From f0a5a8e11e248225fc475164d85a762d2d0743eb Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 20:05:07 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/report.tex b/report.tex
index e4a8802..70df743 100644
--- a/report.tex
+++ b/report.tex
@@ -253,9 +253,9 @@
   \end{frame}
 
   \begin{frame}{Towards first order Z4c}{First order reduction}
-    We can introduce auxiliary variables to reduce a second order PDE system to a first order one. For example, for the wave equation on $1+1$-dimensional flat spacetime:
+    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}
-      \pdvt[2]{\phi} - \pdv[2]{\phi}{x} = 0,
+      \pdvt[2]{\phi} - \tensor{\delta}{^i^j}\pdv{\phi}{x^i}{x^j} = 0,
     \end{equation}
     we can introduce $\pi := \pdvt{\phi}$ and $\psi := \pdv{\phi}{x}$, and rewrite the wave equation as
     \begin{equation}
@@ -271,7 +271,7 @@
         0 & 0 & 0 \\
         0 & 0 & -1 \\
         0 & -1 & 0
-      \end{pmatrix} \qc S=0
+      \end{pmatrix} \qc S=0.
     \end{equation}
   \end{frame}