update: auto commit

report.tex

@@ -253,9 +253,9 @@
   \end{frame}
 
   \begin{frame}{Towards first order Z4c}{First order reduction}
-    We can introduce auxiliary variables to reduce a second order PDE system to a first order one. For example, for the wave equation on $1+1$-dimensional flat spacetime:
+    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}
-      \pdvt[2]{\phi} - \pdv[2]{\phi}{x} = 0,
+      \pdvt[2]{\phi} - \tensor{\delta}{^i^j}\pdv{\phi}{x^i}{x^j} = 0,
     \end{equation}
     we can introduce $\pi := \pdvt{\phi}$ and $\psi := \pdv{\phi}{x}$, and rewrite the wave equation as
     \begin{equation}
@@ -271,7 +271,7 @@
         0 & 0 & 0 \\
         0 & 0 & -1 \\
         0 & -1 & 0
-      \end{pmatrix} \qc S=0
+      \end{pmatrix} \qc S=0.
     \end{equation}
   \end{frame}