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% Macros

\newcommand{\diff}{\ensuremath{\mathrm{d}}}
\newcommand{\ii}{\ensuremath{\mathrm{i}}}

\newcommand{\Vg}{\ensuremath{\frac{V}{\Gamma_{1}}}}
\newcommand{\As}{\ensuremath{A^{\ast}}}

\newcommand{\nabad}{\ensuremath{\nabla_{\rm ad}}}

\newcommand{\kapad}{\ensuremath{\kappa_{\rm ad}}}
\newcommand{\kapS}{\ensuremath{\kappa_{S}}}
\newcommand{\epsad}{\ensuremath{\epsilon_{\rm ad}}}
\newcommand{\epsS}{\ensuremath{\epsilon_{S}}}

\newcommand{\crad}{\ensuremath{c_{\rm rad}}}
\newcommand{\cepsad}{\ensuremath{c_{\epsilon,{\rm ad}}}}
\newcommand{\cepsS}{\ensuremath{c_{\epsilon,S}}}
\newcommand{\cthm}{\ensuremath{c_{\rm thm}}}
\newcommand{\ckap}{\ensuremath{c_{\rm kap}}}

\newcommand{\Rstar}{\ensuremath{R_{\ast}}}
\newcommand{\Mstar}{\ensuremath{M_{\ast}}}
\newcommand{\Lrad}{\ensuremath{L_{\rm rad}}}
\newcommand{\Lstar}{\ensuremath{L_{\ast}}}

\begin{document}

%%% DZIEM %%%%

\section*{DZIEM Formulation}

\subsection*{Radial Adiabatic}

\subsubsection*{Variables}

\begin{align*}
x     &= \frac{r}{\Rstar} \\
y_{1} &= x^{2}\, \frac{\xi_{r}}{r} \\
y_{2} &= x^{2}\, \frac{1}{gr} \left( \frac{p'}{\rho} \right) \\
\end{align*}

\subsubsection*{Differential Equations}

\begin{align*}
x \frac{\diff y_{1}}{\diff x} &=
\left(\Vg - 1 \right) y_{1} +
- \Vg y_{2} \\
x \frac{\diff y_{2}}{\diff x} &=
(c_{1} \omega^{2} + U - \As ) y_{1} +
(3 - U + \As) y_{2}
\end{align*}

\subsubsection*{Boundary Conditions}

\begin{equation*}
\begin{aligned}
y_{1} &= 0
\end{aligned}
\text{as}\ x \rightarrow 0
\end{equation*}

\begin{equation*}
\begin{aligned}
y_{1} - y_{2} &= 0
\end{aligned}
\text{as}\ x \rightarrow 1 \qquad \text{(Zero)}
\end{equation*}

%%

\newpage

\subsection*{Non-Radial Adiabatic}

\subsubsection*{Variables}

\begin{align*}
x     &= \frac{r}{\Rstar} \\
y_{1} &= x^{2-\ell}\, \frac{\xi_{r}}{r} \\
y_{2} &= x^{2-\ell}\, \frac{1}{gr} \left( \frac{p'}{\rho} + \Phi' \right) \\
y_{3} &= x^{2-\ell}\, \frac{\Phi'}{gr} \\
y_{4} &= x^{2-\ell}\, \frac{1}{g} \frac{\diff \Phi'}{\diff r} \\
\end{align*}

\subsubsection*{Differential Equations}

\begin{align*}
x \frac{\diff y_{1}}{\diff x} &=
\left(\Vg - 1 - \ell \right) y_{1} +
\left(\frac{\ell(\ell+1)}{c_{1} \omega^{2}} - \Vg \right) y_{2} +
\Vg y_{3} \\
x \frac{\diff y_{2}}{\diff x} &=
(c_{1} \omega^{2} - \As ) y_{1} +
(3 - U + \As - \ell) y_{2} -
\As y_{3} \\
x \frac{\diff y_{3}}{\diff x} &=
(3 - U - \ell) y_{3} +
y_{4} \\
x \frac{\diff y_{4}}{\diff x} &=
\As U y_{1} +
\Vg U y_{2} +
\left[ \ell(\ell+1) - \Vg U \right] y_{3} -
(U + \ell - 2) y_{4} \\
\end{align*}

\subsubsection*{Boundary Conditions}

\begin{equation*}
\left.
\begin{aligned}
c_{1} \omega^{2} y_{1} - \ell y_{2} &= 0 \\
\ell y_{3} - y_{4} &= 0
\end{aligned}
\right\}
\text{as}\ x \rightarrow 0
\end{equation*}

\begin{equation*}
\left.
\begin{aligned}
y_{1} - y_{2} + y_{3} &= 0 \\
U y_{1} + (\ell + 1) y_{3} + y_{4} &= 0
\end{aligned}
\right\}
\text{as}\ x \rightarrow 1 \qquad \text{(Zero)}
\end{equation*}

\begin{equation*}
\left.
\begin{aligned}
\left\{ 1 + V^{-1} \left[ \frac{\ell(\ell+1)}{\omega^{2}} - 4 - \omega^{2} \right] \right\} y_{1} -
y_{2} +
\left\{ 1 + V^{-1} \left[ \frac{\ell(\ell+1)}{\omega^{2}} - \ell - 1 \right] \right\} y_{3} &= 0 \\
(\ell + 1) y_{3} + y_{4} &= 0
\end{aligned}
\right\} 
\text{as}\ x \rightarrow 1 \qquad \text{(Dziembowski)}
\end{equation*}

%%

%% \newpage

%% \subsection*{Radial Non-Adiabatic}

%% \subsubsection*{Variables}

%% \begin{align*}
%% x     &= \frac{r}{\Rstar} \\
%% y_{1} &= x^{2}\, \frac{\xi_{r}}{r} \\
%% y_{2} &= x^{2}\, \frac{1}{gr} \left( \frac{p'}{\rho} \right)\\
%% y_{5} &= x^{2}\, \frac{\delta S}{c_{p}}\\
%% y_{6} &= x^{-1}\, \frac{\delta \Lrad}{\Lstar}
%% \end{align*}

%% \subsubsection*{Differential Equations}

%% \begin{align*}
%% x \frac{\diff y_{1}}{\diff x} &=
%% \left(\Vg - 1 \right) y_{1} - \Vg y_{2} +
%% \delta y_{5} \\
%% x \frac{\diff y_{2}}{\diff x} &=
%% (c_{1} \omega^{2} + U - \As ) y_{1} +
%% (3 - U + \As ) y_{2} +
%% \delta y_{5} \\
%% x \frac{\diff y_{5}}{\diff x} &=
%% V \left[ \nabad (U - c_{1}\omega^{2}) - U - 4 (\nabad - \nabla) + \ckap \right] y_{1} - V \ckap y_{2} + 
%% \left[ V \nabla (4 - \kapS) + 2 \right] y_{5} -
%% \frac{V \nabla}{\crad} y_{6} \\
%% x \frac{\diff y_{6}}{\diff x} &=
%% - V \cepsad y_{1} +
%% V \cepsad y_{2} + \left[ \cepsS - \ii \omega \cthm\right] y_{5} -
%% y_{6}
%% \end{align*}

%% \subsubsection*{Boundary Conditions}

%% \begin{equation*}
%% \left.
%% \begin{aligned}
%% y_{1} &= 0 \\
%% y_{5} &= 0
%% \end{aligned}
%% \right\}
%% \text{as}\ x \rightarrow 0
%% \end{equation*}

%% \begin{equation*}
%% \left.
%% \begin{aligned}
%% y_{1} - y_{2} &= 0 \\
%% (2 - 4\nabad V) y_{1} + 4 \nabad V y_{2} - 4 \nabad V y_{3} + 4 y_{5} - y_{6} &= 0
%% \end{aligned}
%% \right\}
%% \text{as}\ x \rightarrow 1 \qquad \text{(Zero)}
%% \end{equation*}

%% %%

\newpage

\subsection*{Non-Radial Non-Adiabatic}

\subsubsection*{Variables}

\begin{align*}
x     &= \frac{r}{\Rstar} \\
y_{1} &= x^{2-\ell}\, \frac{\xi_{r}}{r} \\
y_{2} &= x^{2-\ell}\, \frac{1}{gr} \left( \frac{p'}{\rho} + \Phi' \right)\\
y_{3} &= x^{2-\ell}\, \frac{\Phi'}{gr}\\
y_{4} &= x^{2-\ell}\, \frac{1}{g} \frac{\diff \Phi'}{\diff r}\\
y_{5} &= x^{2-\ell}\, \frac{\delta S}{c_{p}}\\
y_{6} &= x^{-1-\ell}\, \frac{\delta \Lrad}{\Lstar}
\end{align*}

\subsubsection*{Differential Equations}

\subsection*{Non-Adiabatic}

\begin{align*}
x \frac{\diff y_{1}}{\diff x} &=
\left(\Vg - 1 - \ell \right) y_{1} +
\left(\frac{\ell(\ell+1)}{c_{1} \omega^{2}} - \Vg \right) y_{2} +
\Vg y_{3} +
\delta y_{5} \\
x \frac{\diff y_{2}}{\diff x} &=
(c_{1} \omega^{2} - \As ) y_{1} +
(3 - U + \As - \ell) y_{2} -
\As y_{3} +
\delta y_{5} \\
x \frac{\diff y_{3}}{\diff x} &=
(3 - U - \ell) y_{3} +
y_{4} \\
x \frac{\diff y_{4}}{\diff x} &=
\As U y_{1} +
\Vg U y_{2} +
\left[ \ell(\ell+1) - \Vg U \right] y_{3} -
(U + \ell - 2) y_{4}
- \delta U y_{5} \\
x \frac{\diff y_{5}}{\diff x} &=
V \left[ \nabad (U - c_{1}\omega^{2}) - 4 (\nabad - \nabla) + \ckap \right] y_{1} + 
V \left[ \frac{\ell(\ell+1)}{c_{1} \omega^{2}} (\nabad - \nabla) - \ckap \right] y_{2} + \mbox{} \\
& 
V \ckap y_{3} + 
V \nabad y_{4} + 
\left[ V \nabla (4 - \kapS) + 2 - \ell \right] y_{5} -
\frac{V \nabla}{\crad} y_{6} \\
x \frac{\diff y_{6}}{\diff x} &=
\left[ \ell(\ell+1)\left( \frac{\nabad}{\nabla} - 1 \right) \crad - V \cepsad \right] y_{1} +
\left[ V \cepsad - \ell(\ell+1)\crad \left( \frac{\nabad}{\nabla} - \frac{3 + \crad'}{c_{1}\omega^{2}} \right) \right] y_{2} + \mbox{} \\
&
\left[ \ell(\ell+1) \frac{\nabad}{\nabla} \crad - V \cepsad \right] y_{3} +
\left[ \cepsS - \frac{\ell(\ell+1)\crad}{\nabla V} - \ii \omega \cthm\right] y_{5} -
\left[ 1 + \ell \right] y_{6}
\end{align*}

\subsubsection*{Boundary Conditions}

\begin{equation*}
\left.
\begin{aligned}
c_{1} \omega^{2} y_{1} - \ell y_{2} &= 0 \\
\ell y_{3} - y_{4} &= 0 \\
y_{5} &= 0
\end{aligned}
\right\}
\text{as}\ x \rightarrow 0
\end{equation*}

\subsubsection*{Outer}

\begin{equation*}
\left.
\begin{aligned}
y_{1} - y_{2} + y_{3} &= 0 \\
U y_{1} + (\ell + 1) y_{3} + y_{4} &= 0 \\
(2 - 4\nabad V) y_{1} + 4 \nabad V y_{2} - 4 \nabad V y_{3} + 4 y_{5} - y_{6} &= 0
\end{aligned}
\right\}
\text{as}\ x \rightarrow 1 \qquad \text{(Zero)}
\end{equation*}

\begin{equation*}
\left.
\begin{aligned}
\left\{ 1 + V^{-1} \left[ \frac{\ell(\ell+1)}{\omega^{2}} - 4 - \omega^{2} \right] \right\} y_{1} -
y_{2} +
\left\{ 1 + V^{-1} \left[ \frac{\ell(\ell+1)}{\omega^{2}} - \ell - 1 \right] \right\} y_{3} &= 0 \\
(\ell + 1) y_{3} + y_{4} &= 0 \\
(2 - 4\nabad V) y_{1} + 4 \nabad V y_{2} - 4 \nabad V y_{3} + 4 y_{5} - y_{6} &= 0
\end{aligned}
\right\}
\text{as}\ x \rightarrow 1 \qquad \text{(Dziembowski)}
\end{equation*}

%%%% JCD FORMULATION %%%%

\newpage

\section*{JCD Formulation}

\subsection*{Radial Adiabatic}

\subsubsection*{Variables}

\begin{align*}
x     &= \frac{r}{\Rstar} \\
y_{1} &= x^{2}\, \frac{\xi_{r}}{r} \\
y_{2} &= x^{2}\, \frac{1}{\sigma^{2} r^{2}} \left( \frac{p'}{\rho} \right) \\
\end{align*}

\subsubsection*{Differential Equations}

\begin{align*}
x \frac{\diff y_{1}}{\diff x} &=
\left(\Vg - 1 \right) y_{1} +
- \Vg c_{1} \omega^{2} y_{2} \\
x \frac{\diff y_{2}}{\diff x} &=
\left(1 + \frac{U - \As}{c_{1} \omega^{2}} \right) y_{1} +
\As y_{2}
\end{align*}

\subsubsection*{Boundary Conditions}

\begin{equation*}
\begin{aligned}
y_{1} &= 0
\end{aligned}
\text{as}\ x \rightarrow 0
\end{equation*}

\begin{equation*}
\begin{aligned}
y_{1} - \frac{y_{2}}{c_{1} \omega^{2}} &= 0
\end{aligned}
\text{as}\ x \rightarrow 1 \qquad \text{(Zero)}
\end{equation*}

%%

\newpage

\subsection*{Non-Radial Adiabatic}

\subsubsection*{Variables}

\begin{equation*}
\left.
\begin{aligned}
x     &= \frac{r}{\Rstar} \\
y_{1} &= x^{2-\ell}\, \frac{\xi_{r}}{r} \\
y_{2} &= x^{2-\ell}\, \frac{\ell(\ell+1)}{r^{2} \sigma^{2}} \left( \frac{p'}{\rho} + \Phi' \right) \\
y_{3} &= - x^{2-\ell}\, \frac{\Phi'}{gr} \\
y_{4} &= - x^{2-\ell}\, r \frac{\diff}{\diff r} \left( \frac{\Phi'}{g r} \right) \\
\end{aligned}
\quad \right\} \ell \neq 0
\end{equation*}

\begin{equation*}
\left.
\begin{aligned}
x     &= \frac{r}{\Rstar} \\
y_{1} &= x^{2}\, \frac{\xi_{r}}{r} \\
y_{2} &= x^{2}\, \frac{1}{r^{2} \sigma^{2}} \left( \frac{p'}{\rho} \right) \\
y_{3} &= - x^{2}\, \frac{\Phi'}{gr} \\
y_{4} &= - x^{2}\, r \frac{\diff}{\diff r} \left( \frac{\Phi'}{g r} \right) \\
\end{aligned}
\quad \right\} \ell = 0
\end{equation*}

\subsubsection*{Differential Equations}

\begin{equation*}
\left.
\begin{aligned}
x \frac{\diff y_{1}}{\diff x} &=
\left(\Vg - 1 - \ell \right) y_{1} +
\left(1 - \Vg \frac{c_{1} \omega^{2}}{\ell(\ell+1)} \right) y_{2} -
\Vg y_{3} \\
x \frac{\diff y_{2}}{\diff x} &=
\left( \ell(\ell+1) - \frac{\ell(\ell+1)}{c_{1} \omega^{2}} \As \right) y_{1} +
(\As - \ell) y_{2} +
\frac{\ell(\ell+1)}{c_{1} \omega^{2}} \As y_{3} \\
x \frac{\diff y_{3}}{\diff x} &=
(2 - \ell) y_{3} + y_{4} \\
x \frac{\diff y_{4}}{\diff x} &=
- \As U y_{1} +
- \Vg U \frac{c_{1} \omega^{2}}{\ell(\ell+1)} y_{2} +
\left[ \ell(\ell+1) + U(\As - 2) \right] y_{3} +
( 3 - 2U - \ell ) y_{4} \\
\end{aligned}
\quad \right\} \ell \neq 0
\end{equation*}

\begin{equation*}
\left.
\begin{aligned}
x \frac{\diff y_{1}}{\diff x} &=
\left(\Vg - 1 \right) y_{1} +
- \Vg c_{1} \omega^{2} y_{2} -
\Vg y_{3} \\
x \frac{\diff y_{2}}{\diff x} &=
(1 - \frac{\As}{c_{1} \omega^{2}} ) y_{1} +
\As y_{2} +
\frac{\As}{c_{1} \omega^{2}} y_{3} \\
x \frac{\diff y_{3}}{\diff x} &=
2 y_{3} + y_{4} \\
x \frac{\diff y_{4}}{\diff x} &=
\As U y_{1} +
\Vg U c_{1} \omega^{2} y_{2} +
U (\As - 2) y_{3} +
(3 - 2 U) y_{4} \\
\end{aligned}
\quad \right\} \ell = 0
\end{equation*}

\subsubsection*{Boundary Conditions}

\begin{equation*}
\left.
\begin{aligned}
c_{1} \omega^{2} y_{1} - \ell y_{2} &= 0 \\
\ell y_{3} - y_{4} &= 0
\end{aligned}
\right\}
\text{as}\ x \rightarrow 0
\end{equation*}

\begin{equation*}
\left.
\begin{aligned}
y_{1} - y_{2} + y_{3} &= 0 \\
U y_{1} + (\ell + 1) y_{3} + y_{4} &= 0
\end{aligned}
\right\}
\text{as}\ x \rightarrow 1 \qquad \text{(Zero)}
\end{equation*}

\begin{equation*}
\left.
\begin{aligned}
\left\{ 1 + V^{-1} \left[ \frac{\ell(\ell+1)}{\omega^{2}} - 4 - \omega^{2} \right] \right\} y_{1} -
y_{2} +
\left\{ 1 + V^{-1} \left[ \frac{\ell(\ell+1)}{\omega^{2}} - \ell - 1 \right] \right\} y_{3} &= 0 \\
(\ell + 1) y_{3} + y_{4} &= 0
\end{aligned}
\right\} 
\text{as}\ x \rightarrow 1 \qquad \text{(Dziembowski)}
\end{equation*}

%%

%% \newpage

%% \subsection*{Radial Non-Adiabatic}

%% \subsubsection*{Variables}

%% \begin{align*}
%% x     &= \frac{r}{\Rstar} \\
%% y_{1} &= x^{2}\, \frac{\xi_{r}}{r} \\
%% y_{2} &= x^{2}\, \frac{1}{gr} \left( \frac{p'}{\rho} \right)\\
%% y_{5} &= x^{2}\, \frac{\delta S}{c_{p}}\\
%% y_{6} &= x^{-1}\, \frac{\delta \Lrad}{\Lstar}
%% \end{align*}

%% \subsubsection*{Differential Equations}

%% \begin{align*}
%% x \frac{\diff y_{1}}{\diff x} &=
%% \left(\Vg - 1 \right) y_{1} - \Vg y_{2} +
%% \delta y_{5} \\
%% x \frac{\diff y_{2}}{\diff x} &=
%% (c_{1} \omega^{2} + U - \As ) y_{1} +
%% (3 - U + \As ) y_{2} +
%% \delta y_{5} \\
%% x \frac{\diff y_{5}}{\diff x} &=
%% V \left[ \nabad (U - c_{1}\omega^{2}) - U - 4 (\nabad - \nabla) + \ckap \right] y_{1} - V \ckap y_{2} + 
%% \left[ V \nabla (4 - \kapS) + 2 \right] y_{5} -
%% \frac{V \nabla}{\crad} y_{6} \\
%% x \frac{\diff y_{6}}{\diff x} &=
%% - V \cepsad y_{1} +
%% V \cepsad y_{2} + \left[ \cepsS - \ii \omega \cthm\right] y_{5} -
%% y_{6}
%% \end{align*}

%% \subsubsection*{Boundary Conditions}

%% \begin{equation*}
%% \left.
%% \begin{aligned}
%% y_{1} &= 0 \\
%% y_{5} &= 0
%% \end{aligned}
%% \right\}
%% \text{as}\ x \rightarrow 0
%% \end{equation*}

%% \begin{equation*}
%% \left.
%% \begin{aligned}
%% y_{1} - y_{2} &= 0 \\
%% (2 - 4\nabad V) y_{1} + 4 \nabad V y_{2} - 4 \nabad V y_{3} + 4 y_{5} - y_{6} &= 0
%% \end{aligned}
%% \right\}
%% \text{as}\ x \rightarrow 1 \qquad \text{(Zero)}
%% \end{equation*}

%% %%

\newpage

\subsection*{Non-Radial Non-Adiabatic}

\subsubsection*{Variables}

\begin{align*}
x     &= \frac{r}{\Rstar} \\
y_{1} &= x^{2-\ell}\, \frac{\xi_{r}}{r} \\
y_{2} &= x^{2-\ell}\, \frac{1}{gr} \left( \frac{p'}{\rho} + \Phi' \right)\\
y_{3} &= x^{2-\ell}\, \frac{\Phi'}{gr}\\
y_{4} &= x^{2-\ell}\, \frac{1}{g} \frac{\diff \Phi'}{\diff r}\\
y_{5} &= x^{2-\ell}\, \frac{\delta S}{c_{p}}\\
y_{6} &= x^{-1-\ell}\, \frac{\delta \Lrad}{\Lstar}
\end{align*}

\subsubsection*{Differential Equations}

\subsection*{Non-Adiabatic}

\begin{align*}
x \frac{\diff y_{1}}{\diff x} &=
\left(\Vg - 1 - \ell \right) y_{1} +
\left(\frac{\ell(\ell+1)}{c_{1} \omega^{2}} - \Vg \right) y_{2} +
\Vg y_{3} +
\delta y_{5} \\
x \frac{\diff y_{2}}{\diff x} &=
(c_{1} \omega^{2} - \As ) y_{1} +
(3 - U + \As - \ell) y_{2} -
\As y_{3} +
\delta y_{5} \\
x \frac{\diff y_{3}}{\diff x} &=
(3 - U - \ell) y_{3} +
y_{4} \\
x \frac{\diff y_{4}}{\diff x} &=
\As U y_{1} +
\Vg U y_{2} +
\left[ \ell(\ell+1) - \Vg U \right] y_{3} -
(U + \ell - 2) y_{4}
- \delta U y_{5} \\
x \frac{\diff y_{5}}{\diff x} &=
V \left[ \nabad (U - c_{1}\omega^{2}) - 4 (\nabad - \nabla) + \ckap \right] y_{1} + 
V \left[ \frac{\ell(\ell+1)}{c_{1} \omega^{2}} (\nabad - \nabla) - \ckap \right] y_{2} + \mbox{} \\
& 
V \ckap y_{3} + 
V \nabad y_{4} + 
\left[ V \nabla (4 - \kapS) + 2 - \ell \right] y_{5} -
\frac{V \nabla}{\crad} y_{6} \\
x \frac{\diff y_{6}}{\diff x} &=
\left[ \ell(\ell+1)\left( \frac{\nabad}{\nabla} - 1 \right) \crad - V \cepsad \right] y_{1} +
\left[ V \cepsad - \ell(\ell+1)\crad \left( \frac{\nabad}{\nabla} - \frac{3 + \crad'}{c_{1}\omega^{2}} \right) \right] y_{2} + \mbox{} \\
&
\left[ \ell(\ell+1) \frac{\nabad}{\nabla} \crad - V \cepsad \right] y_{3} +
\left[ \cepsS - \frac{\ell(\ell+1)\crad}{\nabla V} - \ii \omega \cthm\right] y_{5} -
\left[ 1 + \ell \right] y_{6}
\end{align*}

\subsubsection*{Boundary Conditions}

\begin{equation*}
\left.
\begin{aligned}
c_{1} \omega^{2} y_{1} - \ell y_{2} &= 0 \\
\ell y_{3} - y_{4} &= 0 \\
y_{5} &= 0
\end{aligned}
\right\}
\text{as}\ x \rightarrow 0
\end{equation*}

\subsubsection*{Outer}

\begin{equation*}
\left.
\begin{aligned}
y_{1} - y_{2} + y_{3} &= 0 \\
U y_{1} + (\ell + 1) y_{3} + y_{4} &= 0 \\
(2 - 4\nabad V) y_{1} + 4 \nabad V y_{2} - 4 \nabad V y_{3} + 4 y_{5} - y_{6} &= 0
\end{aligned}
\right\}
\text{as}\ x \rightarrow 1 \qquad \text{(Zero)}
\end{equation*}

\begin{equation*}
\left.
\begin{aligned}
\left\{ 1 + V^{-1} \left[ \frac{\ell(\ell+1)}{\omega^{2}} - 4 - \omega^{2} \right] \right\} y_{1} -
y_{2} +
\left\{ 1 + V^{-1} \left[ \frac{\ell(\ell+1)}{\omega^{2}} - \ell - 1 \right] \right\} y_{3} &= 0 \\
(\ell + 1) y_{3} + y_{4} &= 0 \\
(2 - 4\nabad V) y_{1} + 4 \nabad V y_{2} - 4 \nabad V y_{3} + 4 y_{5} - y_{6} &= 0
\end{aligned}
\right\}
\text{as}\ x \rightarrow 1 \qquad \text{(Dziembowski)}
\end{equation*}

%%%% STRUCTURE COEFFS %%%%

\newpage

\section*{Structure Coefficients}

\subsection*{Mechanical}

\begin{gather*}
V = -\frac{\diff \ln p}{\diff \ln r} \qquad
\As = \frac{1}{\Gamma_{1}} \frac{\diff \ln p}{\diff \ln r} - \frac{\diff \ln \rho}{\diff \ln r} \qquad
U = \frac{\diff \ln M_{r}}{\diff \ln r} \qquad
c_1 = \frac{r^{3}}{\Rstar^{3}} \frac{\Mstar}{M_{r}} \qquad
\Gamma_{1} = \left( \frac{\partial \ln p}{\partial \ln \rho} \right)_{S}
\end{gather*}

\subsection*{Thermal}

\begin{gather*}
\nabla = \frac{\diff \ln T}{\diff \ln p} \qquad
\nabad = \left( \frac{\partial \ln T}{\partial \ln p} \right)_{S} \qquad
\delta = - \left( \frac{\partial \ln \rho}{\partial \ln T} \right)_{p} \qquad \\
\crad = x^{-3} \frac{\Lrad}{\Lstar} \qquad
\crad' = \frac{\diff \ln \crad}{\diff \ln r} \\
\cepsad = x^{-3} \frac{4\pi r^{3} \epsad \rho}{\Lstar} \qquad
\cepsS = x^{-3} \frac{4\pi r^{3} \epsS \rho}{\Lstar} \qquad
\cthm = x^{-3} \frac{4\pi r^{3} c_{p} T \rho}{\Lstar} \sqrt{\frac{G\Mstar}{\Rstar^{3}}} \\
\ckap = \left( \kapad - 4 \nabad \right) V \nabla + \nabad \left( \frac{\diff \ln \nabad}{\diff \ln r} + V \right) \\
\kapad = \left( \frac{\partial \ln \kappa}{\partial \ln p} \right)_{S} \qquad
\kapS = c_{p} T \left( \frac{\partial \ln \kappa}{\partial S} \right)_{p} \\
\epsad = \left( \frac{\partial \epsilon}{\partial \ln p} \right)_{S} \qquad
\epsS = c_{p} T \left( \frac{\partial \epsilon}{\partial S} \right)_{p}
\end{gather*}

\end{document}
