From cdc123997b3304080748e69d6c9ea2e07ac460ba Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 16:05:36 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 10 +++++++++-
 1 file changed, 9 insertions(+), 1 deletion(-)

diff --git a/report.tex b/report.tex
index f5ef8fe..cd301a7 100644
--- a/report.tex
+++ b/report.tex
@@ -238,8 +238,16 @@
     $\implies$ we need to find a first order, strongly hyperbolic formulation of Einstein's equations with constraint damping and compatible with the moving puncture gauge condition.
   \end{frame}
 
-  \begin{frame}{Existing first order formulations}
+  \begin{frame}{Towards first order Z4c}
     \section{Towards first order Z4c}
+    \subsection{Hyperbolicity of first order systems}
+    Say we have a list of variables $u^I$, and a first order PDE system
+    \begin{equation}
+      \pdvt{u^I} + \tensor{A}{^i^I_J} \pdv{u^J}{x^i} = S^I,
+    \end{equation}
+  \end{frame}
+
+  \begin{frame}{Existing first order formulations}
     \subsection{Existing first order formulations}
     \begin{itemize}
       \item GH