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\lhead{22. Dezember 2004}
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\rhead{\bfseries Vorlesung 19}

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\begin{document}
	{\bf Beispiel:} Sei $ U := \left\{ (r,\theta) \mid r > 0, 0 <
	\theta < 2\pi \right\} \subset \setR^2 $ \\
	Sei $ f : U \rightarrow \setR^2 = \setR^2(x,y) \ $ geg. durch $
	\left\{ \begin{array}{l} x = r \cos \theta \\
									y = r \sin \theta 
			\end{array}
			\right. $ 
	$$
		\omega = - \frac{y}{x^2+y^2} dx + \frac{x}{x^2+y^2} dy%\
		\ \textrm{1-Form auf} \ \setR^2 \setminus \left\{0 \right\}
	$$
	Was ist $ f^{*} \omega \ $ ? 
	\begin{eqnarray*}
		f^{*}dx & = &  d(f_1(r,\theta))=d(r\cos\theta)=\cos\theta dr-r\sin\theta d\theta \\ 
		f^{*}dy & = & d(f_2(r,\theta))=d(r\sin\theta)=\sin\theta dr+r\cos\theta d\theta \\
		f^{*} \omega  & = & f^* \left( - \frac{y}{x^2+y^2} \right) f^* dx + f^* \left(
							\frac{x}{x^2+y^2} \right) f^* dy = \\
					  & = & - \frac{r\sin\theta}{r^2} \cos\theta dr +
					 		\frac{r\sin\theta}{r^2}r\sin\theta d\theta +
							\frac{r\cos\theta}{r^2}\sin\theta dr +
							\frac{r\cos\theta}{r^2}r\cos\theta d\theta  = \\
					  & = & \left(-
					 	  \frac{\sin(\theta)\cos(\theta)}{r}+\frac{\sin(\theta)\cos(\theta)}{r}
					 	  \right) dr + (\sin^2(\theta) + \cos^2(\theta))d\theta = d\theta
	\end{eqnarray*}
	%\begin{beh}[Proposition]{}{}
	\begin{beh}[]{}
		Seien $ f: \setR^n \rightarrow \setR^m , \ g: \setR^p \rightarrow \setR^n $ dif\/ferenzierbar.\\
		$ \varphi , \psi $ Dif\/feomorphismen auf $ \setR^m $ $ \Rightarrow $ 
						\begin{enumerate}
								\item $ f^*(\varphi \wedge \psi ) = f^* \varphi \wedge f^* \psi $
								\item $ (f \circ g)^* \varphi = g^* f^* \varphi $
						\end{enumerate}
	\end{beh}
	{\bf Beweis:}
	\begin{enumerate}
	\item Sei $ y_i = f_i (x_1, \ldots , x_n) \ \ i = 1, \ldots , m \\
					 \varphi = \sum\limits_I a_I dy_I \ , \ \psi = \sum\limits_J b_J dy_J $
					 \begin{eqnarray*}
					 \Rightarrow f^*(\varphi \wedge \psi) & = & f^* \left( \sum\limits_{I,J} a_I(y_1, \ldots , y_m) b_J 
					 (y_1 , \ldots , y_m) dy_I
					 \wedge dy_J \right) = \\
					 & = & \sum_{I,J} a_I(f_1, \ldots , f_m) b_J (f_1, \ldots , f_m) df_I \wedge df_J = \\
					 & = & \sum_I a_I ( f_1, \ldots , f_m) df_I \wedge \sum_J b_J (f_1, \ldots f_m) df_J = \\
					 & = & f^* \varphi \wedge f^* \psi 
					 \end{eqnarray*}
	\item \begin{eqnarray*}
					 (f \circ g)^* \varphi & = & \sum_I a_I \big( (f \circ g)_1 , \ldots , (f \circ g)_m \big) d(f \circ g)_I
					 \stackrel{Kettenregel}{=} \\
					 				  & = & \sum_I a_I \left(f_1(g_1 , \ldots , g_n), \ldots , f_m(g_1 , \ldots , g_n)\right) df_I \circ dg
									  = \\
									  & = & g^*\left( \sum_I a_I (f_1 , \ldots , f_m) df_I \right) = \\
									  & = & g^*f^* (\varphi)\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\boxempty
					\end{eqnarray*}
	\end{enumerate}
	%\begin{beh}[Proposition]{}{}
	\begin{beh}[]{}
	Sei $ g: \setR^n \rightarrow \setR \ $ dif\/ferenzierbarbar, d.h. eine 0-Form \\
	$ \Rightarrow dg = \sumie \frac{\partial g_i}{\partial x_i} dx_i $ 
	\end{beh}
	{\bf Beweis:} 
	$\ \forall p \in \setR^n \ \forall v = \sumie v_i \frac{\partial}{\partial x_i}|_p \in T_p \setR^n \ \textrm{ist}
	\\
	dg |_p (v) = dg |_p \left( \sumie v_i \frac{\partial}{\partial x_i} |_p \right) = \sumie v_i dg |_p \left(
	\frac{\partial}{\partial x_i} |_p \right) = \sumie v_i \frac{\partial g}{\partial x_i} (p) = \sumie \frac{\partial g}{\partial
	x_i} (p) dx_i |_p (v) = \sumie \frac{\partial g}{\partial x_i} (p) dx_i|_p \left( \sumie v_j \frac{\partial}{\partial x_j}
	|_p \right) = \sumie \sumje v_j \frac{\partial g}{\partial x_i} (p) \underbrace{dx_i |_p \left(\frac{\partial}{\partial
	x_j}\big|_p\right)}_{=\delta_{ij}} = \sumie v_i \frac{\partial}{\partial x_i}(p) \qquad\qquad\qquad \boxempty $ 
	\\
	{\bf Allgemein:} Sei $ \omega = \sum_{I} a_I dx_I $ eine k-Form auf $ \setR^n  \\ $
	\begin{defi}
	 s $ d \omega := \sum_I da_I \wedge dx_I \ $  ist (k+1)-Form auf $ \setR^n. \  d \omega $ hei"st \underline{"au"seres
	 Dif\/ferential}  oder \underline{"au"sere Ableitung} von $ \omega $.
	\end{defi}
	{\bf Beispiel:} \\
	$ \omega = xyzdx + yzdy + (x+z) dx $ 
	\begin{eqnarray*}\Rightarrow  
	d \omega & = & d(xyz)\wedge dx + d(yz) \wedge dy + d(x+z) \wedge dx =  \\
				& = & yz\ dx \wedge dx + xz\ dy \wedge dx + xy\ dz \wedge dx + \\
				& + & z\ dy \wedge dy + y\ dz \wedge dy + dx \wedge dx+dz \wedge dx = \\
				& = & -xz\ dx \wedge dy - (xy+1)\ dx \wedge dz - y\ dy \wedge dz
	\end{eqnarray*}
	%\begin{beh}[Proposition]{}{}
	\begin{beh}[]{}
	\begin{itemize}
	\item[(a)] $ \varphi_1 , \varphi_2 \ $ k-Formen \\
		$\Rightarrow d(\varphi_1 + \varphi_2) = d\varphi_1 + d\varphi_2 $
	\item[(b)] $ \varphi = \textrm{k-Form, } \ \psi = \textrm{l-Form} \\
					\Rightarrow d(\varphi \wedge \psi) = d \varphi \wedge \psi + (-1)^k \varphi \wedge d\psi $
	\item[(c)] $ dd\varphi=0 $
	\item[(d)] $ f: \setR^m \rightarrow \setR^n $ dif\/fbar, $ \varphi = \ $ k-Form 
	$ \Rightarrow d(f^* \varphi) = f^* d \varphi$
	\end{itemize}
	\end{beh}
	{\bf Beweis:} 
	\begin{description}
	\item{zu (a)} $ \varphi_1 = \sum_{I}\limits a_I dx_I, \quad \varphi_2 = \sum_{I}\limits b_I dx_I \\
						d(\varphi_1+\varphi_2) = d\left( \sum_{I}\limits (a_I + b_I) d_I \right) = \sum_{I}\limits d(a_I +b_I) 
						\wedge dx_I = \sum_{I}\limits (da_I +db_I) \wedge dx_I =\\
						=d(\sum_{I}\limits a_I \wedge dx_I) + d(\sum_{I}\limits b_I \wedge dx_I) = d \varphi_1 + d \varphi_2 $
	\item{zu (b)} $ \varphi = \sum_{I}\limits a_I dx_I \quad \psi = \sum_{J}\limits a_J dx_J \\
						\Rightarrow \varphi \wedge \psi = \sum_{I,J}\limits a_I b_J dx_I \wedge dx_J \\
						\Rightarrow d(\varphi \wedge \psi )=\\= \sum_{I,J} d(a_I b_J) \wedge dx_i \wedge dx_J = \sum_{I,J}\limits b_J da_I \wedge
						dx_I \wedge dx_J + \sum_{I,J}\limits a_I db_J \wedge dx_I \wedge dx_J  
						= \\ =\left(  \sum_{I}\limits da_I \wedge dx_I \right) \wedge \sum_{J}\limits b_J dx_J + (-1)^k \sum_{I}\limits a_I dx_I
						\wedge  \sum_{J}\limits db_J \wedge dx_J = \\ = d\varphi \wedge \psi + (-1)^k \varphi \wedge d \psi $
	\item{zu (c)} \begin{description}
							\item{1. Fall:} $ \varphi = \ $ 0-Form, also $ \varphi : \left\{  \begin{array}{l} 
																													\setR^n   \rightarrow  \setR \\
																													(x_1, \ldots, x_n)   \mapsto  \varphi(x_1, \ldots,
																													x_n)
																													\end{array}
																													 \right. $ \\
								\begin{eqnarray*}
									d(d\varphi) & \stackrel{\textrm{Prop 4}}{=} & d \left( \sumje \frac{\partial \varphi}{\partial x_j} dx_j \right)
									\stackrel{\textrm{Def}}{=} \sumje d \left( \frac{\partial \varphi}{\partial x_j} \right) \wedge dx_j
									\stackrel{\textrm{Prop 4}}{=} \sumje \sumie \left( \frac{\partial^2 \varphi}{\partial x_j \partial x_i} dx_i \right)
									\wedge dx_j = \\
									& = & \sum_{i<j}\limits \left( \frac{\partial^2 \varphi}{\partial x_i \partial x_j} - \frac{\partial^2
									\varphi}{\partial x_i \partial x_j} \right) dx_i \wedge dx_j = 0 
								\end{eqnarray*}
							\item{2. Fall:}  $ \varphi = \sum_{I}\limits a_I dx_I $ k-Form \\
													wegen (a) o.B.d.A. $ \varphi = a(x) dx_I \ $ mit $ a \neq 0 $ \\
													$ d \varphi= d(a dx_I) \stackrel{\textrm{(b)}}{=} da \wedge dx_I + (-1)^0 addx_I $ \\
													Nun ist $ ddx_I = d ( 1 dx_I) = d1 \wedge dx_1 = 0 $ \\
													$ \Rightarrow dd\varphi = d(da \wedge dx_I) \stackrel{\textrm{(b)}}{=} \underbrace{dda \wedge
													dx_I}_{=0 \ nach \ Fall \ 1} + (-1)^0 da \wedge \underbrace{ddx_i}_{= 0} = 0 $
					\end{description}
		\newpage
	\item{zu (d)}
		\begin{description} 
			\item{1. Fall} $ \varphi = 0 $-Form $ \varphi: \setR^n \rightarrow \setR \quad \setR^n = \setR^n (y_1 , \ldots , y_n) $ 
								\begin{eqnarray*}
									f^* d\varphi & \stackrel{Prop 4}{=} & f^* \left( \sumie \frac{\partial \varphi}{\partial y_i} dy_i \right) =
									\sumie \frac{\partial \varphi (f_1, \ldots , f_n)}{\partial y_i}df_i = \\ 
									& = & \sumie \frac{\partial \varphi (f_1 , \ldots , f_n)}{ \partial y_i} \sumje \frac{\partial f_i}{\partial x_j}
									dx_j = \sumje \left( \sumie \frac{\partial	\varphi (f_1 , \ldots , f_n)}{\partial y_i} \frac{\partial
									f_i}{\partial x_j} \right) dx_j =  \\
									& \stackrel{\textrm{Kettenregel}}{=} & \sumje \frac{\partial(\varphi \circ f)}{\partial x_j} dx_j = d(\varphi
									\circ f) = d (f^* \varphi)
								\end{eqnarray*}
			\item{2. Fall} $ \varphi = \sum_I\limits a_I dx_I \quad $ k-Form 
			\begin{eqnarray*}
			d(f^* \varphi) & = & d \left( \sum_I\limits f^*(a_I)f^*(dx_I) \right) = \sum_I\limits d(f^*(a_I)) \wedge f^*dx_I = \\
								& \stackrel{\textrm{1.Fall}}{=} & \sum_I\limits f^* da_I \wedge f^* dx_I  
								 \stackrel{\textrm{Prop 3(1)}}{=}  f^* \left( \sum_I\limits da_I \wedge dx_I \right) = f^*(d\varphi)
			\end{eqnarray*}
		\end{description}
	\end{description}
	\section*{$\mathsection $  15 Dif\/ferentialformen und Vektorfelder auf Mfkt.}
	Sei M = dif\/fbare Mf $\quad p \in M $ \\
	$T_p M = \left\{ [\dot{\alpha}(0)] \mid \alpha:(-\varepsilon , \varepsilon) 
	\rightarrow M, \ \alpha(0)=p \right\} \simeq \setR^n $ \\
	$f: M \rightarrow N $ dif\/fbare Abb. 
	$$
		df_p = T_pf:\left\{ \begin{array}{l}
								T_pM \rightarrow T_{f(p)}N \\
								 \textrm{[}\dot{\alpha}(0)\textrm{]}\mapsto  \textrm{[}\dot{(f \circ \alpha)}(0)\textrm{]}
								\end{array}
						\right.
	$$
	das \underline{Dif\/ferential der Abbildung} f in p. 
	\begin{defi}{}
		Eine \underline{k-Form auf M} ist eine Abbildung 
		$$
			\varphi: \left\{\begin{array}{l}
				M \rightarrow \bigcup\limits_{p \in M} \textrm{Alt}^k (T_pM) \\
				p \mapsto \varphi |_p = \varphi |_p (v_1 , \ldots , v_k)
									\end{array}
									\right.
		$$
	\end{defi}
	Sei $ f: M \rightarrow N \ $ dif\/fbare Abbildung dif\/fbarer Mannigfaltigkeiten \\
	k-Formen kann man mittels $ f $ zur"uckziehen. \\
	Sei $ \psi = \ $ k-Form auf N.
	\begin{defi}{}
		$ f^* \psi |_{p} (v_1, \ldots , v_k) := \psi |_{f(p)}(df_p(v_1) \ldots df_p(v_k)) \quad
		p \in M, v_1 , \ldots , v_k \in T_pM $
	\end{defi}
	Also $ f^* \psi |_p \in \textrm{Alt}^k (T_pM) \ \forall p \textrm{, d.h.} f^*\psi = \ $ k-Form auf M \\
	$ $\\
	Sei $ U \subset M $ of\/fen und $ x: \left\{ \begin{array}{l}
																U \rightarrow V \subset \setR^n \\
																p \mapsto x(p)=(x_1(p), \ldots , x_n(p))
																\end{array}
																\right. 
																$
	eine dif\/fbare Karte. \\
	$ \Rightarrow (x_1, \ldots , x_n) $ die kanonischen Koordinaten von $ V \subset \setR^n ( \textrm{d.h.} \forall x \in V \ \textrm{ist} \ x =
	\sumie x_i e_i) $ \\ $ $\\
	Jede Dif\/ferentialform vom Grad k auf $ V \subset \setR^n $ l"a"st sich eindeutig schreiben als 
	$$ \sum\limits_I a_I dx_I = \sum_{1 \leqslant i_1 < \ldots < i_k \leqslant n}\limits
	a_{i_1 , \ldots, i_k}(x_1, \ldots , x_k) dx_1 \wedge \ldots \wedge dx_k $$ \\
	mit $a_I$ dif\/fbar auf $V$.\\
	%  $\quad \quad a_I= a_{i_1, \ldots , i_k} \ V \rightarrow \setR \textrm{dif\/f'bar}( \infty \textrm{-oft}) $ \\
	$ $\\
	Da $x: U \rightarrow V \ $ dif\/fbare Abbildung von Mfk., so ist $ x^*\left(\sum a_I dx_I \right)$ eine k-Form auf U.\\
	$ $\\
	Nach Def\/inition ist $ \forall p \in M \quad \forall v_1 , \ldots , v_k \in T_pU:$
	$$ x^*\left( \sum_I\limits a_I dx_I \right) (v_1 , \ldots , v_k):= \sum_I\limits a_I(x)dx_I |_{x(p)} 
	\big(dx|_p(v_1), \ldots , dx|_p(v_k) \big)$$
	$ $\\
	Diese k-Form bezeichnet man kurz als:
	$$
	\sum a_I dx_I = \sum a_I(x)dx_I
	$$
	Man identif\/iziert U mit V mittels x in dieser Bezeichnung
	\end{document}