diff --git a/mystyle_beamer.tex b/mystyle_beamer.tex
index 0cfdec5..17140cc 100644
--- a/mystyle_beamer.tex
+++ b/mystyle_beamer.tex
@@ -222,6 +222,7 @@
 
 \newcommand{\form}[1]{\ensuremath{\bm{#1}}}
 \newcommand{\Ric}{\ensuremath{R}}
+\newcommand{\Rscalar}{\ensuremath{R}}
 \newcommand{\curR}{\ensuremath{{R}}}
 \newcommand{\spacecurR}{\ensuremath{\mathcal{R}}}
 \newcommand{\vol}{\ensuremath{\varepsilon}}
diff --git a/report.tex b/report.tex
index ad1392e..62a5c40 100644
--- a/report.tex
+++ b/report.tex
@@ -15,6 +15,182 @@
 \tikzset{zigzag/.style={decorate, decoration=zigzag}}
 \def \L {2.}
 
+\newcommand{\lapse}{\alpha}
+% \newcommand{\shiftu}[1]{\ensuremath{\tensor{\beta}{^{#1}}}}
+% \newcommand{\shiftd}[1]{\ensuremath{\tensor{\beta}{_{#1}}}}
+\newcommand{\shift}[1]{\tensor{\beta}{#1}}
+\newcommand{\pt}{\Partial{_t}}
+% \newcommand{\csmetric}[1]{\tensor{\tilde{\gamma}}{#1}}
+\NewDocumentCommand{\smetric}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{\gamma}}{#2}}
+  {\tensor{\gamma}{#2}}
+}
+\NewDocumentCommand{\Ktf}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{A}}{#2}}
+  {\tensor{A}{#2}}
+}
+\NewDocumentCommand{\Gam}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{\Gamma}}{#2}}
+  {\tensor{\Gamma}{#2}}
+}
+\NewDocumentCommand{\Gamd}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{(\tilde{\Gamma}_\text{d})}{#2}}
+  {\tensor{(\Gamma_\text{d})}{#2}}
+}
+\NewDocumentCommand{\Rictensor}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{R}}{#2}}
+  {\tensor{R}{#2}}
+}
+\RenewDocumentCommand{\spaceD}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{D}}{#2}}
+  {\tensor{D}{#2}}
+}
+\NewDocumentCommand{\constraintZ}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{Z}}{#2}}
+  {\tensor{Z}{#2}}
+}
+\NewDocumentCommand{\constraintM}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{M}}{#2}}
+  {\tensor{M}{#2}}
+}
+\NewDocumentCommand{\dlapse}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{\lapse}}{#2}}
+  {\tensor{\lapse}{#2}}
+}
+\NewDocumentCommand{\dshift}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{\beta}}{#2}}
+  {\tensor{\beta}{#2}}
+}
+\NewDocumentCommand{\dchi}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{\phi}}{#2}}
+  {\tensor{\phi}{#2}}
+}
+\NewDocumentCommand{\dsmetric}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{\psi}}{#2}}
+  {\tensor{\psi}{#2}}
+}
+\NewDocumentCommand{\constraintA}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{\mathcal{A}}}{#2}}
+  {\tensor{\mathcal{A}}{#2}}
+}
+\NewDocumentCommand{\constraintB}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{\mathcal{B}}}{#2}}
+  {\tensor{\mathcal{B}}{#2}}
+}
+\NewDocumentCommand{\constraintC}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{\mathcal{C}}}{#2}}
+  {\tensor{\mathcal{C}}{#2}}
+}
+\NewDocumentCommand{\constraintD}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\tilde{\mathcal{D}}}{#2}}
+  {\tensor{\mathcal{D}}{#2}}
+}
+
+\newcommand{\oc}[1]{\mathring{#1}}
+\newcommand{\olapse}{\oc{\lapse}}
+% \newcommand{\shiftu}[1]{\ensuremath{\tensor{\beta}{^{#1}}}}
+% \newcommand{\shiftd}[1]{\ensuremath{\tensor{\beta}{_{#1}}}}
+\newcommand{\oshift}[1]{\tensor{{\oc{\beta}}}{#1}}
+% \newcommand{\pt}{\Partial{_t}}
+% % \newcommand{\csmetric}[1]{\tensor{\tilde{\gamma}}{#1}}
+\newcommand{\ochi}{\oc{\chi}}
+\NewDocumentCommand{\osmetric}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{{\oc{\tilde{\gamma}}}}{#2}}
+  {\tensor{{\oc{\gamma}}}{#2}}
+}
+\NewDocumentCommand{\oKtf}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{{\oc{\tilde{A}}}}{#2}}
+  {\tensor{{\oc{A}}}{#2}}
+}
+% \NewDocumentCommand{\Gam}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{\Gamma}}{#2}}
+%   {\tensor{\Gamma}{#2}}
+% }
+% \NewDocumentCommand{\Gamd}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{(\tilde{\Gamma}_\text{d})}{#2}}
+%   {\tensor{(\Gamma_\text{d})}{#2}}
+% }
+% \NewDocumentCommand{\Rictensor}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{R}}{#2}}
+%   {\tensor{R}{#2}}
+% }
+% \RenewDocumentCommand{\spaceD}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{D}}{#2}}
+%   {\tensor{D}{#2}}
+% }
+% \NewDocumentCommand{\constraintZ}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{Z}}{#2}}
+%   {\tensor{Z}{#2}}
+% }
+% \NewDocumentCommand{\constraintM}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{M}}{#2}}
+%   {\tensor{M}{#2}}
+% }
+% \NewDocumentCommand{\dlapse}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{\lapse}}{#2}}
+%   {\tensor{\lapse}{#2}}
+% }
+% \NewDocumentCommand{\dshift}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{\beta}}{#2}}
+%   {\tensor{\beta}{#2}}
+% }
+% \NewDocumentCommand{\dchi}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{\phi}}{#2}}
+%   {\tensor{\phi}{#2}}
+% }
+\NewDocumentCommand{\odsmetric}{ s m }{%  s=可选星号，m=索引们
+  \IfBooleanTF{#1}
+  {\tensor{\oc{\tilde{\psi}}{}}{#2}}
+  {\tensor{\oc{\psi}{}}{#2}}
+}
+% \NewDocumentCommand{\constraintA}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{\mathcal{A}}}{#2}}
+%   {\tensor{\mathcal{A}}{#2}}
+% }
+% \NewDocumentCommand{\constraintB}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{\mathcal{B}}}{#2}}
+%   {\tensor{\mathcal{B}}{#2}}
+% }
+% \NewDocumentCommand{\constraintC}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{\mathcal{C}}}{#2}}
+%   {\tensor{\mathcal{C}}{#2}}
+% }
+% \NewDocumentCommand{\constraintD}{ s m }{%  s=可选星号，m=索引们
+%   \IfBooleanTF{#1}
+%   {\tensor{\tilde{\mathcal{D}}}{#2}}
+%   {\tensor{\mathcal{D}}{#2}}
+% }
+
 
 \title{Towards Moving Picture Simulations in a Discontinuous Galerkin Framework}
 \subtitle{A Progress Report}
@@ -479,6 +655,45 @@
     and thus the system is still strongly hyperbolic, and well-posed in the $L^2$ sense.
   \end{frame}
 
+  \begin{frame}{Searching for a first order Z4c}{old workflow: by hand}
+    \subsection{old workflow: by hand}
+     Here is the Z4c evolution system:
+    {\tiny
+      \begin{align}
+        \pt \chi ={}& \frac{2}{3} \chi \left[ \alpha \left( \hat{K} +
+        2\Theta \right) - \spaceD{_i} \shift{^i} \right],\\
+        \pt \smetric*{_i_j} ={} & - 2 \lapse \Ktf*{_i_j} + \shift{^k}
+        \Partial{_k} \smetric*{_i_j} + 2 \smetric*{_{k(i}} \Partial{_{j)}}
+        \shift{^k} - \frac{2}{3} \smetric*{_i_j} \Partial{_k} \shift{^k},\\
+        \pt \hat{K} ={}& - \spaceD{^i} \spaceD{_i} \lapse + \lapse \left[
+          \Ktf*{_i_j} \Ktf*{^i^j} + \frac{1}{3} \left( \hat{K} + 2 \Theta
+        \right)^2 \right] + 4 \pi \lapse \left( S + \rho \right) + \lapse
+        \kappa_1 (1-\kappa_2) \Theta + \shift{^i} \Partial{_i} \hat{K},\\
+        \pt \Ktf*{_i_j} ={} & \chi \left[ - \spaceD{_i} \spaceD{_j} \lapse
+          + \lapse \left( \Rictensor{_i_j} - 8 \pi \tensor{S}{_i_j} \right)
+        \right]^{\text{tf}} + \lapse \left[ \left( \hat{K} + 2 \Theta
+        \right) \Ktf*{_i_j} - 2 \Ktf*{^k_i} \Ktf*{_k_j} \right] +
+        \shift{^k} \Partial{_k} \Ktf*{_i_j} + 2 \Ktf*{_{k(i}}
+        \Partial{_{j)}} \shift{^k} - \frac{2}{3} \Ktf*{_i_j} \Partial{_k}
+        \shift{^k},\\
+        \pt \Gam*{^i} ={} & - 2 \Ktf*{^i^j} \Partial{_j} \lapse + 2 \lapse
+        \left[ \Gam*{^i_j_k} \Ktf*{^j^k} - \frac{3}{2} \Ktf*{^i^j}
+          \Partial{_j} \ln(\chi) - \frac{1}{3} \smetric*{^i^j} \Partial{_j}
+          \left( 2 \hat{K} + \Theta \right) - 8 \pi \smetric*{^i^j}
+        \tensor{S}{_j} \right] + \smetric*{^j^k} \Partial{_j} \Partial{_k}
+        \shift{^i} + \frac{1}{3} \smetric*{^i^j} \Partial{_j} \Partial{_k}
+        \shift{^k} \notag\\
+        & {} + \shift{^j} \Partial{_j} \Gam*{^i} - \Gamd*{^j} \Partial{_j}
+        \shift{^i} + \frac{2}{3} \Gamd*{^i} \Partial{_j} \shift{^j} - 2
+        \lapse \kappa_1 \left[ \Gam*{^i} - \Gamd*{^i} \right], \\
+        \pt \Theta ={} & \frac{1}{2} \lapse \left[ \Rscalar - \Ktf*{_i_j}
+          \Ktf*{^i^j} + \frac{2}{3} \left( \hat{K} + 2 \Theta \right)^2
+        \right] - \lapse \left[ 8 \pi \rho + \kappa_1 (2+\kappa_2) \Theta
+        \right] + \shift{^i} \Partial{_i} \Theta,
+      \end{align}
+    }
+  \end{frame}
+
   \begin{frame}{References}
     \printbibliography
   \end{frame}