units of measurement:
$m$ = meter
$s$ = second
$kg$ = kilogram
$C$ = coulomb

constants:
${\mu_0}$ = magnetic permeability, in units of $\frac{{{kg}} \cdot {{m}}}{{C}^{2}}$
$\gamma$ = heat capacity ratio, in units of $[1]$
${\tilde{\gamma}} = {{\gamma}{-{1}}}$

variables:
${{ g} _i} _j$ = metric tensor, in units of $[1]$
${ n} ^i$ = flux surface normal, in units of $[1]$
$\rho$ = density, in units of $\frac{kg}{{m}^{3}}$
${ v} ^i$ = velocity, in units of ${\frac{1}{s}} {m}$
${{(v)^2}} = {{{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}$ = velocity norm squared, in units of $\frac{{m}^{2}}{{s}^{2}}$
${{ m} ^i} = {{{\rho}} \cdot {{{ v} ^i}}}$ = momentum, in units of $\frac{kg}{{{{m}^{2}}} {{s}}}$
${ B} ^i$ = magnetic field, in units of $\frac{kg}{{{C}} {{s}}}$
${{(B)^2}} = {{{{ B} ^k}} {{{ B} ^l}} {{{{ g} _k} _l}}}$ = magnetic field norm squared, in units of $\frac{{kg}^{2}}{{\left({{{C}} {{s}}}\right)}^{2}}$
$P$ = pressure from fluid, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{P_{mag}}} = {{\frac{1}{{\mu_0}}} {{{\frac{1}{2}}} {{{(B)^2}}}}}$ = pressure due to magnetic field, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{P_{total with mag}}} = {{P} + {{P_{mag}}}}$ = total pressure, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{P_{total with mag}}} = {{P} + {{\frac{1}{{\mu_0}}} {{{\frac{1}{2}}} {{{(B)^2}}}}}}$
${{e_{kin}}} = {{{\frac{1}{2}}} {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}$ = specific kinetic energy, in units of $\frac{{m}^{2}}{{s}^{2}}$
${{e_{int}}} = {\frac{P}{{{\tilde{\gamma}}} \cdot {{\rho}}}}$ = specific internal energy, in units of $\frac{{m}^{2}}{{s}^{2}}$
${{E_{hydro}}} = {{{\rho}} \cdot {{\left({{{e_{int}}} + {{e_{kin}}}}\right)}}}$ = densitized hydro energy, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{E_{hydro}}} = {{{{P}} {{\frac{1}{\tilde{\gamma}}}}} + {{{\frac{1}{2}}} {{{(v)^2}}} \cdot {{\rho}}}}$
${{E_{mag}}} = {{\frac{1}{{\mu_0}}} {{{\frac{1}{2}}} {{{(B)^2}}}}}$ = energy from magnetic field, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{E_{total with mag}}} = {{{{\rho}} \cdot {{\left({{{e_{int}}} + {{e_{kin}}}}\right)}}} + {{E_{mag}}}}$ = total energy, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{E_{total with mag}}} = {{{{\frac{1}{2}}} {{{(B)^2}}} \cdot {{\frac{1}{{\mu_0}}}}} + {{{P}} {{\frac{1}{\tilde{\gamma}}}}} + {{{\frac{1}{2}}} {{{(v)^2}}} \cdot {{\rho}}}}$
${{H_{total with mag}}} = {{{E_{total with mag}}} + {{P_{total with mag}}}}$ = total enthalpy, in units of $\frac{kg}{{{m}} {{{s}^{2}}}}$
${{H_{total with mag}}} = {{{{{(B)^2}}} \cdot {{\frac{1}{{\mu_0}}}}} + {{{P}} {{\frac{1}{\tilde{\gamma}}}}} + {P} + {{{\frac{1}{2}}} {{{(v)^2}}} \cdot {{\rho}}}}$
${{c_s}} = {\sqrt{{\frac{1}{\rho}} {{{\gamma}} \cdot {{P}}}}}$ = speed of sound in units of ${\frac{1}{s}} {m}$
Conservative and primitive variables:
${{ W} ^I} = {\left[\begin{array}{c} \rho\\ { v} ^i\\ P\\ { B} ^i\end{array}\right]}$
${{ U} ^I} = {\left[\begin{array}{c} \rho\\ { m} ^i\\ {E_{total with mag}}\\ { B} ^i\end{array}\right]}$
Partial of conservative quantities wrt primitives:
${\frac{\partial { U} ^I}{\partial { W} ^J}} = {\left[\begin{array}{cccc} \frac{\partial \rho}{\partial \rho}& \frac{\partial \rho}{\partial { v} ^j}& \frac{\partial \rho}{\partial P}& \frac{\partial \rho}{\partial { B} ^j}\\ \frac{\partial { m} ^i}{\partial \rho}& \frac{\partial { m} ^i}{\partial { v} ^j}& \frac{\partial { m} ^i}{\partial P}& \frac{\partial { m} ^i}{\partial { B} ^j}\\ \frac{\partial {E_{total with mag}}}{\partial \rho}& \frac{\partial {E_{total with mag}}}{\partial { v} ^j}& \frac{\partial {E_{total with mag}}}{\partial P}& \frac{\partial {E_{total with mag}}}{\partial { B} ^j}\\ \frac{\partial { B} ^i}{\partial \rho}& \frac{\partial { B} ^i}{\partial { v} ^j}& \frac{\partial { B} ^i}{\partial P}& \frac{\partial { B} ^i}{\partial { B} ^j}\end{array}\right]}$
${\frac{\partial { U} ^I}{\partial { W} ^J}} = {\left[\begin{array}{cccc} \frac{\partial \rho}{\partial \rho}& \frac{\partial \rho}{\partial { v} ^j}& \frac{\partial \rho}{\partial P}& \frac{\partial \rho}{\partial { B} ^j}\\ {\frac{\partial}{\partial \rho}}\left({{{\rho}} \cdot {{{ v} ^i}}}\right)& {\frac{\partial}{\partial { v} ^j}}\left({{{\rho}} \cdot {{{ v} ^i}}}\right)& {\frac{\partial}{\partial P}}\left({{{\rho}} \cdot {{{ v} ^i}}}\right)& {\frac{\partial}{\partial { B} ^j}}\left({{{\rho}} \cdot {{{ v} ^i}}}\right)\\ {\frac{\partial}{\partial \rho}}\left({{{{\frac{1}{2}}} {{\frac{1}{{\mu_0}}}} {{{ B} ^k}} {{{ B} ^l}} {{{{ g} _k} _l}}} + {{{P}} {{\frac{1}{\tilde{\gamma}}}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}}\right)& {\frac{\partial}{\partial { v} ^j}}\left({{{{\frac{1}{2}}} {{\frac{1}{{\mu_0}}}} {{{ B} ^k}} {{{ B} ^l}} {{{{ g} _k} _l}}} + {{{P}} {{\frac{1}{\tilde{\gamma}}}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}}\right)& {\frac{\partial}{\partial P}}\left({{{{\frac{1}{2}}} {{\frac{1}{{\mu_0}}}} {{{ B} ^k}} {{{ B} ^l}} {{{{ g} _k} _l}}} + {{{P}} {{\frac{1}{\tilde{\gamma}}}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}}\right)& {\frac{\partial}{\partial { B} ^j}}\left({{{{\frac{1}{2}}} {{\frac{1}{{\mu_0}}}} {{{ B} ^k}} {{{ B} ^l}} {{{{ g} _k} _l}}} + {{{P}} {{\frac{1}{\tilde{\gamma}}}}} + {{{\frac{1}{2}}} {{\rho}} \cdot {{{ v} ^k}} {{{ v} ^l}} {{{{ g} _k} _l}}}}\right)\\ \frac{\partial { B} ^i}{\partial \rho}& \frac{\partial { B} ^i}{\partial { v} ^j}& \frac{\partial { B} ^i}{\partial P}& \frac{\partial { B} ^i}{\partial { B} ^j}\end{array}\right]}$
${\frac{\partial { U} ^I}{\partial { W} ^J}} = {\left[\begin{array}{cccc} 1& 0& 0& 0\\ { v} ^i& {{\rho}} \cdot {{{{ δ} ^i} _j}}& 0& 0\\ {{\frac{1}{2}}} {{{(v)^2}}}& {{\rho}} \cdot {{{ v} _j}}& \frac{1}{\tilde{\gamma}}& {{{ B} _j}} {{\frac{1}{{\mu_0}}}}\\ 0& 0& 0& {{ δ} ^i} _j\end{array}\right]}$