From 126c88fdc2962d6a44a89feff5e2ae79c9f04c77 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 20:01:16 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 8 ++++++++
 1 file changed, 8 insertions(+)

diff --git a/report.tex b/report.tex
index 898b9ef..ce09a41 100644
--- a/report.tex
+++ b/report.tex
@@ -265,6 +265,14 @@
         \pdvt{\pi} = \pdv{\psi}{x}.
       \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
+    \begin{equation}
+      \tensor{A}{^x^I_J} = \begin{pmatrix}
+        0 & 0 & 0 \\
+        0 & 0 & -1 \\
+        0 & -1 & 0
+      \end{pmatrix}.
+    \end{equation}
   \end{frame}
 
   \begin{frame}{Existing first order formulations}