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{\newpage
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\samepage \begin{equation}U = (R,M_x,M_y,M_z,E)^{T} ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}F^x = (R v^x,M_x v^x + p, M_y v^y, M_z v^z, (E + p)v^x) ^{T } ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}F^y = (R v^y,M_x v^x, M_y v^y + p, M_z v^z, (E + p)v^y) ^{T } ,
\end{equation}
}

{\newpage
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\samepage \begin{equation}F^z = (R v^z,M_x v^x, M_y v^y, M_z v^z + p,(E + p)v^z) ^{T } ,
\end{equation}
}

{\newpage
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\samepage \begin{equation}{\partial{ U}\over \partial{t}} + {\partial\over \partial{x} } (F^x) +
{\partial\over \partial y} (F^{y})+ {\partial\over \partial z} (F^{z}) = 0 .
\end{equation}
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{\newpage
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\samepage \begin{equation}{U^{n+1}}_{i} = {U^{n}}_{i} - \frac{\delta t}{\delta x} (F_{i + \frac{1}{2} } -
F_{i - \frac{1}{2}}) .
\end{equation}
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\samepage \begin{equation}R = \gamma n ,
\end{equation}
}

{\newpage
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\samepage \begin{equation}M^{x} = \gamma^2 ( e + p ) v^{x} ,
\end{equation}
}

{\newpage
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\samepage \begin{equation}M^{y} = \gamma^2 ( e + p ) v^{y} ,
\end{equation}
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{\newpage
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\samepage \begin{equation}M^{z} = \gamma^2 ( e + p ) v^{z} ,
\end{equation}
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\samepage \begin{equation}E = \gamma^2 ( e + p ) - p ,
\end{equation}
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{\newpage
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\samepage \begin{eqnarray}& & \Bigl\lbrack\Gamma v \left(E-Mv\right)-M  \left(1-v^2\right)\Bigr\rbrack^2
 \nonumber \\ 
& & -  \left(1-v^2\right)v^2\left(\Gamma-1\right)^2R^2=0 ,
\end{eqnarray}
}

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