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Sei nun $r < s$ + und $y \in \varphi^{-1}(M \cap B_r(0))$. Dann + ist $|\varphi(y)| \le r$. Also + \[ + |y| = |(y,0)| \le |(y, g(y))| = |\varphi(y)| \le r \tageq \label{eq:1} + .\] + + Damit folgt + \begin{salign*} + \sup \{ \text{dist}(x, T_0M) \colon x \in M \cap B_r(0) \} + &= \sup \{ \text{dist}(\varphi(y), T_0 M) \colon y \in \varphi^{-1}(M \cap B_r(0))\} \\ + &= \sup \left\{ \inf \{ | \varphi(y) - D \varphi(0) z| \colon z \in \R^{n}\} \colon y \in \varphi^{-1}(M \cap B_r(0)) \right\} \\ + &\stackrel{\text{Taylor}}{=} \sup \{ \inf \{ | D\varphi(0)y + o(|y|) - D\varphi(0)z | + \colon z \in \R^{n}\} y \in \varphi^{-1}(M \cap B_r(0))\} \\ + &\le \sup \{ \inf \{ |D\varphi(0)(y - z)| + |o(|y|)| \colon z \in \R^{n}\} + y \in \varphi ^{-1}(M \cap B_r(0))\} \\ + &= \sup \{ |o(|y|)| \colon y \in \varphi^{-1}(M \cap B_r(0)) \} \\ + &\stackrel{\text{(\ref{eq:1})}}{\le } \sup \{ |o(|y|)| \colon y \in B_r(0) \subseteq \R^{n}\} \\ + &= o(r) + .\end{salign*} + Das zeigt die Behauptung. +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Sei $ p \in M$ mit $p = (x, g(x))$ für ein $x \in U$. + Analog zu A1 ist eine Karte von $M$ gegeben als + $\varphi\colon U \to M$ mit $\varphi(x) := \varphi(x, g(x))$ $\forall x \in U$. Dann + ist $T_pM = \text{im}D\varphi(x)$. Im folgenden bezeichne + $E_n \in \R^{n \times n}$ die Einheitsmatrix. Dann gilt für $x \in U$: + \[ + D \varphi(x) = \begin{pmatrix} E_n \\ \nabla g(x)^{t} \end{pmatrix} + \in \R^{m \times n} + .\] Damit folgt + \[ + T_pM = \text{im}D\varphi(x) = + \left\{ \begin{pmatrix} a \\ \nabla g(x)^{t} a \end{pmatrix} \colon a \in \R^{n} \right\} + .\] Betrachte nun + \[ + V\coloneqq \left\{ \begin{pmatrix} - \nabla g(x) \\ 1 \end{pmatrix} t \colon t \in \R \right\} + .\] Es ist für $t \in \R$ und $a \in \R^{n}$: + \[ + \begin{pmatrix} - \nabla g(x) \\ 1 \end{pmatrix} t \cdot + \begin{pmatrix} a \\ \nabla g(x) \cdot a \end{pmatrix} + = t ( - \nabla g(x) \cdot a + \nabla g(x) \cdot a) = 0 + .\] Also ist $V \subseteq T_pM^{\perp} = N_pM$. Da außerdem + $\text{dim } V = 1 = \text{dim } N_{p}M$ folgt $V = N_pM$. + \item Beh.: Für $v \in \R^{n}$ gilt $\text{det}(E_n + vv^{t}) = 1 + |v|^2$. + \begin{proof} + Ergänze $v$ zu einer OB mit $u_2, \ldots, u_n \in \R^{n}$. Es gilt + \[ + (E_n + vv^{t})v = E_n v + v |v|^2 = (1 + |v|^2)v + .\] Weiter gilt für $k \in \{2, \ldots, n\} $: + \[ + (E_n + vv^{t})u_k = u_k + v \underbrace{v^{t}u_k}_{=0} = u_k + ,\] da $v \perp u_k$. Also sind $v, u_2, \ldots, u_n$ Eigenvektoren + von $E_n + v v^{t}$. Damit ist $\text{det}(E_n + v v^{t}) $ das Produkt + der Eigenwerte, also + \[ + \text{det}(E_n + v v^{t}) = (1+|v|^2) 1^{n-1} = 1 + |v|^2 + \tageq \label{eq:2} + .\] + \end{proof} + Es gilt für $x \in U$: + \[ + D^{t}\varphi(x) D\varphi(x) \stackrel{\text{(a)}}{=} + \begin{pmatrix} E_n & \nabla g(x) \end{pmatrix} + \begin{pmatrix} E_n \\ \nabla g(x)^{t} \end{pmatrix} + = E_n + \nabla g(x) \nabla g(x)^{t} + .\] + Es ist bereits $(U, \varphi)$ vollständiges System von Karten für $M$. Damit folgt + \begin{salign*} + \int_{M}^{} f \d{\mathscr{H}^{n}} + &\stackrel{\text{Def.}}{=} \int_{U}^{} (f \circ \varphi) \sqrt{\text{det}(D^{t}\varphi(x)D\varphi(x)) } \d{x} \\ + &\stackrel{\text{(\ref{eq:2})}}{=} \int_{U}^{} f(x, g(x)) \sqrt{1 + |\nabla g(x)|^2} \d{x} + .\end{salign*} + \item Es gilt für $x \in U$: + \[ + \nabla g(x) = -6x + .\] Damit folgt + \[ + (F \cdot \nu)(x, g(x)) = \begin{pmatrix} x_1 \\ 0 \\ 0 \end{pmatrix} + \cdot \begin{pmatrix} - \nabla g(x) \\ 1 \end{pmatrix} + \frac{1}{\sqrt{1 + |\nabla g(x)|^2} } + = \frac{6 x_1^2}{\sqrt{1 + |\nabla g(x)|^2} } + .\] Es ist $U$ beschränkt und $\sup_{x \in U} x_1^2 = 1 < \infty$, also + $x_1^2 \in L^{1}(U)$. Also Transformationssatz mit Polarkoordinaten anwendbar. + Außerdem sind $r^2$ und $\cos^2\varphi$ stetig, also R.-integrierbar mit Hauptsatz + und Integrale stimmen auf kompakten Intervallen überein. + Damit folgt + \begin{salign*} + \int_{M}^{} F \cdot \nu \d{\mathscr{H}^2} + &= \int_{U}^{} (F \cdot \nu)(x, g(x)) \sqrt{1 + |\nabla g(x)|^2} \d{x} \\ + &= \int_{U}^{} 6 x_1^2 \d{x} \\ + &\stackrel{\text{Polarkoord}}{=} + 6 \int_{(0,1)}^{} \d{r} \int_{0}^{2\pi} r^2 \cos^2 \varphi \d{\varphi} \\ + &\stackrel{\mathscr{L}^{n}(\{0, 1\})=0}{=} + 6 \int_{0}^{1} r^2 \d{r} \int_{0}^{2\pi} \cos^2\varphi \d{\varphi} \\ + &= 2 \frac{1}{2}\int_{0}^{2\pi} (\cos^2\varphi + \sin^2\varphi) \d{\varphi} \\ + &= 2\pi + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Sei $\Omega = B_1(0) \subseteq \R^{m}$ offen. Dann + ist $\partial \Omega = S^{m-1} \in C^{1}$ (letzter Zettel). + Dann ist $\nu = \text{id} \in C^{0}(\Omega, S^{m-1})$ + die äußere Normale, da für $x \in S^{m-1}$ und $t>0$: + \[ + |x + t\nu(x)| = |x + tx| = \sqrt{\sum_{i=1}^{n} (x_i (1+t))^2} + = (1+t) |x| = 1+t > 1 + .\] Also $x + t\nu(x) \not\in S^{m-1}$. Da die äußere Normale eindeutig ist, folgt + $\text{id}$ ist die äußere Normale von $S^{m-1}$. + \item Da + $\Omega$ beschränkt, also auch $\partial \Omega$ und da $\partial \Omega$ abgeschlossen, + nach Heine Borel kompakt und da $|x \cdot \nu(x)|$ stetig, ex. ein + $S > 0$, s.d. $|x \cdot \nu(x)| \le S$ $\forall x \in \partial \Omega$. Da + Da $\partial \Omega$ beschränkt, folgt $\mathscr{H}^{m-1}(\partial \Omega) < \infty$, insbesondere + $x \cdot \nu \in L^{1}(\partial \Omega)$. + + Es ist $x \in C^{\infty}(\R^{m}, \R^{m})$ und + \[ + \text{div }x = \sum_{i=1}^{m} \partial_i x_i = \sum_{i=1}^{m} 1 = m + .\] Es gilt nun + \begin{salign*} + m \mathscr{L}^{m}(\Omega) &= \int_{\Omega}^{} m \d{x} \\ + &= \int_{\Omega}^{} \text{div }x \d{x} \\ + &\stackrel{\text{Gauß}}{=} \int_{\partial \Omega}^{} x \cdot \nu(x) \d{\mathscr{H}^{m-1}} + .\end{salign*} + Setze nun $\Omega \coloneqq B_1(0) = B \setminus \partial \Omega$. + Es ist $\mathscr{L}^{m}(\partial \Omega) = 0$, + also folgt $\mathscr{L}^{m}(B) = \mathscr{L}^{m}(\Omega)$. Nun ist $\Omega$ offen + und beschränkt mit $\partial \Omega = S^{m-1} \in C^{1}$. Mit (a) und (b) folgt nun + \begin{salign*} + \mathscr{H}^{m-1}(S^{m-1}) &= \int_{S^{m-1}}^{} \d{\mathscr{H}^{m-1}} \\ + &= \int_{S^{m-1}}^{} |x|^2 \d{\mathscr{H}^{m-1}} \\ + &\stackrel{\text{(a)}}{=} \int_{S^{m-1}}^{} x \cdot \nu(x) \d{\mathscr{H}^{m-1}} \\ + &\stackrel{\text{(b)}}{=} m \mathscr{L}^{m}(\Omega) \\ + &= m \mathscr{L}^{m}(B) + .\end{salign*} + \item Es ist $S^2$ beschränkt und $x_1^{4}$ auf $S^2$ beschränkt, also + $x_1^{4} \in L^{1}(S^2)$. Damit folgt mit der selben Argumentation wie + in 2c im letzten Schritt: + \begin{salign*} + \int_{S^2}^{} x_1^{4} \d{\mathscr{H}^2} &= + \int_{S^2}^{} \begin{pmatrix} x_1^{3} \\ 0 \\ 0 \end{pmatrix} + \cdot \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \d{\mathscr{H}^2} \\ + &\stackrel{\text{(a)}}{=} \int_{S^2}^{} \begin{pmatrix} x_1^{3} \\ 0 \\ 0 \end{pmatrix} + \cdot \nu(x)\d{\mathscr{H}^2} \\ + &\stackrel{\text{Gauß}}{=} \int_{B_1(0)}^{} \text{div } + \begin{pmatrix} x_1^{3} \\ 0 \\ 0 \end{pmatrix} \d{x} \\ + &= \int_{B_1(0)}^{} 3x_1^2 \d{x} \\ + &= 3 \int_{0}^{1} \d{r} \int_{0}^{2\pi} r^2 \cos^2\varphi \d{\varphi} \\ + &= \pi + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Da $\varphi, \psi \in C^{2}(\Omega) \cap C^{1}(\overline{\Omega})$. Damit sind + $\nabla \varphi, \nabla \psi \in C^{1}(\Omega, \R^{n}) \cap C^{1}(\overline{\Omega},\R^{n})$. + Da $\nabla \varphi, \nu$ und $\nabla \psi$ stetig und + $\Omega$, also inbesondere $\partial \Omega$ beschränkt, ist + $\nabla \varphi \cdot \nu, \nabla \psi \cdot \nu, + \varphi \nabla \psi \cdot \nu, \psi \nabla \varphi \cdot \nu \in L^{1}(\partial \Omega)$. + \begin{enumerate}[a)] + \item Es gilt direkt + \[ + \int_{\Omega}^{} \Delta \varphi \d{x} + = \int_{\Omega}^{} \text{div}(\nabla \varphi) \d{x} + \quad \stackrel{\text{Gauß}}{=} \quad + \int_{\partial \Omega}^{} \nabla \varphi \cdot \nu \d{\mathscr{H}^{n-1}} + = \int_{\partial \Omega}^{} \partial_{\nu} \varphi \d{\mathscr{H}^{n-1}} + .\] + \item Es gilt für $i \in \{1, \ldots, n\} $: + \begin{salign*} + \partial_i \varphi (\partial_i \psi) = (\partial_i \psi)(\partial_i \varphi) + + \varphi \partial_i \partial_i \psi + = (\nabla \psi)_i (\nabla \varphi)_i + \varphi \partial_i (\nabla \psi)_i + .\end{salign*} + Damit folgt + \begin{salign*} + \text{div}(\varphi(\nabla \psi)) + = \sum_{i=1}^{n} \partial_i \varphi(\partial_i \psi) + = \nabla \varphi \cdot \nabla \psi + \varphi \Delta \psi + .\end{salign*} + Die Aussage folgt nun ganz analog zu (a) mit Satz von Gauß. + \item Es gilt + \begin{salign*} + \text{div}(\varphi \nabla \psi - \psi \nabla \varphi) + &= \text{div}(\varphi \nabla \psi) - \text{div}(\psi \nabla \varphi) \\ + &\stackrel{\text{(b)}}{=} \varphi \Delta \psi + \nabla \varphi \cdot \nabla \psi + - \psi \Delta \varphi - \nabla \varphi \cdot \nabla \psi \\ + &= \varphi \Delta \psi - \psi \Delta \varphi + .\end{salign*} + Die Aussage folgt nun ganz analog zu (a) mit Satz von Gauß. + \end{enumerate} +\end{aufgabe} + +\end{document}