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a/ws2020/ana/uebungen/ana10.tex b/ws2020/ana/uebungen/ana10.tex new file mode 100644 index 0000000..daf8a04 --- /dev/null +++ b/ws2020/ana/uebungen/ana10.tex @@ -0,0 +1,177 @@ +\documentclass[uebung]{../../../lecture} + +\title{Analysis 3: Übungsblatt 10} +\author{Leon Burgard, Christian Merten} +\usepackage[]{mathrsfs} +\newcommand{\tageq}{\stepcounter{equation}\tag{\theequation}} + +\begin{document} + +\punkte + +\begin{aufgabe} + Sei $\emptyset \neq \Omega \subseteq \R^{n}$. + \begin{itemize} + \item ,,$\implies$'': Sei $\Omega $ $n$-dimensionale $C^{1}$-Mannigfaltigkeit der + Dimension $n$ und $x \in M$ beliebig. Dann ex. eine Umgebung $\Omega \subseteq \R^{n}$ von + $x$ und eine Abbildung $f \in C^{1}(\Omega, \R^{0})$, mit + $f^{-1}(0) = M \cap \Omega$. Da $\R^{0} = \{0\} $ folgt + \[ + M \cap \Omega = f^{-1}(0) = \Omega + .\] Also folgt $\Omega \subseteq M$. Da $\Omega$ Umgebung, ex. ein $\epsilon > 0$, s.d. + $B_{\epsilon}(x) \subseteq \Omega \subseteq M$. Da $x$ beliebig, folgt $M$ offen. + \item ,,$\impliedby$'': Sei $\Omega$ offen und $x \in M$ beliebig. Dann $\exists \epsilon > 0$, s.d. + $B_{\epsilon}(x) \subseteq M$. Setze $\Omega \coloneqq B_{\epsilon}(x)$. Dann ist + $\Omega \subseteq \R^{n}$ eine Umgebung von $x$. Setze weiter + $f\colon \Omega \to \R^{0}, x \mapsto 0$. Dann ist $f$ konstant, insbesondere + $f \in C^{1}(\Omega, \R^{0})$ und $f^{-1}(0) = \Omega = M \cap \Omega$, da $\Omega \subseteq M$. + Außerdem ist $Df(x) = 0$ $\forall x \in \Omega$, also $\text{rang} Df(x) = 0 = n - n$ + $\forall x \in \Omega$. Also $M$ $C^{1}$-Mannigfaltigkeit der Dimension $n$. + \end{itemize} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Sei $x \in S^{n-1}$. Dann setze $\Omega \coloneqq \R^{n} \setminus \{0\} $. Dann + ist $\Omega$ offen, da $\Omega^{c} = \{0\} $ abgeschlossen in $\R^{n}$. Außerdem + ist $x \neq 0$, also $x \in \Omega$ und damit $\Omega$ Umgebung von $x$. + + Setze weiter $f\colon \Omega \to \R$ mit $g(x) \coloneqq |x| -1$. Dann + gilt, da $0 \not\in S^{n-1}$: + \[ + f^{-1}(0) = \{ x \in \R^{n} \mid |x| = 1\} + = S^{n-1} = S^{n-1} \cap (\R^{n} \setminus \{0\} ) = S^{n-1} \cap \Omega + .\] + Außerdem ist für $x \in \Omega$: $f(x) = \sqrt{\sum_{k=1}^{n} x_k} - 1$ als + Komposition differenzierbarer Abbildungen differenzierbar auf $\Omega$. Außerdem + ist für $i \in \{1, \ldots, n\} $ und $x \in \Omega$ + \[ + \frac{\partial f(x)}{\partial x_i} = \frac{x_i}{|x|} + \] stetig also $f \in C^{1}(\Omega, \R)$ und + \[ + D f(x) = \frac{x^{t}}{|x|} \neq 0 + .\] Dann folgt also $\text{rang}D f(x) = 1$ $\forall x \in \Omega$. + + Also folgt $S^{n-1}$ $n-1$-dimensionale $C^{1}$-Mfkt. + \item Sei $x \in K^{n-1} \setminus \{0\} $. Dann setze $\Omega \coloneqq \R^{n} \setminus \{0\} $. + Dann ist analog zu (a), $\Omega$ eine Umgebung von $x$. Setze weiter + $f\colon \Omega \to \R, x \mapsto \sum_{k=1}^{x_k^2} - x_n^2$. Dann ist + $f$ differenzierbar und für $i \in \{ 1, \ldots, n-1\} $ gilt + \[ + \frac{\partial f}{ \partial x_i} = 2 x_i \qquad \frac{\partial f}{\partial x_n} = - 2x_n + .\] Also folgt + \[ + D f(x) = 2 \begin{pmatrix} x_1 & x_2 & \ldots & x_{n-1} & - x_n \end{pmatrix} \neq 0 + \quad \forall x \in \Omega + .\] Also $\text{rang}Df(x) = 1$ $\forall x \in \Omega$. Zuletzt gilt + \[ + f^{-1}(0) = K^{n-1} \setminus \{0\} = (K^{n-1} \setminus \{0\}) \cap \Omega + .\] Also ist $K^{n-1}$ eine $n-1$ dim. $C^{1}$-Mfkt. + \item Ang.: $K^{n-1}$ ist eine $C^{1}$-Mfkt. Dann betrachte $x := 0$. Dann ex. + eine Umgebung $\Omega \subseteq \R^{n}$ von $x$ und $U \subseteq \R^{n-1}$ und + ein $g \in C^{1}(U, \R^{1})$ mit $K^{n-1} \cap \Omega = \pi (\text{graph }g)$. + Da $\Omega$ Umgebung + von $x$, ex. ein $\epsilon > 0$, s.d. $B_{\epsilon}(0) \subseteq \Omega$. + + Setze + nun $a \coloneqq \frac{\epsilon}{2}$ und $b \coloneqq \sqrt{\frac{a^2}{n-1}} $. + Für $z \in \pi(\text{graph }g)$ ex. ein $k \in \{1, \ldots, n\} $ mit + $z_k = g(z_1, \ldots, z_{k-1}, z_{k+1}, \ldots, z_n)$. + Dann + setze für $i \in \{ 1 \ldots, n-1\} $: $y_i := b$ und + $y_n \coloneqq a$. Setze nun + \[ + \tilde{y}_i \coloneqq \begin{cases} + y_i & i \neq k \\ + - y_i & i = k + \end{cases} \qquad \forall i \in \{ 1, \ldots, n\} + .\] Dann folgt da $\tilde{y}_i^2 = y_i^2$ $\forall i \in \{1, \ldots, n\} $: + \[ + |y| = |\tilde{y}| = \sqrt{(n-1) b^2 + a^2} = \sqrt{2 a^2} = \frac{\epsilon}{\sqrt{2} } + < \epsilon + .\] Außerdem + \[ + \sum_{i=1}^{n-1} y_i^2 = \sum_{i=1}^{n-1} \tilde{y}_i^2 = (n-1) b^2 = a^2 = + y_n^2 = \tilde{y}_n^2 + .\] Also ist $y, \tilde{y} \in \Omega \cap K^{n-1}$. Sei nun nach Umnummerierung + der Koordinaten o.E. $k = n$. Dann ex. ein $u \in U \subseteq \R^{n-1}$, + s.d. $(u, g(u)) = y$. Das heißt $u_1, \ldots, u_{n-1} = b$ und + $g(u) = a$. Aber da $\tilde{y} \in K^{n-1} \cap \Omega$ ist auch + $(u, -a) \in \text{graph }g$, insbesondere folgt + \[ + -a = g(u) = a + \] aber das ist ein Widerspruch zu $g$ Abbildung. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Sei $v \in T_{\xi}M$. Dann ex. $\gamma \in C^{1}((-\epsilon, \epsilon), M)$ mit + $\gamma(0) = \xi$ und $\gamma'(0) = v$. Dann ist $F \circ \gamma(0) = F(\xi)$. + Es gilt $\forall t \in (- \epsilon, \epsilon)\colon \gamma(t) \in M$. Da + $B_r(\xi)$ offen und $\gamma \in C^{1}((- \epsilon, \epsilon), M)$ ex. + ein $0 < \delta < \epsilon$, s.d. $\forall t \in (-\delta, \delta)\colon + \gamma(t) \in M \cap B_r(\xi)$. Also gilt + $\forall t \in (-\delta, \delta)\colon F \circ \gamma(t) \ge F \circ \gamma(0)$. Also + ist $0$ ein lokales Minimum von $F \circ \gamma$. Damit folgt + mit der Kettenregel + \begin{salign*} + 0 = (F \circ \gamma)(0)' + = (D F)(\xi) \gamma'(0) + = (\nabla F)(\xi)^{t} v + = \langle \nabla F(\xi), v \rangle + .\end{salign*} + Also folgt $\nabla F(\xi) \in T_{\xi}M^{\perp} = N_{\xi}M$. + \item Das gegebene $f$ erfüllt gerade die Eigenschaften aus Satz 5.6, d.h. es folgt + $N_{\xi}M = \text{span} \langle \nabla f_1(\xi), \ldots, \nabla f_{m-n}(\xi) \rangle$. + Da nach (a) $\nabla F(\xi) \in N_{\xi}M$, ex. also $y_i \in \R$ s.d. + \[ + \nabla F(\xi) = \sum_{i=1}^{m-n} y_i \nabla f_i(\xi) + .\] + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $f \in \mathscr{S}(\R)$ und $\lambda > 0$. Dann ist + $\widehat{f} \in \mathscr{\R}$ und für ein beliebiges Polynom $p \in C^{\infty}(\R)$ + auch $p \widehat{f} \in \mathscr{S}(\R)$ nach Definition von $\mathscr{S}(\R)$. Falls + $p(x) \neq 0$ $\forall x \in \R$ ist ebenfalls $\frac{\widehat{f}}{p} \in \mathscr{S}(\R)$. + Mit Quotientenregel folgt, dass + \[ + \frac{\mathrm{d}^{k}}{\d{x^{k}}}\left( \frac{\widehat{f}(x)}{-x^2 - \lambda} \right) + = \frac{\sum_{k=1}^{s} p_k(x) \frac{\mathrm{d}^{k}}{\d{x^{k}}}\widehat{f}(x)}{(-x^2 - \lambda)^{2k}} + \tageq \label{eq:1} + \] für Polynome $p_k \in C^{\infty}$. + Da $(-x^2 - \lambda)^{k} \neq 0$ für $k \in \N$ und $x \in \R$, folgt mit (\ref{eq:1}), dass + \[ + \frac{\widehat{f}}{-x^2 - \lambda} \in \mathscr{S}(\R) + .\] + Setze nun $u\colon \R \to \R$ mit + $u(x) \coloneqq \mathcal{F}^{-1}\left( \frac{\widehat{f}}{-x^2 - \lambda} \right)(x) $ für $x \in \R$. + Dann ist $u \in \mathscr{S}(\R)$ und es gilt: + %\begin{salign*} + % &\quad \; u(x) = \mathcal{F}^{-1}\left( \frac{\widehat{f}}{-x^2 - \lambda} \right)(x) \qquad \forall x \in \R\\ + % \stackrel{\mathcal{F} \text{ inj.}}{\iff}& \widehat{u}(x) = \frac{\widehat{f}}{-x^2 - \lambda} + % \qquad \forall x \in \R\\ + % \stackrel{-x^2 - \lambda \neq 0}{\iff}& \widehat{u}(x) (-x^2 - \lambda) = \widehat{f}(x) + % \qquad \forall x \in \R\\ + % \iff& ix \widehat{u'}(x) - \lambda \widehat{u}(x) = \widehat{f}(x) \qquad \forall x \in \R\\ + % \iff& \widehat{u''}(x) - \lambda \widehat{u}(x) = \widehat{f}(x) \qquad \forall x \in \R\\ + % \iff& \mathcal{F}(u'' - \lambda u)(x) = \mathcal{F}f(x) \qquad \forall x \in \R\\ + % \stackrel{\mathcal{F} \text{ inj.}}{\iff}& u''(x) - \lambda u(x) = f(x) \qquad \forall x \in \R + %.\end{salign*} + \begin{alignat*}{4} + &\quad & u(x) = \mathcal{F}^{-1}\left( \frac{\widehat{f}}{-x^2 - \lambda} \right)(x) &\qquad &\forall x \in \R\\ + \stackrel{\mathcal{F} \text{ inj.}}{\iff}& & \widehat{u}(x) = \frac{\widehat{f}}{-x^2 - \lambda} + & &\forall x \in \R\\ + \stackrel{-x^2 - \lambda \neq 0}{\iff}& &\widehat{u}(x) (-x^2 - \lambda) = \widehat{f}(x) + & &\forall x \in \R\\ + \iff& & ix \widehat{u'}(x) - \lambda \widehat{u}(x) = \widehat{f}(x) &&\forall x \in \R\\ + \iff& & \widehat{u''}(x) - \lambda \widehat{u}(x) = \widehat{f}(x) &&\forall x \in \R\\ + \iff& & \mathcal{F}(u'' - \lambda u)(x) = \mathcal{F}f(x) &&\forall x \in \R\\ + \stackrel{\mathcal{F} \text{ inj.}}{\iff}& & u''(x) - \lambda u(x) = f(x) &&\forall x \in \R + \end{alignat*} + Das zeigt die Existenz und die Eindeutigkeit. +\end{aufgabe} + +\end{document}