From 26fe34b8d198062b7c7515bd204910fc658951f4 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 22:24:22 -0400
Subject: [PATCH] update: auto commit

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

diff --git a/report.tex b/report.tex
index f51aa0d..c961f7e 100644
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   \end{frame}
 
   \begin{frame}{Towards first order Z4c}{First order reduction}
+    \subsection{First order reduction: wave equation}
     We can introduce auxiliary variables to reduce a second order PDE system to a first order one. For example, for the wave equation on flat spacetime:
     \begin{equation}
       \Partial{t}^2 {\phi} - \tensor{\delta}{^i^j} \Partial{i} \Partial{j} \phi = 0,