From 5a0c786aa0183d3cb84c9703f486b61faa97e0a3 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 20:10:30 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/report.tex b/report.tex
index 19793c3..5d379c5 100644
--- a/report.tex
+++ b/report.tex
@@ -262,7 +262,7 @@
       \begin{cases}
         \Partial{t}{\phi} = \pi, \\
         \Partial{t}{\tensor{\psi}{_i}} = \Partial{i} \pi, \\
-        \Partial{t}{\pi} = \pdv{\tensor{\psi}{_i}}{x}.
+        \Partial{t}{\pi} = \tensor{\delta}{^i^j} \Partial{i} \tensor{\psi}{_j}.
       \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