Riemannian geometry identity:
${\Gamma^i}_{ki} = \frac{1}{2 g} g_{,k} = ln(\sqrt{g})_{,k}$
$ = \frac{(\sqrt{g})_{,k}}{\sqrt{g}}$
Euler Fluid Equations
variables:
$\rho =$ density
$v_i =$ velocity
$m_i = \rho v_i = $ momentum
$P =$ pressure
$\gamma =$ heat capacity ratio
$\tilde\gamma = \gamma - 1$
$E_{kin} = \frac{1}{2} \rho v^k v_k$
$E_{int} = \frac{P}{\tilde\gamma}$
$E_{total} = E_{kin} + E_{int}$
$H_{total} = E_{total} + P = \frac{1}{2} \rho v^2 + \frac{\gamma}{\tilde\gamma} P$
Let $IJK...$ spanning $\{ \rho, v^i, E_{total} \}$
Conservative and primitive variables:
$W^I = \left[\matrix{ \rho \\ v_i \\ P }\right]$
$U^I = \left[\matrix{ \rho \\ \rho v_i \\ E_{total} }\right]$
$\frac{\partial U^I}{\partial W^J} = \downarrow I \overset{\rightarrow J}{\left[\matrix{
1 & 0 & 0 \\
v_i & \rho \delta_i^j & 0 \\
\frac{1}{2} v^2 & \rho v^j & 1/\tilde\gamma
}\right]}$
$\frac{\partial W^I}{\partial U^J} = \downarrow I \overset{\rightarrow J}{ \left[\matrix{
1 & 0 & 0 \\
-v_i / \rho & \delta_i^j / \rho & 0 \\
\frac{1}{2} \tilde\gamma v^2 & -\tilde\gamma v^j & \tilde\gamma
}\right]}$
Flux term:
$F^{Ij} = \left[\matrix{ \rho v^j \\ \rho v_i v^j + \delta_i^j P \\ v^j H_{total} }\right]$
Flux derivative wrt primitives:
$\frac{\partial F^{Ij}}{\partial W^M} = \downarrow I \overset{\rightarrow M}{ \left[\matrix{
v^j & \rho g^{jm} & 0 \\
v_i v^j & \rho (\delta_i^m v^j + v_i g^{jm}) & \delta_i^j \\
\frac{1}{2} v^j v^2 & g^{jm} H_{total} + \rho v^j v^m & \frac{\gamma}{\tilde{\gamma}} v^j
}\right] }$
Euler fluid equations, curved geometry:
$U^I + {F^{Ij}}_{;j} = 0$
$\rho_{,t} + (\rho v^j)_{;j} = 0$
$(\rho v_i)_{,t} + (\rho v_i v^j + \delta_i^j P)_{;j} = 0$
$E_{total,t} + (v^j H_{total})_{;j} = 0$
$\rho_{,t} + (\rho v^j)_{;j} = 0$
$(\rho v_i)_{,t} + (\rho v_i v^j)_{;j} + \delta_i^j P_{,j} = 0$
$E_{total,t} + (v^j H_{total})_{;j} = 0$
$\rho_{,t} + (\rho v^j)_{,j} + {\Gamma^j}_{kj} \rho v^k = 0$
$(\rho v_i)_{,t} + (\rho v_i v^j)_{,j} + \delta_i^j P_{,j} - {\Gamma^k}_{ij} \rho v_k v^j + {\Gamma^j}_{kj} \rho v_i v^k = 0$
$E_{total,t} + (v^j H_{total})_{,j} + {\Gamma^j}_{kj} v^k H_{total} = 0$
$\rho_{,t} + \rho_{,j} v^j + \rho v_{k,j} g^{kj} + \rho v_k {g^{kj}}_{,j} = -{\Gamma^j}_{kj} \rho v^k$
$(\rho v_i)_{,t} + \rho_{,j} v_i v^j + \rho v_{i,j} v^j + \rho v_i v_{k,j} g^{kj} + \rho v_i v_k {g^{kj}}_{,j} + \delta_i^j P_{,j} = {\Gamma^k}_{ij} \rho v_k v^j - {\Gamma^j}_{kj} \rho v_i v^k$
$E_{total,t} + v_{k,j} g^{kj} H_{total} + v_k {g^{kj}}_{,j} H_{total} + v^j (\frac{1}{2} \rho_{,j} v^2 + \rho v^k v_{k,j} + \rho v_k v_l {g^{kl}}_{,j} + \frac{\gamma}{\tilde\gamma} P_{,j}) = -{\Gamma^j}_{kj} v^k H_{total}$
$\left[\matrix{
\rho \\ \rho v_i \\ E_{total}
}\right]_{,t}
+ \left[\matrix{
v^j &
\rho g^{jm} &
0 \\
v_i v^j &
\rho ( v^j \delta_i^m + v_i g^{jm} ) &
\delta_i^j \\
\frac{1}{2} v^j v^2 &
g^{jm} H_{total} + \rho v^j v^m &
\frac{\gamma}{\tilde\gamma} v^j
}\right]
\left[\matrix{
\rho \\ v_m \\ P
}\right]_{,j} = \left[\matrix{
-\rho v_k {g^{kj}}_{,j} - {\Gamma^j}_{kj} \rho v^k \\
-\rho v_i v_k {g^{kj}}_{,j} + {\Gamma^k}_{ij} \rho v_k v^j - {\Gamma^j}_{kj} \rho v_i v^k \\
-v_k {g^{kj}}_{,j} H_{total} - \rho v^j v_k v_l {g^{kl}}_{,j} - {\Gamma^j}_{kj} v^k H_{total}
}\right]$
$\left[\matrix{
\rho \\ \rho v_i \\ E_{total}
}\right]_{,t}
+ \left[\matrix{
v^j &
\rho g^{jm} &
0 \\
v_i v^j &
\rho ( v^j \delta_i^m + v_i g^{jm} ) &
\delta_i^j \\
\frac{1}{2} v^j v^2 &
g^{jm} H_{total} + \rho v^j v^m &
\frac{\gamma}{\tilde\gamma} v^j
}\right]
\left[\matrix{
\rho \\ v_m \\ P
}\right]_{,j} = \left[\matrix{
\rho v^m g_{mn,j} g^{nj} - {\Gamma^j}_{kj} \rho v^k \\
\rho v_i v^m g_{mn,j} g^{nj} + {\Gamma^k}_{ij} \rho v_k v^j - {\Gamma^j}_{kj} \rho v_i v^k \\
v_k g^{km} g_{mn,j} g^{nj} H_{total} + \rho v^j v^m v^n g_{mn,j} - {\Gamma^j}_{kj} v^k H_{total}
}\right]$
$\left[\matrix{
\rho \\ \rho v_i \\ E_{total}
}\right]_{,t}
+ \left[\matrix{
v^j &
\rho g^{jm} &
0 \\
v_i v^j &
\rho ( v^j \delta_i^m + v_i g^{jm} ) &
\delta_i^j \\
\frac{1}{2} v^j v^2 &
g^{jm} H_{total} + \rho v^j v^m &
\frac{\gamma}{\tilde\gamma} v^j
}\right]
\left[\matrix{
\rho \\ v_m \\ P
}\right]_{,j} = \left[\matrix{
2 \rho v_k {\Gamma^k}_{mn} g^{mn} - {\Gamma^j}_{kj} \rho v^k \\
2 \rho v_i v_k {\Gamma^k}_{mn} g^{mn} + {\Gamma^k}_{ij} \rho v_k v^j - {\Gamma^j}_{kj} \rho v_i v^k \\
2 v_k {\Gamma^k}_{mn} g^{mn} H_{total} + 2 \rho v_k {\Gamma^k}_{mn} v^m v^n - {\Gamma^j}_{kj} v^k H_{total}
}\right]$
This only seems like it gets worse, because the indexes of the velocity and the derivative no longer oppose,
so the metric has to be introduced at every derivative (unless the derivatives are supposed to be contravariant).
Likewise, whereas the metric was introduced next to the $P$ in the conatravariant velocity form, it isn't there but it now is everywhere else.
And all fluid implementations always work around both of these by simply choosing a anholonomic normalized form such that the metric is identity ...
and instead reintroduce the missing information as commutation coefficients.