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\documentclass{article}
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\usepackage{graphicx}
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\input apj
\title{Test Perturbations}
\author{}
\date{\today,~ $ $Revision: 1.2 $ $}
\begin{document}
\maketitle

In the following, $\emf_i^\sigma$ for $i=1,2,3$ and $\sigma=0,1,...,4$
denote fully three-dimensional arrays and angular brackets denote volume
averages.
\EQ
\alpha_{11}
=\left\langle
\left(\emf_1^1-\emf_1^0\right)B^{-1}\cos kz
+\left(\emf_1^2-\emf_1^0\right)B^{-1}\sin kz
\right\rangle
\EN
\EQ
\alpha_{21}
=\left\langle
\left(\emf_2^1-\emf_2^0\right)B^{-1}\cos kz
+\left(\emf_2^2-\emf_2^0\right)B^{-1}\sin kz
\right\rangle
\EN
\EQ
\alpha_{12}
=\left\langle
\left(\emf_1^3-\emf_1^0\right)B^{-1}\cos kz
+\left(\emf_1^4-\emf_1^0\right)B^{-1}\sin kz
\right\rangle
\EN
\EQ
\alpha_{22}
=\left\langle
\left(\emf_2^3-\emf_2^0\right)B^{-1}\cos kz
+\left(\emf_2^4-\emf_2^0\right)B^{-1}\sin kz
\right\rangle
\EN
\EQ
\eta_{113}
=\left\langle
-\left(\emf_1^1-\emf_1^0\right)B^{-1}k^{-1}\sin kz
+\left(\emf_1^2-\emf_1^0\right)B^{-1}k^{-1}\cos kz
\right\rangle
\EN
\EQ
\eta_{213}
=\left\langle
-\left(\emf_2^1-\emf_2^0\right)B^{-1}k^{-1}\sin kz
+\left(\emf_2^2-\emf_2^0\right)B^{-1}k^{-1}\cos kz
\right\rangle
\EN
\EQ
\eta_{123}
=\left\langle
-\left(\emf_1^3-\emf_1^0\right)B^{-1}k^{-1}\sin kz
+\left(\emf_1^4-\emf_1^0\right)B^{-1}k^{-1}\cos kz
\right\rangle
\EN
\EQ
\eta_{223}
=\left\langle
-\left(\emf_2^3-\emf_2^0\right)B^{-1}k^{-1}\sin kz
+\left(\emf_2^4-\emf_2^0\right)B^{-1}k^{-1}\cos kz
\right\rangle
\EN

The final result can be split into 3 separate volume averages, i.e.\
\EQA
\alpha_{p1}B
=\bra{\emf_p^1\cos kz}
+\bra{\emf_p^2\sin kz}
-\bra{\emf_p^0(\cos kz+\sin kz)}
\\
\alpha_{p2}B
=\bra{\emf_p^3\cos kz}
+\bra{\emf_p^4\sin kz}
-\bra{\emf_p^0(\cos kz+\sin kz)}
\\
\eta_{p13}kB=
-\bra{\emf_p^1\sin kz}
+\bra{\emf_p^2\cos kz}
+\bra{\emf_p^0(\sin kz-\cos kz)}
\\
\eta_{p23}kB=
-\bra{\emf_p^3\sin kz}
+\bra{\emf_p^4\cos kz}
+\bra{\emf_p^0(\sin kz-\cos kz)}
\ENA

%\begin{figure}[t!]\begin{center}
%\includegraphics[width=\columnwidth]{pcoolheat}
%\end{center}\caption[]{
%Equilibria
%}\label{pcoolheat}\end{figure}

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\vfill\bigskip\noindent\tiny\begin{verbatim}
$Header: /var/cvs/brandenb/f90/pencil-code/doc/implementation/testperturb/notes.tex,v 1.2 2008-05-20 07:09:48 brandenb Exp $
\end{verbatim}

\end{document}
