ \frac{1}{\rho} m^i {m^j}_{;j}
+ {g^{ij}}_{;j} P
+ g^{ij} \tilde\gamma (
E_{total,j}
+ \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
- \frac{1}{\rho} {m^k}_{;j} m_k
- \frac{1}{2} \frac{1}{\rho} m^k m^l g_{kl;j}
)
\\
(-\frac{1}{\rho^2} \rho_{,j} m^j + \frac{1}{\rho} {m^j}_{;j}) H_{total}
+ \frac{1}{\rho} m^j (
\gamma E_{total,j}
+ \tilde\gamma \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
- \tilde\gamma \frac{1}{\rho} {m^k}_{;j} m_k
- \tilde\gamma \frac{1}{2} \frac{1}{\rho} m^k m^l g_{kl;j}
)
\end{matrix}\right]$
$ = \left[\begin{matrix}
{m^j}_{;j} \\
-\frac{1}{\rho^2} \rho_{,j} m^i m^j
+ \frac{1}{\rho} {m^i}_{;j} m^j
+ \frac{1}{\rho} m^i {m^j}_{;j}
+ g^{ij} \tilde\gamma (
E_{total,j}
+ \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
- \frac{1}{\rho} {m^k}_{;j} m_k
)
\\
(-\frac{1}{\rho^2} \rho_{,j} m^j + \frac{1}{\rho} {m^j}_{;j}) H_{total}
+ \frac{1}{\rho} m^j (
\gamma E_{total,j}
+ \tilde\gamma \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
- \tilde\gamma \frac{1}{\rho} {m^k}_{;j} m_k
)
\end{matrix}\right]$
$= \left[\begin{matrix}
{m^j}_{,j} + {\Gamma^j}_{jk} m^k \\
-\frac{1}{\rho^2} \rho_{,j} m^i m^j
+ \frac{1}{\rho} ({m^i}_{,j} + {\Gamma^i}_{jk} m^k) m^j
+ \frac{1}{\rho} m^i ({m^j}_{,j} + {\Gamma^j}_{jk} m^k)
+ g^{ij} \tilde\gamma (
E_{total,j}
+ \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
- \frac{1}{\rho} m_k {m^k}_{,j}
)
\\
(-\frac{1}{\rho^2} \rho_{,j} m^j + \frac{1}{\rho} ({m^j}_{,j} + {\Gamma^j}_{jk} m^k)) H_{total}
+ \frac{1}{\rho} m^j (
\gamma E_{total,j}
+ \tilde\gamma \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
- \tilde\gamma \frac{1}{\rho} m_k {m^k}_{,j}
)
\end{matrix}\right]$
Moving connections outside, and isolating the volume gradient times the conserved quantities:
$= \left[\begin{matrix}
{m^j}_{,j} \\
- \frac{1}{\rho^2} m^i m^j \rho_{,j}
+ \frac{1}{\rho} m^j {m^i}_{,j}
+ \frac{1}{\rho} m^i {m^j}_{,j}
+ g^{ij} \tilde\gamma (
E_{total,j}
+ \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
- \frac{1}{\rho} m_k {m^k}_{,j}
)
\\
- \frac{1}{\rho^2} m^j H_{total} \rho_{,j}
+ \frac{1}{\rho} m^j \tilde\gamma \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
+ \frac{1}{\rho} H_{total} {m^j}_{,j}
- \frac{1}{\rho} m^j \tilde\gamma \frac{1}{\rho} m_k {m^k}_{,j}
+ \frac{1}{\rho} m^j \gamma E_{total,j}
\end{matrix}\right]
+ \left[\begin{matrix}
{\Gamma^j}_{jk} m^k \\
\frac{1}{\rho} m^i {\Gamma^j}_{jk} m^k
+ {\Gamma^j}_{jk} g^{ik} P
\\
\frac{1}{\rho} {\Gamma^j}_{jk} m^k H_{total}
\end{matrix}\right]
+ \left[\begin{matrix}
0 \\
\frac{1}{\rho} {\Gamma^i}_{jk} m^k m^j
- {\Gamma^j}_{jk} g^{ik} P
\\
0
\end{matrix}\right]$
Introducing metric partials, to write the flux covariant derivative in terms of the flux partial derivative:
$= \left[\begin{matrix}
{m^j}_{,j} \\
- \frac{1}{\rho^2} m^i m^j \rho_{,j}
+ \frac{1}{\rho} m^j {m^i}_{,j}
+ \frac{1}{\rho} m^i {m^j}_{,j}
+ {g^{ij}}_{,j} P
+ g^{ij} \tilde\gamma (
E_{total,j}
+ \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
- \frac{1}{\rho} m_k {m^k}_{,j}
- \frac{1}{2} \frac{1}{\rho} m^k m^l g_{kl,j}
)
\\
- \frac{1}{\rho^2} m^j H_{total} \rho_{,j}
+ \frac{1}{\rho} m^j \tilde\gamma \frac{1}{2} \frac{1}{\rho^2} \rho_{,j} m^2
+ \frac{1}{\rho} H_{total} {m^j}_{,j}
- \frac{1}{\rho} m^j \tilde\gamma \frac{1}{\rho} m_k {m^k}_{,j}
+ \frac{1}{\rho} m^j \gamma E_{total,j}
- \frac{1}{\rho} m^j \tilde\gamma \frac{1}{2} \frac{1}{\rho} m^k m^l g_{kl,j}
\end{matrix}\right]
+ \left[\begin{matrix}
{\Gamma^j}_{jk} m^k \\
\frac{1}{\rho} m^i {\Gamma^j}_{jk} m^k
+ {\Gamma^j}_{jk} g^{ik} P
\\
\frac{1}{\rho} {\Gamma^j}_{jk} m^k H_{total}
\end{matrix}\right]
+ \left[\begin{matrix}
0 \\
\frac{1}{\rho} {\Gamma^i}_{jk} m^k m^j
- {\Gamma^j}_{jk} g^{ik} P
- {g^{ij}}_{,j} P
+ \tilde\gamma \frac{1}{2} \frac{1}{\rho} m^k m^l g_{kl,j} g^{ij}
\\
\frac{1}{\rho} m^j \tilde\gamma \frac{1}{2} \frac{1}{\rho} m^k m^l g_{kl,j}
\end{matrix}\right]$
$= \frac{\sqrt{g}}{\sqrt{g}} {F^{Ij}}_{,j}
+ (
\frac{\sqrt{g}_{,j}}{\sqrt{g}}
+ {c_{kj}}^k
) F^{Ij}
+ \left[\begin{matrix}
0 \\
\frac{1}{\rho} {\Gamma^i}_{jk} m^k m^j
- {\Gamma^j}_{jk} g^{ik} P
+ ({\Gamma^i}_{jk} g^{jk} + {\Gamma^j}_{jk} g^{ki}) P
- \frac{1}{\rho} \tilde{\gamma} m^k m^l \Gamma_{kjl} g^{ij}
\\
\frac{1}{\rho^2} \tilde{\gamma} m^j m^k m^l \Gamma_{kjl}
\end{matrix}\right]$
$= \frac{\sqrt{g}}{\sqrt{g}} {F^{Ij}}_{,j}
+ \frac{\sqrt{g}_{,j}}{\sqrt{g}} F^{Ij}
+ ({\Gamma^k}_{kj} - {\Gamma^k}_{jk})
\cdot \left[\begin{matrix}
m^j \\
\frac{1}{\rho} m^i m^j + g^{ij} P \\
v^j H_{total}
\end{matrix}\right]
+ \left[\begin{matrix}
0 \\
{\Gamma^i}_{jk} (
\frac{1}{\rho} m^k m^j
+ g^{jk} P
)
- \frac{1}{\rho} \tilde{\gamma} m^k m^l \Gamma_{kjl} g^{ij}
\\
\frac{1}{\rho^2} \tilde{\gamma} m^j m^k m^l \Gamma_{jkl}
\end{matrix}\right]$
$= \frac{1}{\sqrt{g}} \left(
\sqrt{g} F^{Ij}
\right)_{,j}
+ \left[\begin{matrix}
{c_{kj}}^k m^j \\
{c_{kj}}^k (\frac{1}{\rho} m^i m^j + g^{ij} P)
+ {\Gamma^i}_{jk} (\frac{1}{\rho} m^k m^j + g^{jk} P)
- \frac{1}{\rho} \tilde{\gamma} m^k m^l \Gamma_{kjl} g^{ij}
\\
{c_{kj}}^k v^j H_{total}
+ \frac{1}{\rho^2} \tilde{\gamma} m^j m^k m^l \Gamma_{jkl}
\end{matrix}\right]$
Euler fluid equations, curved geometry:
${U^I}_{,t} + {F^{Ij}}_{;j} = 0$
${U^I}_{,t} + \frac{1}{\sqrt{g}} \left(
\sqrt{g} F^{Ij}
\right)_{,j}
= \left[\begin{matrix}
0 \\
- {\Gamma^i}_{jk} (\frac{1}{\rho} m^k m^j + g^{jk} P)
+ \frac{1}{\rho} \tilde{\gamma} m^k m^l \Gamma_{kjl} g^{ij}
\\
- \frac{1}{\rho^2} \tilde{\gamma} m^j m^k m^l \Gamma_{jkl}
\end{matrix}\right]
- {c_{kj}}^k F^{Ij}
$
TODO above this line replace F as a function of $g^{ij}$ with F as a function fo $n_i$
Maxima has a much harder time calculating the eigenvalues/vectors with those $\delta^{ij}$'s replaced with $g^{ij}$'s.
But I can get around that by using acoustic tensor to find the eigen-decomposition of the flux in curved coordinates:
TODO...?
Eigen decomposition of flux jacobian:
$\frac{\partial F^I}{\partial U^J} = R_F \Lambda_F L_F$
$\frac{\partial F}{\partial U} = \frac{\partial F}{\partial W} \frac{\partial W}{\partial U}$
$\frac{\partial W}{\partial U} \frac{\partial F}{\partial W} \frac{\partial W}{\partial U} \frac{\partial U}{\partial W} = \frac{\partial W}{\partial U} ( R_F \Lambda_F L_F ) \frac{\partial U}{\partial W}$
$\frac{\partial W}{\partial U} \frac{\partial F}{\partial W} = (\frac{\partial W}{\partial U} R_F) \Lambda_F (L_F \frac{\partial U}{\partial W})$
Let $A + I v_n = \frac{\partial W}{\partial U} \frac{\partial F}{\partial W}$, $\Lambda_A = \Lambda_F + I v_n$
$A + I v_n = (\frac{\partial W}{\partial U} R_F) (\Lambda_A + I v_n) (L_F \frac{\partial U}{\partial W})$
$A = (\frac{\partial W}{\partial U} R_F) \Lambda_A (L_F \frac{\partial U}{\partial W})$
Let $R_A = \frac{\partial W}{\partial U} R_F$, $L_A = L_F \frac{\partial U}{\partial W}$
$A = R_A \Lambda_A L_A$
$\frac{\partial W^I}{\partial U^K} \cdot \frac{\partial F^K}{\partial W^J}$
$= \downarrow N \overset{\rightarrow I}{\left[\begin{matrix}
1 & 0 & 0 \\
-\frac{1}{\rho} v^i & \frac{1}{\rho} \delta^i_k & 0 \\
\frac{1}{2} \tilde\gamma v^2 & -\tilde\gamma v_k & \tilde\gamma
\end{matrix}\right]} \cdot \downarrow K \overset{\rightarrow J}{\left[\begin{matrix}
v^l n_l & \rho n_j & 0 \\
v^k v^l n_l & \rho (v^l n_l \delta^k_j + v^k n_j) & n^k \\
\frac{1}{2} v^2 v^l n_l & n_j H_{total} + \rho v^l n_l v_j & \frac{\gamma}{\tilde{\gamma}} v^l n_l
\end{matrix}\right]}$
$= \downarrow N \overset{\rightarrow M}{\left[\begin{matrix}
v^l n_l & \rho n_j & 0 \\
0 & v^l n_l \delta^i_j & \frac{1}{\rho} n^i \\
0 & \gamma P n_j & v^l n_l
\end{matrix}\right]}$
$= {A^I}_J + v^l n_l \delta^I_J$
...for acoustic tensor:
${A^I}_J = \downarrow I \overset{\rightarrow J}{\left[\begin{matrix}
0 & \rho n_j & 0 \\
0 & 0 & \frac{1}{\rho} n^i \\
0 & \gamma P n_j & 0
\end{matrix}\right]}$
Acoustic tensor eigenvalues:
$\left|\begin{matrix}
-\lambda & \rho n_j & 0 \\
0 & -\lambda \delta^i_j & \frac{1}{\rho} n^i \\
0 & \gamma P n_j & -\lambda
\end{matrix}\right| = 0$
$\left|\begin{matrix}
-\lambda & \rho n_x & \rho n_y & \rho n_z & 0 \\
0 & -\lambda & 0 & 0 & \frac{1}{\rho} n^x \\
0 & 0 & -\lambda & 0 & \frac{1}{\rho} n^y \\
0 & 0 & 0 & -\lambda & \frac{1}{\rho} n^z \\
0 & n_x \gamma P & n_y \gamma P & n_z \gamma P & -\lambda
\end{matrix}\right| = 0$
$-\lambda \cdot \left|\begin{matrix}
-\lambda & 0 & 0 & \frac{1}{\rho} n^x \\
0 & -\lambda & 0 & \frac{1}{\rho} n^y \\
0 & 0 & -\lambda & \frac{1}{\rho} n^z \\
n_x \gamma P & n_y \gamma P & n_z \gamma P & -\lambda
\end{matrix}\right| = 0$
$-\lambda \cdot \left(
-\lambda \cdot \left|\begin{matrix}
-\lambda & 0 & \frac{1}{\rho} n^y \\
0 & -\lambda & \frac{1}{\rho} n^z \\
n_y \gamma P & n_z \gamma P & -\lambda
\end{matrix}\right|
- \frac{1}{\rho} n^x \cdot \left|\begin{matrix}
0 & -\lambda & 0 \\
0 & 0 & -\lambda \\
n_x \gamma P & n_y \gamma P & n_z \gamma P
\end{matrix}\right|
\right) = 0$
$-\lambda \cdot \left(
-\lambda \cdot \left(
-\lambda \left|\begin{matrix}
-\lambda & \frac{1}{\rho} n^z \\
n_z \gamma P & -\lambda
\end{matrix}\right|
+ \frac{1}{\rho} n^y \cdot \left|\begin{matrix}
0 & -\lambda \\
n_y \gamma P & n_z \gamma P
\end{matrix}\right|
\right)
- \frac{1}{\rho} n^x \cdot
n_x \gamma P \cdot
\left|\begin{matrix}
-\lambda & 0 \\
0 & -\lambda
\end{matrix}\right|
\right) = 0$
$-\lambda \cdot \left(
-\lambda \cdot \left(
-\lambda (
\lambda^2
- \frac{\gamma P}{\rho} n^z n_z
)
+ \lambda \frac{\gamma P}{\rho} n^y n_y
\right)
- \lambda^2 \frac{\gamma P}{\rho} n^x n_x
\right) = 0$
$-\lambda^3 \cdot \left(
\lambda^2
- \frac{\gamma P}{\rho} n^x n_x
- \frac{\gamma P}{\rho} n^y n_y
- \frac{\gamma P}{\rho} n^z n_z
\right) = 0$
$-\lambda^3 \cdot \left(\lambda^2 - \frac{\gamma P}{\rho} n^k n_k \right) = 0$
Assume $n^k n_k = 1$?
But what if it's not? What if $n_k = \partial_k$ and $|n_k| = 1$, then $|n^k| = |g^{kl}|$...
This is the case if we write the flux in terms of the coordinate by replacing $n_k$ with $\delta_k^l$ and $n^k$ with $g^{kl}$.
So I will instead say $n^k n_k = n^2$.
$-\lambda^3 \cdot \left(\lambda^2 - \frac{\gamma P}{\rho} n^2 \right) = 0$
$-\lambda^3
(\lambda + |n| \sqrt{\frac{\gamma P}{\rho}})
(\lambda - |n| \sqrt{\frac{\gamma P}{\rho}})
= 0$
So $\lambda = \{
-|n| \sqrt{\frac{\gamma P}{\rho}},
0, 0, 0,
|n| \sqrt{\frac{\gamma P}{\rho}}
\}$
$\lambda = \{
-|n| c_s,
0, 0, 0,
|n| c_s
\}$
Acoustic tensor right eigenvectors:
For $\lambda = 0$:
$\left[\begin{matrix}
0 & \rho n_x & \rho n_y & \rho n_z & 0 \\
0 & 0 & 0 & 0 & \frac{1}{\rho} n^x \\
0 & 0 & 0 & 0 & \frac{1}{\rho} n^y \\
0 & 0 & 0 & 0 & \frac{1}{\rho} n^z \\
0 & n_x \gamma P & n_y \gamma P & n_z \gamma P & 0
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
...for $n_x \ne 0$:
$\left[\begin{matrix}
0 & 1 & n_y / n_x & n_z / n_x & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$X^{v_x} + n_y / n_x X^{v_y} + n_z / n_x X^{v_z} = 0$
$X^P = 0$
Let $u_1 = X^\rho, u_2 = X^{v_y}, u_3 = X^{v_z}$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = \left[\begin{matrix}
1 & 0 & 0 \\
0 & -n_y / n_x & -n_z / n_x \\
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0 \\
\end{matrix}\right] \left[\begin{matrix}
u_1 \\ u_2 \\ u_3
\end{matrix}\right]$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = \left[\begin{matrix}
1 & 0 & 0 \\
0 & -n_y & -n_z \\
0 & n_x & 0 \\
0 & 0 & n_x \\
0 & 0 & 0 \\
\end{matrix}\right] \left[\begin{matrix}
u_1 \\ u_2 \\ u_3
\end{matrix}\right]$
...for $n_x = 0, n_y \ne 0$:
$\left[\begin{matrix}
0 & 0 & 1 & n_z / n_y & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$X^{v_y} + n_z / n_y X^{v_z} = 0$
$X^P = 0$
Let $u_1 = X^\rho, u_2 = X^{v_z}, u_3 = X^{v_x}$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = \left[\begin{matrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & -n_z / n_y & 0\\
0 & 1 & 0 \\
0 & 0 & 0 \\
\end{matrix}\right] \left[\begin{matrix}
u_1 \\ u_2 \\ u_3
\end{matrix}\right]$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = \left[\begin{matrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & -n_z & 0 \\
0 & n_y & 0 \\
0 & 0 & 0 \\
\end{matrix}\right] \left[\begin{matrix}
u_1 \\ u_2 \\ u_3
\end{matrix}\right]$
...for $n_x = 0, n_y = 0, n_z \ne 0$:
$\left[\begin{matrix}
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$X^{v_z} = 0$
$X^P = 0$
Let $u_1 = X^\rho, u_2 = X^{v_x}, u_3 = X^{v_y}$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = \left[\begin{matrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{matrix}\right] \left[\begin{matrix}
u_1 \\ u_2 \\ u_3
\end{matrix}\right]$
For $\lambda = 0$, and with orthonormal basis $\{n^j, (n_2)^j, (n_3)^j\}$, such that $|n_2|^2 = |n_3|^3 = |n|^2 = n^i n_i$ and $\epsilon_{ijk} n^i (n_2)^j (n_3)^k = 1, n^i (n_2)_i = 0, n^i (n_3)_i = 0$.
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = \left[\begin{matrix}
1 & 0 & 0 \\
0 & n_2^x & n_3^x \\
0 & n_2^y & n_3^y \\
0 & n_2^z & n_3^z \\
0 & 0 & 0 \\
\end{matrix}\right] \left[\begin{matrix}
u_1 \\ u_2 \\ u_3
\end{matrix}\right]$
For $\lambda = |n| \sqrt{\frac{\gamma P}{\rho}}$:
$\left[\begin{matrix}
-|n| \sqrt{\frac{\gamma P}{\rho}} & \rho n_x & \rho n_y & \rho n_z & 0 \\
0 & -|n| \sqrt{\frac{\gamma P}{\rho}} & 0 & 0 & \frac{1}{\rho} n^x \\
0 & 0 & -|n| \sqrt{\frac{\gamma P}{\rho}} & 0 & \frac{1}{\rho} n^y \\
0 & 0 & 0 & -|n| \sqrt{\frac{\gamma P}{\rho}} & \frac{1}{\rho} n^z \\
0 & n_x \gamma P & n_y \gamma P & n_z \gamma P & -|n| \sqrt{\frac{\gamma P}{\rho}}
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$\left[\begin{matrix}
-|n| \sqrt{\frac{\gamma P}{\rho^3}} & n_x & n_y & n_z & 0 \\
0 & 1 & 0 & 0 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x \\
0 & 0 & 1 & 0 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y \\
0 & 0 & 0 & 1 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z \\
0 & n_x & n_y & n_z & -|n| \sqrt{\frac{1}{\gamma P \rho}}
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$\left[\begin{matrix}
-|n| \sqrt{\frac{\gamma P}{\rho^3}} & 0 & 0 & 0 &
\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x n_x
+ \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y n_y
+ \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z n_z
\\
0 & 1 & 0 & 0 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x \\
0 & 0 & 1 & 0 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y \\
0 & 0 & 0 & 1 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z \\
0 & 0 & 0 & 0 & -|n| \sqrt{\frac{1}{\gamma P \rho}}
+ \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x n_x
+ \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y n_y
+ \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z n_z
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$\left[\begin{matrix}
-|n| \sqrt{\frac{\gamma P}{\rho^3}} & 0 & 0 & 0 & |n| \sqrt{{1}{\gamma P \rho}} \\
0 & 1 & 0 & 0 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x \\
0 & 0 & 1 & 0 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y \\
0 & 0 & 0 & 1 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z \\
0 & 0 & 0 & 0 & 0
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$\left[\begin{matrix}
1 & 0 & 0 & 0 & -\frac{\rho}{\gamma P} \\
0 & 1 & 0 & 0 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x \\
0 & 0 & 1 & 0 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y \\
0 & 0 & 0 & 1 & -\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z \\
0 & 0 & 0 & 0 & 0
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
Let $u = X^P$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = u \left[\begin{matrix}
\frac{\rho}{\gamma P} \\
\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x \\
\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y \\
\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z \\
1
\end{matrix}\right]$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = u \left[\begin{matrix}
\sqrt{\frac{\rho^3}{\gamma P}} \\
\frac{n^x}{|n|} \\
\frac{n^y}{|n|} \\
\frac{n^z}{|n|} \\
\sqrt{\gamma P \rho}
\end{matrix}\right]$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = u \left[\begin{matrix}
\frac{\rho}{c_s} \\
\frac{n^x}{|n|} \\
\frac{n^y}{|n|} \\
\frac{n^z}{|n|} \\
\rho c_s
\end{matrix}\right]$
For $\lambda = -|n| \sqrt{\frac{\gamma P}{\rho}}$:
$\left[\begin{matrix}
|n| \sqrt{\frac{\gamma P}{\rho}} & \rho n_x & \rho n_y & \rho n_z & 0 \\
0 & |n| \sqrt{\frac{\gamma P}{\rho}} & 0 & 0 & \frac{1}{\rho} n^x \\
0 & 0 & |n| \sqrt{\frac{\gamma P}{\rho}} & 0 & \frac{1}{\rho} n^y \\
0 & 0 & 0 & |n| \sqrt{\frac{\gamma P}{\rho}} & \frac{1}{\rho} n^z \\
0 & n_x \gamma P & n_y \gamma P & n_z \gamma P & |n| \sqrt{\frac{\gamma P}{\rho}}
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$\left[\begin{matrix}
|n| \sqrt{\frac{\gamma P}{\rho^3}} & n_x & n_y & n_z & 0 \\
0 & 1 & 0 & 0 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x \\
0 & 0 & 1 & 0 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y \\
0 & 0 & 0 & 1 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z \\
0 & n_x & n_y & n_z & |n| \sqrt{\frac{1}{\gamma P \rho}}
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$\left[\begin{matrix}
|n| \sqrt{\frac{\gamma P}{\rho^3}} & 0 & 0 & 0 &
-\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x n_x
-\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y n_y
-\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z n_z
\\
0 & 1 & 0 & 0 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x \\
0 & 0 & 1 & 0 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y \\
0 & 0 & 0 & 1 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z \\
0 & 0 & 0 & 0 & |n| \sqrt{\frac{1}{\gamma P \rho}}
-\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x n_x
-\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y n_y
-\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z n_z
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$\left[\begin{matrix}
|n| \sqrt{\frac{\gamma P}{\rho^3}} & 0 & 0 & 0 & -|n| \sqrt{\frac{1}{\gamma P \rho}} \\
0 & 1 & 0 & 0 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x \\
0 & 0 & 1 & 0 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y \\
0 & 0 & 0 & 1 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z \\
0 & 0 & 0 & 0 & 0
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
$\left[\begin{matrix}
1 & 0 & 0 & 0 & -\frac{\rho}{\gamma P} \\
0 & 1 & 0 & 0 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x \\
0 & 0 & 1 & 0 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y \\
0 & 0 & 0 & 1 & \frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z \\
0 & 0 & 0 & 0 & 0
\end{matrix}\right] \left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = 0$
Let $u = X^P$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = u \left[\begin{matrix}
\frac{\rho}{\gamma P} \\
-\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^x \\
-\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^y \\
-\frac{1}{|n|} \sqrt{\frac{1}{\gamma P \rho}} n^z \\
1
\end{matrix}\right]$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = u \left[\begin{matrix}
\sqrt{\frac{\rho^3}{\gamma P}} \\
-\frac{n^x}{|n|} \\
-\frac{n^y}{|n|} \\
-\frac{n^z}{|n|} \\
\sqrt{\gamma P \rho}
\end{matrix}\right]$
$\left[\begin{matrix}
X^\rho \\ X^{v_x} \\ X^{v_y} \\ X^{v_z} \\ X^P
\end{matrix}\right] = u \left[\begin{matrix}
\frac{\rho}{c_s} \\
-\frac{n^x}{|n|} \\
-\frac{n^y}{|n|} \\
-\frac{n^z}{|n|} \\
\rho c_s
\end{matrix}\right]$
Right eigenvector matrix from collected right eigenvectors:
$\left[\begin{matrix}
\frac{\rho}{c_s} & 1 & 0 & 0 & \frac{\rho}{c_s} \\
-\frac{n^x}{|n|} & 0 & n_2^x & n_3^x & \frac{n^x}{|n|} \\
-\frac{n^y}{|n|} & 0 & n_2^y & n_3^y & \frac{n^y}{|n|} \\
-\frac{n^z}{|n|} & 0 & n_2^z & n_3^z & \frac{n^z}{|n|} \\
\rho c_s & 0 & 0 & 0 & \rho c_s
\end{matrix}\right]$
Left eigenvector matrix from inverse of right eigenvector matrix:
$\left[\begin{matrix}
0 & -\frac{n_x}{2 |n|} & -\frac{n_y}{2 |n|} & -\frac{n_z}{2 |n|} & \frac{1}{2 \rho c_s} \\
1 & 0 & 0 & 0 & -\frac{1}{(c_s)^2} \\
0 & (n_2)_x & (n_2)_y & (n_2)_z & 0 \\
0 & (n_3)_x & (n_3)_y & (n_3)_z & 0 \\
0 & \frac{n_x}{2 |n|} & \frac{n_y}{2 |n|} & \frac{n_z}{2 |n|} & \frac{1}{2 \rho c_s}
\end{matrix}\right]$
Scale right eigenvector columns by $\{ \frac{c_s}{\rho}, 1, \frac{1}{\rho}, \frac{1}{\rho}, \frac{c_s}{\rho} \}$...
$V_A^R = \left[\begin{matrix}
1 & 1 & 0 & 0 & 1 \\
-\frac{c_s n^x}{\rho |n|} & 0 & \frac{1}{\rho} n_2^x & \frac{1}{\rho} n_3^x & \frac{c_s n^x}{\rho |n|} \\
-\frac{c_s n^y}{\rho |n|} & 0 & \frac{1}{\rho} n_2^y & \frac{1}{\rho} n_3^y & \frac{c_s n^y}{\rho |n|} \\
-\frac{c_s n^z}{\rho |n|} & 0 & \frac{1}{\rho} n_2^z & \frac{1}{\rho} n_3^z & \frac{c_s n^z}{\rho |n|} \\
(c_s)^2 & 0 & 0 & 0 & (c_s)^2
\end{matrix}\right]$
Scale left eigenvector rows by $\{ \frac{\rho}{c_s}, 1, \rho, \rho, \frac{\rho}{c_s} \}$...
$V_A^L = \left[\begin{matrix}
0 & -\frac{\rho n_x}{2 c_s |n|} & -\frac{\rho n_y}{2 c_s |n|} & -\frac{\rho n_z}{2 c_s |n|} & \frac{1}{2 (c_s)^2} \\
1 & 0 & 0 & 0 & -\frac{1}{(c_s)^2} \\
0 & \rho (n_2)_x & \rho (n_2)_y & \rho (n_2)_z & 0 \\
0 & \rho (n_3)_x & \rho (n_3)_y & \rho (n_3)_z & 0 \\
0 & \frac{\rho n_x}{2 c_s |n|} & \frac{\rho n_y}{2 c_s |n|} & \frac{\rho n_z}{2 c_s |n|} & \frac{1}{2 (c_s)^2}
\end{matrix}\right]$
Eigensystem of acoustic matrix:
${(\Lambda_A)^I}_K = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix}
-c_s |n| & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & c_s |n|
\end{matrix}\right]}$
${(V^R_A)^I}_J = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix}
1 & 1 & 0 & 0 & 1 \\
-\frac{c_s n^x}{\rho |n|} & 0 & \frac{1}{\rho} n_2^x & \frac{1}{\rho} n_3^x & \frac{c_s n^x}{\rho |n|} \\
-\frac{c_s n^y}{\rho |n|} & 0 & \frac{1}{\rho} n_2^y & \frac{1}{\rho} n_3^y & \frac{c_s n^y}{\rho |n|} \\
-\frac{c_s n^z}{\rho |n|} & 0 & \frac{1}{\rho} n_2^z & \frac{1}{\rho} n_3^z & \frac{c_s n^z}{\rho |n|} \\
(c_s)^2 & 0 & 0 & 0 & (c_s)^2
\end{matrix}\right]}$
${(V^L_A)^I}_J = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix}
0 & -\frac{\rho n_x}{2 c_s |n|} & -\frac{\rho n_y}{2 c_s |n|} & -\frac{\rho n_z}{2 c_s |n|} & \frac{1}{2 (c_s)^2} \\
1 & 0 & 0 & 0 & -\frac{1}{(c_s)^2} \\
0 & \rho (n_2)_x & \rho (n_2)_y & \rho (n_2)_z & 0 \\
0 & \rho (n_3)_x & \rho (n_3)_y & \rho (n_3)_z & 0 \\
0 & \frac{\rho n_x}{2 c_s |n|} & \frac{\rho n_y}{2 c_s |n|} & \frac{\rho n_z}{2 c_s |n|} & \frac{1}{2 (c_s)^2}
\end{matrix}\right]}$
Now $\frac{\partial F^I}{\partial U^J}$
$ = \frac{\partial F^I}{\partial W^K} \cdot \frac{\partial W^K}{\partial U^J}$
$ = \frac{\partial U}{\partial W} \frac{\partial W}{\partial U} \frac{\partial F}{\partial W} \frac{\partial W}{\partial U}$
$ = \frac{\partial U}{\partial W} (A + v^l n_l I) \frac{\partial W}{\partial U}$
The eigenvalues of $F$ are $\Lambda_A + v^l n_l I = \left[\begin{matrix}
v^l n_l - c_s |n| & 0 & 0 & 0 & 0 \\
0 & v^l n_l & 0 & 0 & 0 \\
0 & 0 & v^l n_l & 0 & 0 \\
0 & 0 & 0 & v^l n_l & 0 \\
0 & 0 & 0 & 0 & v^l n_l + c_s |n|
\end{matrix}\right]$
The right eigenvectors of $F$ are ${(V^R_F)^I}_J = \frac{\partial U^I}{\partial W^K} \cdot {(V^R_A)^K}_J$
$= \downarrow I \overset{\rightarrow K}{\left[\begin{matrix}
1 & 0 & 0 & 0 & 0 \\
v^x & \rho & 0 & 0 & 0 \\
v^y & 0 & \rho & 0 & 0 \\
v^z & 0 & 0 & \rho & 0 \\
\frac{1}{2} v^2 & \rho v_x & \rho v_y & \rho v_z & \frac{1}{\tilde\gamma}
\end{matrix}\right]} \cdot \downarrow K \overset{\rightarrow J}{ \left[\begin{matrix}
1 & 1 & 0 & 0 & 1 \\
-\frac{c_s n^x}{\rho |n|} & 0 & \frac{1}{\rho} n_2^x & \frac{1}{\rho} n_3^x & \frac{c_s n^x}{\rho |n|} \\
-\frac{c_s n^y}{\rho |n|} & 0 & \frac{1}{\rho} n_2^y & \frac{1}{\rho} n_3^y & \frac{c_s n^y}{\rho |n|} \\
-\frac{c_s n^z}{\rho |n|} & 0 & \frac{1}{\rho} n_2^z & \frac{1}{\rho} n_3^z & \frac{c_s n^z}{\rho |n|} \\
(c_s)^2 & 0 & 0 & 0 & (c_s)^2
\end{matrix}\right]}$
$ = \downarrow I \overset{\rightarrow J}{\left[\begin{matrix}
1 & 1 & 0 & 0 & 1 \\
v^x - c_s \frac{n^x}{|n|} & v^x & (n_2)^x & (n_3)^x & v^x + c_s \frac{n^x}{|n|} \\
v^y - c_s \frac{n^y}{|n|} & v^y & (n_2)^y & (n_3)^y & v^y + c_s \frac{n^y}{|n|} \\
v^z - c_s \frac{n^z}{|n|} & v^z & (n_2)^z & (n_3)^z & v^z + c_s \frac{n^z}{|n|} \\
\frac{v^2}{2} + \frac{1}{\tilde\gamma} (c_s)^2 - c_s v^k \frac{n_k}{|n|} & \frac{1}{2} v^2 & v^k (n_2)_k & v^k (n_3)_k & \frac{v^2}{2} + \frac{1}{\tilde\gamma} (c_s)^2 + c_s v^k \frac{n_k}{|n|}
\end{matrix}\right]}$
$ = \downarrow I \overset{\rightarrow J}{\left[\begin{matrix}
1 & 1 & 0 & 0 & 1 \\
v^x - c_s \frac{n^x}{|n|} & v^x & (n_2)^x & (n_3)^x & v^x + c_s \frac{n^x}{|n|} \\
v^y - c_s \frac{n^y}{|n|} & v^y & (n_2)^y & (n_3)^y & v^y + c_s \frac{n^y}{|n|} \\
v^z - c_s \frac{n^z}{|n|} & v^z & (n_2)^z & (n_3)^z & v^z + c_s \frac{n^z}{|n|} \\
h_{total} - c_s v^k \frac{n_k}{|n|} & \frac{1}{2} v^2 & v^k (n_2)_k & v^k (n_3)_k & h_{total} + c_s v^k \frac{n_k}{|n|}
\end{matrix}\right]}$
The left eigenvectors of $F$ are ${(V^L_F)^I}_J = {(V^L_A)^I}_K \cdot \frac{\partial W^K}{\partial U^J}$
$= \downarrow I \overset{\rightarrow K}{ \left[\begin{matrix}
0 & -\frac{\rho n_x}{2 c_s |n|} & -\frac{\rho n_y}{2 c_s |n|} & -\frac{\rho n_z}{2 c_s |n|} & \frac{1}{2 (c_s)^2} \\
1 & 0 & 0 & 0 & -\frac{1}{(c_s)^2} \\
0 & \rho (n_2)_x & \rho (n_2)_y & \rho (n_2)_z & 0 \\
0 & \rho (n_3)_x & \rho (n_3)_y & \rho (n_3)_z & 0 \\
0 & \frac{\rho n_x}{2 c_s |n|} & \frac{\rho n_y}{2 c_s |n|} & \frac{\rho n_z}{2 c_s |n|} & \frac{1}{2 (c_s)^2}
\end{matrix}\right]}
\cdot \downarrow K \overset{\rightarrow J}{ \left[\begin{matrix}
1 & 0 & 0 & 0 & 0 \\
-\frac{1}{\rho} v^x & \frac{1}{\rho} & 0 & 0 & 0 \\
-\frac{1}{\rho} v^y & 0 & \frac{1}{\rho} & 0 & 0 \\
-\frac{1}{\rho} v^z & 0 & 0 & \frac{1}{\rho} & 0 \\
\frac{1}{2} \tilde\gamma v^2 & -\tilde\gamma v_x & -\tilde\gamma v_y & -\tilde\gamma v_z & \tilde\gamma
\end{matrix}\right]}$
$= \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix}
\frac{1}{2 c_s} (\tilde\gamma \frac{v^2}{2 c_s} + v^k \frac{n_k}{|n|} ) &
\frac{1}{2 c_s} ( -\tilde\gamma \frac{1}{c_s} v_x - \frac{n_x}{|n|} ) &
\frac{1}{2 c_s} ( -\tilde\gamma \frac{1}{c_s} v_y - \frac{n_y}{|n|} ) &
\frac{1}{2 c_s} ( -\tilde\gamma \frac{1}{c_s} v_z - \frac{n_z}{|n|} ) &
\frac{\tilde\gamma}{2 (c_s)^2} \\
1 - \frac{\tilde\gamma v^2}{2 (c_s)^2} &
\frac{ \tilde\gamma v_x }{(c_s)^2} &
\frac{ \tilde\gamma v_y }{(c_s)^2} &
\frac{ \tilde\gamma v_z }{(c_s)^2} &
-\frac{ \tilde\gamma }{(c_s)^2} \\
-v^l (n_2)_l &
(n_2)_x &
(n_2)_y &
(n_2)_z &
0 \\
-v^l (n_3)_l &
(n_3)_x &
(n_3)_y &
(n_3)_z &
0 \\
\frac{1}{2 c_s} (\tilde\gamma \frac{v^2}{2 c_s} - v^k \frac{n_k}{|n|}) &
\frac{1}{2 c_s} (-\tilde\gamma \frac{1}{c_s} v_x + \frac{n_x}{|n|}) &
\frac{1}{2 c_s} (-\tilde\gamma \frac{1}{c_s} v_y + \frac{n_y}{|n|}) &
\frac{1}{2 c_s} (-\tilde\gamma \frac{1}{c_s} v_z + \frac{n_z}{|n|}) &
\frac{\tilde\gamma}{2 (c_s)^2} \\
\end{matrix}\right]}$