From 393466c42a29419021f1da079f65970468172d1b Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Fri, 10 Apr 2026 01:10:26 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 11 +++++++++++
 1 file changed, 11 insertions(+)

diff --git a/report.tex b/report.tex
index 31c1532..72a957f 100644
--- a/report.tex
+++ b/report.tex
@@ -700,6 +700,17 @@ $}
   where all the second order spatial derivatives are highlighted in red.
   \end{frame}
 
+  \begin{frame}{Searching for a first order Z4c}{old workflow: by hand}
+    We introduce the following auxiliary variables:
+    \begin{equation}
+  \dlapse{_i} \definedby \Partial{_i} \lapse \qc
+  \dshift{^i_j} \definedby \Partial{_j} \shift{^i} \qc
+  \dchi*{_i} \definedby \spaceD*{_i} \ln(\chi) \qc
+  \dsmetric*{_i_j_k} \definedby \Partial{_i} \smetric*{_j_k},
+\end{equation}
+
+  \end{frame}
+
   \begin{frame}{References}
     \printbibliography
   \end{frame}