relations
${Z} = {{E} + {{{\frac{1}{{{{{2}} {{\rho}}}} {{\mu}}}}} {{({{{{{B_x}}^{2}} + {{{B_y}}^{2}}} + {{{B_z}}^{2}}})}}}}$
${P} = {{p} + {{{\frac{1}{{{2}} {{\mu}}}}} {{({{{{{B_x}}^{2}} + {{{B_y}}^{2}}} + {{{B_z}}^{2}}})}}}}$
${p} = {{{{{({{\gamma} - {1}})}} {{\rho}}}} {{({{E} - {{{\frac{1}{2}}} {{({{{{{v_x}}^{2}} + {{{v_y}}^{2}}} + {{{v_z}}^{2}}})}}}})}}}$
${{c}^{2}} = {{\frac{1}{\rho}}{({{{\gamma}} \cdot {{p}}})}}$

continuity
${{{\partial \rho}\over{\partial t}} + {{{{ \partial\over{\partial x}}\left({{{\rho}} \cdot {{{v_x}}}}\right)} + {{ \partial\over{\partial y}}\left({{{\rho}} \cdot {{{v_y}}}}\right)}} + {{ \partial\over{\partial z}}\left({{{\rho}} \cdot {{{v_z}}}}\right)}}} = {0}$

momentum
${{{{ \partial\over{\partial t}}\left({{{\rho}} \cdot {{{v_x}}}}\right)} + {{{{ \partial\over{\partial x}}\left({{{{{{\rho}} \cdot {{{v_x}}}}} {{{v_x}}}} - {{{{{\frac{1}{\mu}}} {{{B_x}}}}} {{{B_x}}}}}\right)} + {{ \partial\over{\partial y}}\left({{{{{{\rho}} \cdot {{{v_x}}}}} {{{v_y}}}} - {{{{{\frac{1}{\mu}}} {{{B_x}}}}} {{{B_y}}}}}\right)}} + {{ \partial\over{\partial z}}\left({{{{{{\rho}} \cdot {{{v_x}}}}} {{{v_z}}}} - {{{{{\frac{1}{\mu}}} {{{B_x}}}}} {{{B_z}}}}}\right)}}} + {{\partial P}\over{\partial x}}} = {{{{{\frac{-1}{\mu}}} {{{B_x}}}}} {{({{{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}}} + {{\partial {B_z}}\over{\partial z}}})}}}$
${{{{ \partial\over{\partial t}}\left({{{\rho}} \cdot {{{v_y}}}}\right)} + {{{{ \partial\over{\partial x}}\left({{{{{{\rho}} \cdot {{{v_y}}}}} {{{v_x}}}} - {{{{{\frac{1}{\mu}}} {{{B_y}}}}} {{{B_x}}}}}\right)} + {{ \partial\over{\partial y}}\left({{{{{{\rho}} \cdot {{{v_y}}}}} {{{v_y}}}} - {{{{{\frac{1}{\mu}}} {{{B_y}}}}} {{{B_y}}}}}\right)}} + {{ \partial\over{\partial z}}\left({{{{{{\rho}} \cdot {{{v_y}}}}} {{{v_z}}}} - {{{{{\frac{1}{\mu}}} {{{B_y}}}}} {{{B_z}}}}}\right)}}} + {{\partial P}\over{\partial y}}} = {{{{{\frac{-1}{\mu}}} {{{B_y}}}}} {{({{{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}}} + {{\partial {B_z}}\over{\partial z}}})}}}$
${{{{ \partial\over{\partial t}}\left({{{\rho}} \cdot {{{v_z}}}}\right)} + {{{{ \partial\over{\partial x}}\left({{{{{{\rho}} \cdot {{{v_z}}}}} {{{v_x}}}} - {{{{{\frac{1}{\mu}}} {{{B_z}}}}} {{{B_x}}}}}\right)} + {{ \partial\over{\partial y}}\left({{{{{{\rho}} \cdot {{{v_z}}}}} {{{v_y}}}} - {{{{{\frac{1}{\mu}}} {{{B_z}}}}} {{{B_y}}}}}\right)}} + {{ \partial\over{\partial z}}\left({{{{{{\rho}} \cdot {{{v_z}}}}} {{{v_z}}}} - {{{{{\frac{1}{\mu}}} {{{B_z}}}}} {{{B_z}}}}}\right)}}} + {{\partial P}\over{\partial z}}} = {{{{{\frac{-1}{\mu}}} {{{B_z}}}}} {{({{{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}}} + {{\partial {B_z}}\over{\partial z}}})}}}$

magnetic field
${{{\partial {B_x}}\over{\partial t}} + {{{{ \partial\over{\partial x}}\left({{{{{v_x}}} \cdot {{{B_x}}}} - {{{{v_x}}} \cdot {{{B_x}}}}}\right)} + {{ \partial\over{\partial y}}\left({{{{{v_y}}} \cdot {{{B_x}}}} - {{{{v_x}}} \cdot {{{B_y}}}}}\right)}} + {{ \partial\over{\partial z}}\left({{{{{v_z}}} \cdot {{{B_x}}}} - {{{{v_x}}} \cdot {{{B_z}}}}}\right)}}} = { {-{{v_x}}} {{({{{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}}} + {{\partial {B_z}}\over{\partial z}}})}}}$
${{{\partial {B_y}}\over{\partial t}} + {{{{ \partial\over{\partial x}}\left({{{{{v_x}}} \cdot {{{B_y}}}} - {{{{v_y}}} \cdot {{{B_x}}}}}\right)} + {{ \partial\over{\partial y}}\left({{{{{v_y}}} \cdot {{{B_y}}}} - {{{{v_y}}} \cdot {{{B_y}}}}}\right)}} + {{ \partial\over{\partial z}}\left({{{{{v_z}}} \cdot {{{B_y}}}} - {{{{v_y}}} \cdot {{{B_z}}}}}\right)}}} = { {-{{v_y}}} {{({{{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}}} + {{\partial {B_z}}\over{\partial z}}})}}}$
${{{\partial {B_z}}\over{\partial t}} + {{{{ \partial\over{\partial x}}\left({{{{{v_x}}} \cdot {{{B_z}}}} - {{{{v_z}}} \cdot {{{B_x}}}}}\right)} + {{ \partial\over{\partial y}}\left({{{{{v_y}}} \cdot {{{B_z}}}} - {{{{v_z}}} \cdot {{{B_y}}}}}\right)}} + {{ \partial\over{\partial z}}\left({{{{{v_z}}} \cdot {{{B_z}}}} - {{{{v_z}}} \cdot {{{B_z}}}}}\right)}}} = { {-{{v_z}}} {{({{{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}}} + {{\partial {B_z}}\over{\partial z}}})}}}$

energy total
${{{ \partial\over{\partial t}}\left({{{\rho}} \cdot {{Z}}}\right)} + {{{{{{({{{{\rho}} \cdot {{Z}}} + {p}})}} {{{v_x}}}} - {{{{{\frac{1}{\mu}}} {{({{{{{{v_x}}} \cdot {{{B_x}}}} + {{{{v_y}}} \cdot {{{B_y}}}}} + {{{{v_z}}} \cdot {{{B_z}}}}})}}}} {{{B_x}}}}} + {{{{({{{{\rho}} \cdot {{Z}}} + {p}})}} {{{v_y}}}} - {{{{{\frac{1}{\mu}}} {{({{{{{{v_x}}} \cdot {{{B_x}}}} + {{{{v_y}}} \cdot {{{B_y}}}}} + {{{{v_z}}} \cdot {{{B_z}}}}})}}}} {{{B_y}}}}}} + {{{{({{{{\rho}} \cdot {{Z}}} + {p}})}} {{{v_z}}}} - {{{{{\frac{1}{\mu}}} {{({{{{{{v_x}}} \cdot {{{B_x}}}} + {{{{v_y}}} \cdot {{{B_y}}}}} + {{{{v_z}}} \cdot {{{B_z}}}}})}}}} {{{B_z}}}}}}} = {{{{{\frac{-1}{\mu}}} {{({{{{{{v_x}}} \cdot {{{B_x}}}} + {{{{v_y}}} \cdot {{{B_y}}}}} + {{{{v_z}}} \cdot {{{B_z}}}}})}}}} {{({{{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}}} + {{\partial {B_z}}\over{\partial z}}})}}}$

all
${{{\partial \rho}\over{\partial t}} + {{{{\partial \rho}\over{\partial x}}} {{{v_x}}}} + {{{\rho}} \cdot {{{\partial {v_x}}\over{\partial x}}}} + {{{{\partial \rho}\over{\partial y}}} {{{v_y}}}} + {{{\rho}} \cdot {{{\partial {v_y}}\over{\partial y}}}} + {{{{\partial \rho}\over{\partial z}}} {{{v_z}}}} + {{{\rho}} \cdot {{{\partial {v_z}}\over{\partial z}}}}} = {0}$
${{\frac{1}{\mu}}{({-{({{{{2}} {{{B_x}}} \cdot {{{\partial {B_x}}\over{\partial x}}}} + {{{{B_x}}} \cdot {{{\partial {B_y}}\over{\partial y}}}} + {{{{B_x}}} \cdot {{{\partial {B_z}}\over{\partial z}}}} + {{{{B_y}}} \cdot {{{\partial {B_x}}\over{\partial y}}}} + {{{{{{{{{{{{{{{B_z}}} \cdot {{{\partial {B_x}}\over{\partial z}}}} - {{{\mu}} \cdot {{{\partial P}\over{\partial x}}}}} - {{{\mu}} \cdot {{\rho}} \cdot {{{\partial {v_x}}\over{\partial t}}}}} - {{{\mu}} \cdot {{{v_x}}} \cdot {{{\partial \rho}\over{\partial t}}}}} - {{{\mu}} \cdot {{{{v_x}}^{2}}} {{{\partial \rho}\over{\partial x}}}}} - {{{2}} {{\mu}} \cdot {{{v_x}}} \cdot {{\rho}} \cdot {{{\partial {v_x}}\over{\partial x}}}}} - {{{\mu}} \cdot {{{v_x}}} \cdot {{\rho}} \cdot {{{\partial {v_y}}\over{\partial y}}}}} - {{{\mu}} \cdot {{{v_x}}} \cdot {{\rho}} \cdot {{{\partial {v_z}}\over{\partial z}}}}} - {{{\mu}} \cdot {{{v_x}}} \cdot {{{v_y}}} \cdot {{{\partial \rho}\over{\partial y}}}}} - {{{\mu}} \cdot {{{v_x}}} \cdot {{{v_z}}} \cdot {{{\partial \rho}\over{\partial z}}}}} - {{{\mu}} \cdot {{{v_y}}} \cdot {{\rho}} \cdot {{{\partial {v_x}}\over{\partial y}}}}} - {{{\mu}} \cdot {{{v_z}}} \cdot {{\rho}} \cdot {{{\partial {v_x}}\over{\partial z}}}}}})}})}} = {{\frac{1}{\mu}}{({-{{{{B_x}}} \cdot {{({{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}} + {{\partial {B_z}}\over{\partial z}}})}}}})}}$
${{\frac{1}{\mu}}{({-{({{{{{B_x}}} \cdot {{{\partial {B_y}}\over{\partial x}}}} + {{{{B_y}}} \cdot {{{\partial {B_x}}\over{\partial x}}}} + {{{2}} {{{B_y}}} \cdot {{{\partial {B_y}}\over{\partial y}}}} + {{{{B_y}}} \cdot {{{\partial {B_z}}\over{\partial z}}}} + {{{{{{{{{{{{{{{B_z}}} \cdot {{{\partial {B_y}}\over{\partial z}}}} - {{{\mu}} \cdot {{{\partial P}\over{\partial y}}}}} - {{{\mu}} \cdot {{\rho}} \cdot {{{\partial {v_y}}\over{\partial t}}}}} - {{{\mu}} \cdot {{{v_x}}} \cdot {{\rho}} \cdot {{{\partial {v_y}}\over{\partial x}}}}} - {{{\mu}} \cdot {{{v_x}}} \cdot {{{v_y}}} \cdot {{{\partial \rho}\over{\partial x}}}}} - {{{\mu}} \cdot {{{v_y}}} \cdot {{{\partial \rho}\over{\partial t}}}}} - {{{\mu}} \cdot {{{{v_y}}^{2}}} {{{\partial \rho}\over{\partial y}}}}} - {{{\mu}} \cdot {{{v_y}}} \cdot {{\rho}} \cdot {{{\partial {v_x}}\over{\partial x}}}}} - {{{2}} {{\mu}} \cdot {{{v_y}}} \cdot {{\rho}} \cdot {{{\partial {v_y}}\over{\partial y}}}}} - {{{\mu}} \cdot {{{v_y}}} \cdot {{\rho}} \cdot {{{\partial {v_z}}\over{\partial z}}}}} - {{{\mu}} \cdot {{{v_y}}} \cdot {{{v_z}}} \cdot {{{\partial \rho}\over{\partial z}}}}} - {{{\mu}} \cdot {{{v_z}}} \cdot {{\rho}} \cdot {{{\partial {v_y}}\over{\partial z}}}}}})}})}} = {{\frac{1}{\mu}}{({-{{{{B_y}}} \cdot {{({{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}} + {{\partial {B_z}}\over{\partial z}}})}}}})}}$
${{\frac{1}{\mu}}{({-{({{{{{B_x}}} \cdot {{{\partial {B_z}}\over{\partial x}}}} + {{{{B_y}}} \cdot {{{\partial {B_z}}\over{\partial y}}}} + {{{{B_z}}} \cdot {{{\partial {B_x}}\over{\partial x}}}} + {{{{B_z}}} \cdot {{{\partial {B_y}}\over{\partial y}}}} + {{{{{{{{{{{{{{2}} {{{B_z}}} \cdot {{{\partial {B_z}}\over{\partial z}}}} - {{{\mu}} \cdot {{{\partial P}\over{\partial z}}}}} - {{{\mu}} \cdot {{\rho}} \cdot {{{\partial {v_z}}\over{\partial t}}}}} - {{{\mu}} \cdot {{{v_x}}} \cdot {{\rho}} \cdot {{{\partial {v_z}}\over{\partial x}}}}} - {{{\mu}} \cdot {{{v_x}}} \cdot {{{v_z}}} \cdot {{{\partial \rho}\over{\partial x}}}}} - {{{\mu}} \cdot {{{v_y}}} \cdot {{\rho}} \cdot {{{\partial {v_z}}\over{\partial y}}}}} - {{{\mu}} \cdot {{{v_y}}} \cdot {{{v_z}}} \cdot {{{\partial \rho}\over{\partial y}}}}} - {{{\mu}} \cdot {{{v_z}}} \cdot {{{\partial \rho}\over{\partial t}}}}} - {{{\mu}} \cdot {{{{v_z}}^{2}}} {{{\partial \rho}\over{\partial z}}}}} - {{{\mu}} \cdot {{{v_z}}} \cdot {{\rho}} \cdot {{{\partial {v_x}}\over{\partial x}}}}} - {{{\mu}} \cdot {{{v_z}}} \cdot {{\rho}} \cdot {{{\partial {v_y}}\over{\partial y}}}}} - {{{2}} {{\mu}} \cdot {{{v_z}}} \cdot {{\rho}} \cdot {{{\partial {v_z}}\over{\partial z}}}}}})}})}} = {{\frac{1}{\mu}}{({-{{{{B_z}}} \cdot {{({{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}} + {{\partial {B_z}}\over{\partial z}}})}}}})}}$
${{{\partial {B_x}}\over{\partial t}} + {{{{\partial {v_y}}\over{\partial y}}} {{{B_x}}}} + {{{{{{v_y}}} \cdot {{{\partial {B_x}}\over{\partial y}}}} - {{{{\partial {v_x}}\over{\partial y}}} {{{B_y}}}}} - {{{{v_x}}} \cdot {{{\partial {B_y}}\over{\partial y}}}}} + {{{{\partial {v_z}}\over{\partial z}}} {{{B_x}}}} + {{{{{{v_z}}} \cdot {{{\partial {B_x}}\over{\partial z}}}} - {{{{\partial {v_x}}\over{\partial z}}} {{{B_z}}}}} - {{{{v_x}}} \cdot {{{\partial {B_z}}\over{\partial z}}}}}} = {-{{{{v_x}}} \cdot {{({{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}} + {{\partial {B_z}}\over{\partial z}}})}}}}$
${{{\partial {B_y}}\over{\partial t}} + {{{{\partial {v_x}}\over{\partial x}}} {{{B_y}}}} + {{{{{{v_x}}} \cdot {{{\partial {B_y}}\over{\partial x}}}} - {{{{\partial {v_y}}\over{\partial x}}} {{{B_x}}}}} - {{{{v_y}}} \cdot {{{\partial {B_x}}\over{\partial x}}}}} + {{{{\partial {v_z}}\over{\partial z}}} {{{B_y}}}} + {{{{{{v_z}}} \cdot {{{\partial {B_y}}\over{\partial z}}}} - {{{{\partial {v_y}}\over{\partial z}}} {{{B_z}}}}} - {{{{v_y}}} \cdot {{{\partial {B_z}}\over{\partial z}}}}}} = {-{{{{v_y}}} \cdot {{({{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}} + {{\partial {B_z}}\over{\partial z}}})}}}}$
${{{\partial {B_z}}\over{\partial t}} + {{{{\partial {v_x}}\over{\partial x}}} {{{B_z}}}} + {{{{{{v_x}}} \cdot {{{\partial {B_z}}\over{\partial x}}}} - {{{{\partial {v_z}}\over{\partial x}}} {{{B_x}}}}} - {{{{v_z}}} \cdot {{{\partial {B_x}}\over{\partial x}}}}} + {{{{\partial {v_y}}\over{\partial y}}} {{{B_z}}}} + {{{{{{v_y}}} \cdot {{{\partial {B_z}}\over{\partial y}}}} - {{{{\partial {v_z}}\over{\partial y}}} {{{B_y}}}}} - {{{{v_z}}} \cdot {{{\partial {B_y}}\over{\partial y}}}}}} = {-{{{{v_z}}} \cdot {{({{{\partial {B_x}}\over{\partial x}} + {{\partial {B_y}}\over{\partial y}} + {{\partial {B_z}}\over{\partial z}}})}}}}$
${{\frac{1}{\mu}}{({{{{{\partial \rho}\over{\partial t}}} {{Z}} {{\mu}}} + {{{\rho}} \cdot {{{\partial Z}\over{\partial t}}} {{\mu}}} + {{{\rho}} \cdot {{Z}} {{{v_z}}} \cdot {{\mu}}} + {{{{{{p}} {{{v_z}}} \cdot {{\mu}}} - {{{{v_x}}} \cdot {{{B_x}}} \cdot {{{B_z}}}}} - {{{{v_y}}} \cdot {{{B_y}}} \cdot {{{B_z}}}}} - {{{{v_z}}} \cdot {{{{B_z}}^{2}}}}} + {{{\rho}} \cdot {{Z}} {{{v_y}}} \cdot {{\mu}}} + {{{{{{p}} {{{v_y}}} \cdot {{\mu}}} - {{{{v_x}}} \cdot {{{B_x}}} \cdot {{{B_y}}}}} - {{{{v_y}}} \cdot {{{{B_y}}^{2}}}}} - {{{{v_z}}} \cdot {{{B_z}}} \cdot {{{B_y}}}}} + {{{\rho}} \cdot {{Z}} {{{v_x}}} \cdot {{\mu}}} + {{{{{{p}} {{{v_x}}} \cdot {{\mu}}} - {{{{v_x}}} \cdot {{{{B_x}}^{2}}}}} - {{{{v_y}}} \cdot {{{B_y}}} \cdot {{{B_x}}}}} - {{{{v_z}}} \cdot {{{B_z}}} \cdot {{{B_x}}}}}})}} = {{\frac{1}{\mu}}{({-{({{{{{B_x}}} \cdot {{{v_x}}} \cdot {{{\partial {B_x}}\over{\partial x}}}} + {{{{B_x}}} \cdot {{{v_x}}} \cdot {{{\partial {B_y}}\over{\partial y}}}} + {{{{B_x}}} \cdot {{{v_x}}} \cdot {{{\partial {B_z}}\over{\partial z}}}} + {{{{B_y}}} \cdot {{{v_y}}} \cdot {{{\partial {B_x}}\over{\partial x}}}} + {{{{B_y}}} \cdot {{{v_y}}} \cdot {{{\partial {B_y}}\over{\partial y}}}} + {{{{B_y}}} \cdot {{{v_y}}} \cdot {{{\partial {B_z}}\over{\partial z}}}} + {{{{B_z}}} \cdot {{{v_z}}} \cdot {{{\partial {B_x}}\over{\partial x}}}} + {{{{B_z}}} \cdot {{{v_z}}} \cdot {{{\partial {B_y}}\over{\partial y}}}} + {{{{B_z}}} \cdot {{{v_z}}} \cdot {{{\partial {B_z}}\over{\partial z}}}}})}})}}$