Slides erweitert.

10 files changed, 424 insertions, 92 deletions

diff --git a/vorlesungen/08_dgl/Makefile b/vorlesungen/08_dgl/Makefile
index f646db6..7a3960a 100644
--- a/vorlesungen/08_dgl/Makefile
+++ b/vorlesungen/08_dgl/Makefile
@@ -14,7 +14,7 @@ IMAGES = vektorfelder-1.pdf
 
 
 MathSem-08-dgl.pdf:	MathSem-08-dgl.tex $(SOURCES) $(IMAGES)
-	pdflatex MathSem-08-dgl.tex > /dev/null
+	pdflatex --synctex=1 MathSem-08-dgl.tex > /dev/null
 
 dgl-handout.pdf:	dgl-handout.tex $(SOURCES) $(IMAGES)
 	pdflatex dgl-handout.tex > /dev/null
diff --git a/vorlesungen/08_dgl/common.tex b/vorlesungen/08_dgl/common.tex
index 75b8586..fbf3ad9 100644
--- a/vorlesungen/08_dgl/common.tex
+++ b/vorlesungen/08_dgl/common.tex
@@ -14,3 +14,14 @@
 \date[]{}
 \newboolean{presentation}
 
+\newcommand{\gSL}[2]{\ensuremath{\text{SL}(#1, \mathbb{#2})}}
+\newcommand{\gSO}[1]{\ensuremath{\text{SO}(#1)}}
+\newcommand{\gGL}[2]{\ensuremath{\text{GL}(#1, \mathbb #2)}}
+
+\newcommand{\asl}[2]{\ensuremath{\mathfrak{sl}(#1, \mathbb{#2})}}
+\newcommand{\aso}[1]{\ensuremath{\mathfrak{so}(#1)}}
+\newcommand{\agl}[2]{\ensuremath{\mathfrak{gl}(#1, \mathbb #2)}}
+
+\DeclareMathOperator{\Spur}{Spur}
+
+
diff --git a/vorlesungen/08_dgl/slides.tex b/vorlesungen/08_dgl/slides.tex
index 48b01ff..30ee52f 100644
--- a/vorlesungen/08_dgl/slides.tex
+++ b/vorlesungen/08_dgl/slides.tex
@@ -17,10 +17,16 @@
 % 6. Strömungslinien = Pfade für Lie-Theorie, A lokal, exp(Ax) global
 % 7. Beispiele so(2), Jordan-Block, vielleicht [0 1; 1 0]
 
+\section{Einführung}
+\folie{10/repetition.tex}
+\section{Woher kommt $\exp(At)$?}
+\subsection{Taylor-Reihen}
 \folie{10/taylor.tex}
-\folie{5/potenzreihenmethode.tex}
+\folie{10/potenzreihenmethode.tex}
+\subsection{Ableitung von $\exp(At)$}
+\folie{10/ableitung-exp.tex}
+\section{Lösen einer Matrix-DGL}
 \folie{10/n-zu-1.tex}
-\folie{10/matrix-vektor-dgl.tex}
+\folie{10/matrix-dgl.tex}
+\section{Was bedeutet $\exp(At)$?}
 \folie{10/so2.tex}
-
-%\folie{10/eindimensional.tex}
diff --git a/vorlesungen/common/packages.tex b/vorlesungen/common/packages.tex
index d71438b..7e044ed 100644
--- a/vorlesungen/common/packages.tex
+++ b/vorlesungen/common/packages.tex
@@ -12,6 +12,7 @@
 \usepackage{lmodern}
 \usepackage{amsmath}
 \usepackage{amssymb}
+\usepackage{nccmath}
 \usepackage{mathtools}
 \usepackage{adjustbox}
 \usepackage{multimedia}
diff --git a/vorlesungen/slides/10/ableitung-exp.tex b/vorlesungen/slides/10/ableitung-exp.tex
new file mode 100644
index 0000000..10ce191
--- /dev/null
+++ b/vorlesungen/slides/10/ableitung-exp.tex
@@ -0,0 +1,60 @@
+%
+% ableitung-exp.tex -- Ableitung von exp(x)
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+\begin{frame}[t]
+  \setlength{\abovedisplayskip}{5pt}
+  \setlength{\belowdisplayskip}{5pt}
+  %\frametitle{Ableitung von $\exp(x)$}
+  %\vspace{-20pt}
+  \begin{columns}[t,onlytextwidth]
+    \begin{column}{0.48\textwidth}
+      \begin{block}{Ableitung von $\exp(at)$}
+        \begin{align*}
+          \frac{d}{dt} \exp(at)
+          &=
+          \frac{d}{dt} \sum_{k=0}^{\infty} a^k \frac{t^k}{k!}
+          \\
+          &\uncover<2->{
+            = \sum_{k=0}^{\infty} a^k\frac{kt^{k-1}}{k(k-1)!}
+          }
+          \\
+          &\uncover<3->{
+            = a \sum_{k=1}^{\infty}
+            a^{k-1}\frac{t^{k-1}}{(k-1)!}
+          }
+          \\
+          &\uncover<4->{
+            = a \exp(at)
+          }
+        \end{align*}
+      \end{block}
+    \end{column}
+    \begin{column}{0.48\textwidth}
+      \uncover<5->{
+        \begin{block}{Ableitung von $\exp(At)$}
+          \begin{align*}
+            \frac{d}{dt} \exp(At)
+            &=
+            \frac{d}{dt} \sum_{k=0}^{\infty} A^k \frac{t^k}{k!}
+            \\
+            &=
+            \sum_{k=0}^{\infty} A^k\frac{kt^{k-1}}{k(k-1)!}
+            \\
+            &=
+            A \sum_{k=1}^{\infty} A^{k-1}\frac{t^{k-1}}{(k-1)!}
+            \\
+            &=
+            A \exp(At)
+          \end{align*}
+        \end{block}
+      }
+    \end{column}
+  \end{columns}  
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/10/matrix-vektor-dgl.tex b/vorlesungen/slides/10/matrix-dgl.tex
index f7bd995..ae68fb1 100644
--- a/vorlesungen/slides/10/matrix-vektor-dgl.tex
+++ b/vorlesungen/slides/10/matrix-dgl.tex
@@ -1,5 +1,5 @@
 %
-% matrix-vektor-dgl.tex -- DGL mit Matrix-Koeffizienten und Vektor-Variablen
+% matrix-dgl.tex -- Matrix-Differentialgleichungen
 %
 % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 % Erstellt durch Roy Seitz
diff --git a/vorlesungen/slides/10/potenzreihenmethode.tex b/vorlesungen/slides/10/potenzreihenmethode.tex
new file mode 100644
index 0000000..1715134
--- /dev/null
+++ b/vorlesungen/slides/10/potenzreihenmethode.tex
@@ -0,0 +1,91 @@
+%
+% potenzreihenmethode.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Bearbeitet durch Roy Seitz
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Potenzreihenmethode}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Lineare Differentialgleichung}
+\begin{align*}
+x'&=ax&&\Rightarrow&x'-ax&=0
+\\
+x(0)&=C
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Potenzreihenansatz}
+\begin{align*}
+x(t)
+&=
+a_0+ a_1t + a_2t^2 + \dots
+\\
+x(0)&=a_0=C
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Lösung}
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcr}
+\uncover<3->{ x'(t)}
+	\uncover<5->{
+	&=&\phantom{(} a_1\phantom{\mathstrut-aa_0)}
+	&+& 2a_2\phantom{\mathstrut-aa_1)}t
+		&+& 3a_3\phantom{\mathstrut-aa_2)}t^2
+			&+& 4a_4\phantom{\mathstrut-aa_3)}t^3
+				&+& \dots}\\
+\uncover<3->{-ax(t)}
+	\uncover<6->{
+	&=&\mathstrut-aa_0 \phantom{)}
+	&-&  aa_1\phantom{)}t
+		&-&  aa_2\phantom{)}t^2
+			&-&  aa_3\phantom{)}t^3
+				&-& \dots}\\[2pt]
+\hline
+\\[-10pt]
+\uncover<3->{0}
+	\uncover<7->{
+	&=&(a_1-aa_0)
+	&+& (2a_2-aa_1)t
+		&+& (3a_3-aa_2)t^2
+			&+& (4a_4-aa_3)t^3
+				&+& \dots}\\
+\end{array}
+\]
+\begin{align*}
+\uncover<4->{
+a_0&=C}\uncover<8->{,
+\quad
+a_1=aa_0=aC}\uncover<9->{,
+\quad
+a_2=\frac12a^2C}\uncover<10->{,
+\quad
+a_3=\frac16a^3C}\uncover<11->{,
+\ldots,
+a_k=\frac1{k!}a^kC}
+\hspace{3cm}
+\\
+\uncover<4->{
+\Rightarrow x(t) &= C}\uncover<8->{+Cat}\uncover<9->{ + C\frac12(at)^2}
+\uncover<10->{ + C \frac16(at)^3}
+\uncover<11->{ + \dots+C\frac{1}{k!}(at)^k+\dots}
+\ifthenelse{\boolean{presentation}}{
+\only<12>{
+=
+C\sum_{k=0}^\infty \frac{(at)^k}{k!}}
+}{}
+\uncover<13->{=
+C\exp(at)}
+\end{align*}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/10/repetition.tex b/vorlesungen/slides/10/repetition.tex
new file mode 100644
index 0000000..c45d47b
--- /dev/null
+++ b/vorlesungen/slides/10/repetition.tex
@@ -0,0 +1,151 @@
+%
+% intro.tex -- Repetition Lie-Gruppen und -Algebren
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+
+\begin{frame}[t]
+  \setlength{\abovedisplayskip}{5pt}
+  \setlength{\belowdisplayskip}{5pt}
+  \frametitle{Repetition}
+  \vspace{-20pt}
+  \begin{columns}[t,onlytextwidth]
+    \begin{column}{0.48\textwidth}
+      \uncover<1->{
+        \begin{block}{Lie-Gruppe}
+          Kontinuierliche Matrix-Gruppe $G$ mit bestimmter Eigenschaft
+        \end{block}
+      }
+      \uncover<3->{
+        \begin{block}{Ein-Parameter-Untergruppe}
+          Darstellung der Lie-Gruppe $G$:
+          \[
+          \gamma \colon \mathbb R \to G
+          : \quad
+          t \mapsto \gamma(t),
+          \]
+          so dass
+          \[ \gamma(s + t) = \gamma(t) \gamma(s). \]
+        \end{block}
+      }
+    \end{column}
+    \begin{column}{0.48\textwidth}
+      \uncover<2->{
+        \begin{block}{Beispiel}
+          Volumen-erhaltende Abbildungen:
+          \[ \gSL2R= \{A \in M_2 \,|\, \det(A) = 1\} .\]
+          \begin{align*}
+            \uncover<4->{ \gamma_x(t) }
+            &
+            \uncover<4->{= \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} }
+            \\
+            \uncover<5->{ \gamma_y(t) }
+            &
+            \uncover<5->{= \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix} }
+            \\
+            \uncover<6->{ \gamma_h(t)}
+            &
+            \uncover<6->{= \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix} }
+          \end{align*}
+        \end{block}
+      }
+    \end{column}
+  \end{columns}
+\end{frame}
+
+
+\begin{frame}[t]
+  \setlength{\abovedisplayskip}{5pt}
+  \setlength{\belowdisplayskip}{5pt}
+  \frametitle{Repetition}
+  \vspace{-20pt}
+  \begin{columns}[t,onlytextwidth]
+    \begin{column}{0.48\textwidth}
+      \uncover<1->{
+        \begin{block}{Lie-Algebra aus Lie-Gruppe}
+          Ableitungen der Ein-Parameter-Untergruppen:
+          \begin{align*}
+            G       &\to      \mathcal A    \\
+            \gamma  &\mapsto  \dot\gamma(0)
+          \end{align*}
+          \uncover<3->{
+            Lie-Klammer als Produkt: 
+            \[ [A, B] = AB - BA \in \mathcal A \]
+          }
+        \end{block}
+      }
+      \uncover<7->{\vspace*{-4ex}
+        \begin{block}{Lie-Gruppe aus Lie-Algebra}
+          Lösung der Differentialgleichung:
+          \[
+          \dot\gamma(t) = A\gamma(t)
+          \quad \text{mit} \quad
+          A = \dot\gamma(0)
+          \]
+          \[
+          \Rightarrow \gamma(t) = \exp(At)
+          \]
+        \end{block}
+      }
+    \end{column}
+    \begin{column}{0.48\textwidth}
+      \uncover<2->{
+        \begin{block}{Beispiel}
+          Lie-Algebra von \gSL2R:
+          \[ \asl2R = \{ A \in M_2 \,|\, \Spur(A) = 0 \} \]
+        \end{block}
+        }
+        \begin{align*}
+          \uncover<4->{ X(t) }
+          &
+          \uncover<4->{= \begin{pmatrix} 0 & t \\ 0 & 0 \end{pmatrix} }
+          \\
+          \uncover<5->{ Y(t) }
+          &
+          \uncover<5->{= \begin{pmatrix} 0 & 0 \\ t & 0 \end{pmatrix} }
+          \\
+          \uncover<6->{ H(t) }
+          &
+          \uncover<6->{= \begin{pmatrix} t & 0 \\ 0 & -t \end{pmatrix} }
+        \end{align*}
+
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}[t]
+  \setlength{\abovedisplayskip}{5pt}
+  \setlength{\belowdisplayskip}{5pt}
+  \frametitle{Repetition}
+  \vspace{-20pt}
+  \begin{block}{Offene Fragen}
+    \begin{itemize}[<+->]
+      \item Woher kommt die Exponentialfunktion?
+      \begin{fleqn} 
+        \[
+        \exp(At)
+        =
+        1 
+        + At 
+        + A^2\frac{t^2}{2}
+        + A^3\frac{t^3}{3!} 
+        + \ldots
+        \]
+      \end{fleqn}
+      \item Wie löst man eine Matrix-DGL?
+      \begin{fleqn} 
+        \[ 
+        \dot\gamma(t) = A\gamma(t),
+        \qquad
+        \gamma(t) \in G \subset M_n
+        \]
+      \end{fleqn}
+      \item Was bedeutet $\exp(At)$?
+    \end{itemize}
+  \end{block}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/10/so2.tex b/vorlesungen/slides/10/so2.tex
index e3f74ae..b63a67e 100644
--- a/vorlesungen/slides/10/so2.tex
+++ b/vorlesungen/slides/10/so2.tex
@@ -7,14 +7,6 @@
 % !TeX spellcheck = de_CH
 \bgroup
 
-\newcommand{\gSL}[2]{\ensuremath{\text{SL}(#1, \mathbb{#2})}}
-\newcommand{\gSO}[1]{\ensuremath{\text{SO}(#1)}}
-\newcommand{\gGL}[2]{\ensuremath{\text{GL}(#1, \mathbb #2)}}
-
-\newcommand{\asl}[2]{\ensuremath{\mathfrak{sl}(#1, \mathbb{#2})}}
-\newcommand{\aso}[1]{\ensuremath{\mathfrak{so}(#1)}}
-\newcommand{\agl}[2]{\ensuremath{\mathfrak{gl}(#1, \mathbb #2)}}
-
 \begin{frame}[t]
 \setlength{\abovedisplayskip}{5pt}
 \setlength{\belowdisplayskip}{5pt}
diff --git a/vorlesungen/slides/10/taylor.tex b/vorlesungen/slides/10/taylor.tex
index 920470f..25745f5 100644
--- a/vorlesungen/slides/10/taylor.tex
+++ b/vorlesungen/slides/10/taylor.tex
@@ -10,12 +10,19 @@
 \begin{frame}[t]
   \setlength{\abovedisplayskip}{5pt}
   \setlength{\belowdisplayskip}{5pt}
-  \frametitle{Beispiel $\sin x$}
+  \frametitle{Beispiel $\sin(x)$}
   \vspace{-20pt}
-  %\onslide<+->
-  \begin{block}{Taylor-Approximationen von $\sin x$}
+  \begin{block}{Taylor-Approximationen von $\sin(x)$}
     \begin{align*}
-      p_n(x)
+      p_{
+      \only<1>{0}
+      \only<2>{1}
+      \only<3>{2}
+      \only<4>{3}
+      \only<5>{4}
+      \only<6>{5}
+      \only<7->{n}
+      }(x)
       &= 
       \uncover<1->{0}
       \uncover<2->{+ x}
@@ -74,121 +81,134 @@
   \end{center}
 \end{frame}
 
-
 \begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Taylor-Reihen}
-\vspace{-20pt}
-\onslide<+->
-  \begin{block}{Polynom-Approximationen von $f(t)$}
-    \vspace{-15pt}
-    \begin{align*}
-      p_n(t) 
-      &=
-      f(0) 
-      + f'(0) t 
-      + f''(0)\frac{t^2}{2} 
-      + f^{(3)}(0)\frac{t^3}{3!} 
-      + \ldots 
-      + f^{(n)}(0) \frac{t^n}{n!}
-      =
-      \sum_{k=0}^{n} f^{(k)} \frac{t^k}{k!}
-    \end{align*}
-  \end{block}
-  \begin{block}{Die ersten $n$ Ableitungen von $f(0)$ und $p_n(0)$ sind gleich!}
-    \vspace{-15pt}
+  \setlength{\abovedisplayskip}{5pt}
+  \setlength{\belowdisplayskip}{5pt}
+  \frametitle{Taylor-Reihen}
+  \vspace{-20pt}
+    \begin{block}{Polynom-Approximationen von $f(t)$}
+      \begin{align*}
+        p_n(t) 
+        &=
+        f(0)
+        \uncover<2->{ + f' (0) t }
+        \uncover<3->{ + f''(0)\frac{t^2}{2} }
+        \uncover<4->{ + \ldots + f^{(n)}(0) \frac{t^n}{n!} }
+        \uncover<5->{ = \sum_{k=0}^{n} f^{(k)} \frac{t^k}{k!} }
+      \end{align*}
+    \end{block}
+  \uncover<6->{
+    \begin{block}{Erste $n$ Ableitungen von $f(0)$ und $p_n(0)$ sind gleich!}}
     \begin{align*}
-      p'_n(t)
-      &=
-      f'(0) 
-      + f''(0)t 
-      + f^{(3)}(0) \frac{t^2}{2!}
-      + \mathcal O(t^3)
-      &\Rightarrow&&
-      p'_n(0) = f'(0)
+      \uncover<6->{ p'_n(t) }
+      &
+      \uncover<7->{
+        = f'(0) 
+        + f''(0)t 
+        + \mathcal O(t^2)
+      }
+      &\uncover<8->{\Rightarrow}&&
+      \uncover<8->{p'_n(0) = f'(0)}
       \\
-      p''_n(0)
-      &=
-      f''(0) + f^{(3)}(0)t + \ldots + f^{(n)}(0) \frac{t^{n-2}}{(n-2)!}
-      &\Rightarrow&&
-      p''_n(0) = f''(0)
-    \end{align*}
-  \end{block}
-  \begin{block}{Für unendlich lange Polynome stimmen alle Ableitungen überein!}
-    \vspace{-15pt}
-    \begin{align*}
-      \lim_{n\to \infty} p_n(t)
-      =
-      f(t)
+      \uncover<9->{ p''_n(t) }
+      &
+      \uncover<10->{
+        = f''(0) 
+        + \mathcal O(t)
+      }
+      &\uncover<11->{\Rightarrow}&&
+      \uncover<11->{ p''_n(0) = f''(0) }
     \end{align*}
   \end{block}
+  \uncover<12->{
+    \begin{block}{Für alle praktisch relevanten Funktionen $f(t)$ gilt:}
+      \begin{align*}
+        \lim_{n\to \infty} p_n(t)
+        =
+        f(t)
+      \end{align*}
+    \end{block}
+  }
 \end{frame}
 
 
 \begin{frame}[t]
   \setlength{\abovedisplayskip}{5pt}
   \setlength{\belowdisplayskip}{5pt}
-  \frametitle{Beispiel $\exp x$}
-  \vspace{-20pt}
-  %\onslide<+->
-  \begin{block}{Taylor-Approximationen von $\exp x$}
+%  \frametitle{Beispiel $e^t$}
+%  \vspace{-20pt}
+  \begin{block}{Taylor-Approximationen von $e^{at}$}
     \begin{align*}
-      p_n(x) 
-      = 
+      p_{
+        \only<1>{0}
+        \only<2>{1}
+        \only<3>{2}
+        \only<4>{3}
+        \only<5>{4}
+        \only<6>{5}
+        \only<7->{n}
+      }(t)
+      &=
       1
-      \uncover<1->{+ x}
-      \uncover<2->{+ \frac{x^2}{2}}
-      \uncover<3->{+ \frac{x^3}{3!}}
-      \uncover<4->{+ \frac{x^4}{4!}}
-      \uncover<5->{+ \frac{x^5}{5!}}
-      \uncover<6->{+ \frac{x^6}{6!}}
-      \uncover<7->{+ \ldots
-                   = \sum_{k=0}^{n} \frac{x^k}{k!}}
+      \uncover<2->{+ a t}
+      \uncover<3->{+ a^2 \frac{t^2}{2}}
+      \uncover<4->{+ a^3 \frac{t^3}{3!}}
+      \uncover<5->{+ a^4 \frac{t^4}{4!}}
+      \uncover<6->{+ a^5 \frac{t^5}{5!}}
+      \uncover<7->{+ a^6 \frac{t^6}{6!}}
+      \uncover<8->{+ \ldots
+                   = \sum_{k=0}^{n} a^k \frac{t^k}{k!}}
+      \\
+      &
+      \uncover<9->{= \exp(at)}
     \end{align*}
   \end{block}
   \begin{center}
     \begin{tikzpicture}[>=latex,thick,scale=1.3]
-      \draw[->] (-4.0, 0.0) -- (4.0,0.0) coordinate[label=$x$];
+      \draw[->] (-4.0, 0.0) -- (4.0,0.0) coordinate[label=$t$];
       \draw[->] ( 0.0,-0.5) -- (0.0,2.5);
       \clip (-3,-0.5) rectangle (3,2.5);
       \draw[domain=-4:1, samples=50, smooth, blue]
         plot ({\x}, {exp(\x)})
-        node[above right] {$\exp(x)$};
+        node[above right] {$\exp(t)$};
       \uncover<1>{
-      \draw[domain=-4:1.5, samples=10, smooth, red]
-        plot ({\x}, {1 + \x})
-        node[below right] {$p_1(x)$};}
+      \draw[domain=-4:4, samples=12, smooth, red]
+        plot ({\x}, {1})
+        node[below right] {$p_0(t)$};}
       \uncover<2>{
+      \draw[domain=-4:1.5, samples=10, smooth, red]
+      plot ({\x}, {1 + \x})
+      node[below right] {$p_1(t)$};}
+      \uncover<3>{
       \draw[domain=-4:1, samples=50, smooth, red]
         plot ({\x}, {1 + \x + \x*\x/2})
-        node[below right] {$p_2(x)$};}
-      \uncover<3>{
+        node[below right] {$p_2(t)$};}
+      \uncover<4>{
       \draw[domain=-4:1, samples=50, smooth, red]
         plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6})
-        node[below right] {$p_3(x)$};}
-      \uncover<4>{
+        node[below right] {$p_3(t)$};}
+      \uncover<5>{
       \draw[domain=-4:0.9, samples=50, smooth, red]
         plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24})
-        node[below left] {$p_4(x)$};}
-      \uncover<5>{
+        node[below left] {$p_4(t)$};}
+      \uncover<6>{
       \draw[domain=-4:0.9, samples=50, smooth, red]
       plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24
         + \x*\x*\x*\x*\x/120})
-      node[below left] {$p_5(x)$};}
-      \uncover<6>{
+      node[below left] {$p_5(t)$};}
+      \uncover<7>{
       \draw[domain=-4:0.9, samples=50, smooth, red]
       plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24
         + \x*\x*\x*\x*\x/120
         + \x*\x*\x*\x*\x*\x/720})
-      node[below left] {$p_6(x)$};}
-      \uncover<7>{
+      node[below left] {$p_6(t)$};}
+      \uncover<8->{
       \draw[domain=-4:0.9, samples=50, smooth, red]
       plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24
         + \x*\x*\x*\x*\x/120
         + \x*\x*\x*\x*\x*\x/720
         + \x*\x*\x*\x*\x*\x*\x/5040})
-      node[below left] {$p_7(x)$};}
+      node[below left] {$p_7(t)$};}
     \end{tikzpicture}
   \end{center}
 \end{frame}