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\samepage \begin{equation}\bar{D}_j\bar{K}^{ij} = 0.
			\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bar{\nabla}^2\psi = -\frac{1}{8}\psi^{-7}\bar{K}_{ij}\bar{K}^{ij}
			\end{equation}
}

{\newpage
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\samepage \begin{equation}{\partial\psi\over\partial r} = {1 - \psi\over r},
\end{equation}
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{\newpage
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\samepage \begin{equation}% latex2html id marker 73
n^i_\alpha\bar{D}_i\psi\biggl|_{B_\alpha} =
         -\frac{\psi}{2r_\alpha}\biggr|_{B_\alpha},
\end{equation}
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\samepage \begin{eqnarray}% latex2html id marker 111
\partial_t g_{ij} & = & - 2 \alpha K_{ij} + D_i \beta_j + D_j \beta_i \\ 
\label{evolk.eqn}
\partial_t K^i_j  & = & {\cal L}_{\beta} K^i_j - D^i D_j \alpha +
\alpha \left( R^i_j + K K^i_j \right)
\label{evol.eqn}
\end{eqnarray}
}

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{\newpage
\clearpage
\samepage \begin{equation}{\cal L}_{\beta} K^i_j = \beta^k D_k K^i_j - K^k_j D_k \beta^i
+ K^i_k D_j \beta_k
\label{lie.term}
\end{equation}
}

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\samepage \begin{tabular}% latex2html id marker 133
{ll}
\hline
Grid Function & Number of independent components \\  \hline
$g_{ij}$  & $6\times 2$ independent components(two = n+1 timestep) \\ 
$g^{ij}$  & 6 independent components(required for various stages) \\ 
$K^i_j$   & $9\times 2$ components (multiply by two for n+1 timestep) \\ 
$\Gamma^i_{jk}$ & 18 components (stored instead of $g_{ij,k}$) \\ 
$R_{ij}$  & 6 components (Raised when needed in eqn.~(\ref{evolk.eqn})) \\ 
$\alpha$  & scalar function \\ 
$\beta^i$ & 3 components ($\beta_i$ can be obtained using $g_{ij}$) \\ 
\hline
\end{tabular}
}

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\samepage \begin{equation}\Gamma^i_{jk} = \frac{1}{2} g^{il} \left( g_{lj,k} + g_{lk,j} - g_{jk,l}
\right)
\label{connect.1}
\end{equation}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{eqnarray}\Gamma^i_{jk} & = & \frac{1}{2} g^{il} g_{lj,k} \nonumber \\ 
& = & \frac{1}{2} \left(g^{i1} g_{1j,k} + g^{i2} g_{2j,k} + g^{i3} g_{3j,k}
\right) \label{connect.2}\\ 
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\Gamma^i_{jk} & = & \Gamma^i_{jk} + \frac{1}{2} g^{il} g_{lk,j} \nonumber \\ 
& = & \Gamma^i_{jk} + \frac{1}{2} \left(g^{i1} g_{1k,j} + g^{i2} g_{2k,j}
+ g^{i3} g_{3k,j} \right) \label{connect.3}\\ 
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\Gamma^i_{jk} & = & \Gamma^i_{jk} - \frac{1}{2} g^{il} g_{jk,l} \nonumber \\ 
& = & \Gamma^i_{jk} - \frac{1}{2} \left(g^{i1} g_{jk,1} + g^{i2} g_{jk,2}
+ g^{i3} g_{jk,3} \right) \label{connect.4}
\end{eqnarray}
}

\stepcounter{section}
{\newpage
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\samepage \begin{equation}R_{ij} = \Gamma^a_{ij,a} - \Gamma^a_{ia,j} +
\Gamma^a_{ab} \Gamma^b_{ij} - \Gamma^a_{bj} \Gamma^b_{ia}
\label{ricci.1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}R_{ij} & = & \Gamma^a_{ij,a} \nonumber \\ 
& = & \Gamma^1_{ij,1} + \Gamma^2_{ij,2} + \Gamma^3_{ij,3}
\label{ricci.2}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}R_{ij} & = & R_{ij} - \Gamma^a_{ia,j} \nonumber \\ 
& = & R_{ij} - \left(\Gamma^1_{i1,j} + \Gamma^2_{i2,j} + \Gamma^3_{i3,j}
\right)
\label{ricci.3}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}R_{ij} & = & R_{ij} + \Gamma^a_{ab} \Gamma^b_{ij} - \Gamma^a_{bj} \Gamma^b_{ia}
\label{ricci.4}
\end{eqnarray}
}

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\samepage \begin{eqnarray}% latex2html id marker 262
\partial_t g_{ij} & = & - 2 \alpha K_{ij} + D_i \beta_j + D_j \beta_i. 
\end{eqnarray}
}

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{\newpage
\clearpage
\samepage \begin{eqnarray}\label{gshift.eqn}
D_i \beta_j = g_{jk} \left( \partial_i \beta^k - \Gamma^k_{il}\beta^l \right).
\end{eqnarray}
}

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{\newpage
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\samepage \begin{eqnarray}% latex2html id marker 280
\nonumber
\partial_t K^i_j  & = & {\cal L}_{\beta} K^i_j - D^i D_j \alpha +
\alpha \left( R^i_j + K K^i_j \right)
\end{eqnarray}
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