diff --git a/report.tex b/report.tex
index 3d3e7c3..c8126f5 100644
--- a/report.tex
+++ b/report.tex
@@ -12,8 +12,6 @@
 
 \usepackage{appendixnumberbeamer}
 
-\newcommand{\Schwarzschild}{Schwarzschild}
-
 \tikzset{zigzag/.style={decorate, decoration=zigzag}}
 \def \L {2.}
 
@@ -295,7 +293,7 @@
 
 
   \begin{frame}{Methods on evolving black holes: moving puncture}
-    In the isotropic coordinate for \Schwarzschild{} black hole, the spacial metric
+    In the isotropic coordinate for Schwarzschild black hole, the spatial metric
     \begin{equation}
       \dd{l^2} = \left( 1+\frac{M}{2r} \right)^4 (\dd{r^2} + r^2 \dd{\Omega^2})
     \end{equation}
@@ -313,7 +311,7 @@
 
   \begin{frame}{Methods on evolving black holes: moving puncture}
 
-    In the moving puncture mothod, we won't cut out the black hole singularity, but choose a gauge condition that makes the singularity invisible to the numerical evolution.
+    In the moving puncture method, we won't cut out the black hole singularity, but choose a gauge condition that makes the singularity invisible to the numerical evolution.
     \begin{figure}
       \centering
       \includegraphics[width=0.7\textwidth, trim=4bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_penrose_1.png}
@@ -354,7 +352,7 @@
     \begin{equation}
       \left(v_a, \pdvt{u}\right) + \left(v_a, A^i \pdv{u}{x^i}\right) = (v_a, S),
     \end{equation}
-    which reduce to a linear system of $\{ u_a = (v_a, u)\}$ after intergrating by parts. Boundary terms are replaced by numerical fluxes.
+    which reduce to a linear system of $\{ u_a = (v_a, u)\}$ after integrating by parts. Boundary terms are replaced by numerical fluxes.
 
     In \texttt{nmesh}, we use Lagrange polynomials over Gauss-Legendre points on each element.
     \begin{center}
@@ -465,7 +463,7 @@
     \vspace{-0.5cm}
     \begin{itemize}
       \item Z4 and GH evolve the physical metric, and thus not compatible with the moving puncture method.
-      \item The rests are using the conformal metric.
+      \item The rests use the conformal metric.
       \begin{itemize}
         \item Z4c damps the constraints better than BSSN, as the constraint violations propagate at the speed of light and thus move out.
         \item Unstable in CCZ4 for black hole spacetime was reported with some parameters.
@@ -527,7 +525,7 @@
     \begin{itemize}
       \item GH
       \begin{itemize}
-        \item Not compatible with moving puncture mothod
+        \item Not compatible with moving puncture method
       \end{itemize}
       \item FOCCZ4
       \begin{itemize}
@@ -712,7 +710,7 @@ $}
           \dsmetric*{_i_j_k} \definedby \Partial{_i} \smetric*{_j_k},
     $}
     \]
-    and replace all the spatical derivatives of the original variables with the new auxiliary variables, we got this
+    and replace all the spatial derivatives of the original variables with the new auxiliary variables, we got this
     \[
     \scalebox{0.47}{$\displaystyle
     \begin{aligned}
@@ -859,18 +857,18 @@ $}
     Another way to speed up is, maybe we can start from FOCCZ4:
     \begin{itemize}
       \item FOCCZ4 is already strongly hyperbolic
-      \item FOCCZ4 only differs from our naive first order reduction of Z4c by some constraint terms, to be specific, sone terms proportional to $\tensor{\tilde{Z}}{^i}$ or it's derivatives. We can remove these differing terms and watch how the hyperbolicity changes, and find out how to recover the hyperbolicity
+      \item FOCCZ4 only differs from our naive first order reduction of Z4c by some constraint terms, to be specific, some terms proportional to $\tensor{\tilde{Z}}{^i}$ or it's derivatives. We can remove these differing terms and watch how the hyperbolicity changes, and find out how to recover the hyperbolicity
       \item We also add constraint damping terms, which will break the hyperbolicity, and then find out how to recover the hyperbolicity.
     \end{itemize}
 
-    In short, we are looking for a series of hyprbolicity-preserving modifications to FOCCZ4, to get a first order reduction of Z4c with constraint damping. This is working in progress.
+    In short, we are looking for a series of hyperbolicity-preserving modifications to FOCCZ4, to get a first order reduction of Z4c with constraint damping. This is working in progress.
   \end{frame}
 
   \begin{frame}{Searching for a first order Z4c}{Current status}
     \begin{itemize}
       \item What we have:
       \begin{itemize}
-        \item A mathematica note book that takes readable tensor equations as input, and automatically computes the principal symbol matrix
+        \item A Mathematica notebook that takes readable tensor equations as input, and automatically computes the principal symbol matrix
       \end{itemize}
       \item What we are doing:
       \begin{itemize}