From 419f18e66e2836630ab4d010482b70da82c02c24 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 20:14:32 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/report.tex b/report.tex
index 5d379c5..5fe17e0 100644
--- a/report.tex
+++ b/report.tex
@@ -261,16 +261,16 @@
     \begin{equation}
       \begin{cases}
         \Partial{t}{\phi} = \pi, \\
-        \Partial{t}{\tensor{\psi}{_i}} = \Partial{i} \pi, \\
-        \Partial{t}{\pi} = \tensor{\delta}{^i^j} \Partial{i} \tensor{\psi}{_j}.
+        \Partial{t}{\pi} = \tensor{\delta}{^i^j} \Partial{i} \tensor{\psi}{_j},
+        \Partial{t}{\tensor{\psi}{_i}} = \Partial{i} \pi.
       \end{cases}
     \end{equation}
-    write $u^I = (\phi, \pi, \psi)$, then the system can be written in the form of~\eqref{first-order-PDE} with
+    write $u^I = (\phi, \pi, \tensor{\psi}{_1}, \tensor{\psi}{_2}, \tensor{\psi}{_3})$, then the system can be written in the form of~\eqref{first-order-PDE} with
     \begin{equation}
-      \tensor{A}{^x^I_J} = \begin{pmatrix}
-        0 & 0 & 0 \\
-        0 & 0 & -1 \\
-        0 & -1 & 0
+      \tensor{A}{^1^I_J} = \begin{pmatrix}
+        0 & 0 & 0 & 0 & 0 \\
+        0 & 0 & -1 & 0 & 0 \\
+        0 & -1 & 0 & 0 & 0
       \end{pmatrix} \qc S=0.
     \end{equation}
   \end{frame}