From 2e96c2b95143f7efc119a9c0b15ebc7f4c6a3046 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Fri, 10 Apr 2026 01:15:09 -0400
Subject: [PATCH] update: auto commit

---
 mystyle_beamer.tex |  2 +-
 report.tex         | 22 +++++++++++-----------
 2 files changed, 12 insertions(+), 12 deletions(-)

diff --git a/mystyle_beamer.tex b/mystyle_beamer.tex
index 17140cc..c6f2613 100644
--- a/mystyle_beamer.tex
+++ b/mystyle_beamer.tex
@@ -165,7 +165,7 @@
 \newcommand{\AD}[1]{\tensor{\mathcal{D}}{_{#1}}}
 \newcommand{\ADd}{\mathcal{D}}
 \newcommand{\connection}[1]{\mathcal{#1}}
-\newcommand{\Partial}[1]{\tensor{\partial}{_{#1}}}%普通导数算符
+\newcommand{\Partial}[1]{\tensor{\partial}{#1}}%普通导数算符
 \newcommand{\tPartial}[1]{\tensor{\tilde{\partial}}{_{#1}}}
 \newcommand{\Fd}[2][\tau]{\ensuremath{\frac{\mathrm{D_F}#2}{\dd{#1}}}}%费米导数
 \newcommand{\Fdd}[2][\tau]{\ensuremath{\mathrm{D_F}#2/\dd{#1}}}%行内费米导数
diff --git a/report.tex b/report.tex
index 72a957f..bf33ff8 100644
--- a/report.tex
+++ b/report.tex
@@ -566,14 +566,14 @@
     \subsection{First order reduction: wave equation as an example}
     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}
-      \Partial{t}^2 {\phi} - \tensor{\delta}{^i^j} \Partial{i} \Partial{j} \phi = 0,
+      \pt^2 {\phi} - \tensor{\delta}{^i^j} \Partial{_i} \Partial{_j} \phi = 0,
     \end{equation}
-    we can introduce $\pi := \Partial{t} \phi$ and $\tensor{\psi}{_i} := \Partial{_i} \phi$, and rewrite the wave equation as
+    we can introduce $\pi := \pt \phi$ and $\tensor{\psi}{_i} := \Partial{_i} \phi$, and rewrite the wave equation as
     \begin{equation}
       \begin{cases}
-        \Partial{t}{\phi} = \pi, \\
-        \Partial{t}{\pi} = \tensor{\delta}{^i^j} \Partial{i} \tensor{\psi}{_j},\\
-        \Partial{t}{\tensor{\psi}{_i}} = \Partial{i} \pi.
+        \pt{\phi} = \pi, \\
+        \pt{\pi} = \tensor{\delta}{^i^j} \Partial{_i} \tensor{\psi}{_j},\\
+        \pt{\tensor{\psi}{_i}} = \Partial{_i} \pi.
       \end{cases}
     \end{equation}
     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
@@ -614,11 +614,11 @@
   \begin{frame}{First order reduction: wave equation as an example}{constraints during the reduction}
     During the first order reduction, we introduce new variables. A solution to the new system is a solution to the original system if and only if the new variables satisfy some constraints. For example, in the wave equation case, we have the constraints
     \begin{equation}
-      \mathcal{C}_i \definedby \tensor{\psi}{_i} - \Partial{i} \phi = 0.
+      \mathcal{C}_i \definedby \tensor{\psi}{_i} - \Partial{_i} \phi = 0.
     \end{equation}
     The evolution of the constraints is given by
     \begin{equation}
-      \Partial{t} \mathcal{C}_i = \Partial{t} \psi_i - \Partial{i} \Partial{t} \phi = \Partial{i} \pi - \Partial{i} \pi = 0,
+      \pt \mathcal{C}_i = \pt \psi_i - \Partial{_i} \pt \phi = \Partial{_i} \pi - \Partial{_i} \pi = 0,
     \end{equation}
     which means that if the constraints are satisfied initially, they will be satisfied for all time. However, in numerical simulations, there will always be some constraint violation due to numerical errors, and thus it's better to add some constraint damping terms to the evolution equations to suppress the growth of constraint violation.
   \end{frame}
@@ -627,14 +627,14 @@
     For example, we can add a constraint damping term $-\gamma \mathcal{C}_i$ to the evolution equation of $\tensor{\psi}{_i}$,
     \begin{equation}
       \begin{cases}
-        \Partial{t}{\phi} = \pi, \\
-        \Partial{t}{\pi} = \tensor{\delta}{^i^j} \Partial{i} \tensor{\psi}{_j},\\
-        \Partial{t}{\tensor{\psi}{_i}} = \Partial{i} \pi - \gamma \left( \tensor{\psi}{_i} - \Partial{i} \phi \right),
+        \pt{\phi} = \pi, \\
+        \pt{\pi} = \tensor{\delta}{^i^j} \Partial{_i} \tensor{\psi}{_j},\\
+        \pt{\tensor{\psi}{_i}} = \Partial{_i} \pi - \gamma \left( \tensor{\psi}{_i} - \Partial{_i} \phi \right),
       \end{cases}
     \end{equation}
     where $\gamma > 0$ is a constant. Then the evolution of the constraints becomes
     \begin{equation}
-      \Partial{t} \mathcal{C}_i = -\gamma \mathcal{C}_i,
+      \pt \mathcal{C}_i = -\gamma \mathcal{C}_i,
     \end{equation}
     which means that the constraint violation will decay exponentially with time, and thus the system is more stable for numerical simulations.
   \end{frame}