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I.A.: $n = 1$ trivial. + + Sei $n \in \N$ beliebig und Beh. gezeigt für $n-1$. Dann gilt für $x \in \R^{n}$ und + $\nu \in \N_0$ mit $\tilde{x} = (x_2, \ldots, x_n)^{T} \in \R^{n-1}$: + \begin{salign*} + (x_1 + x_2 \ldots + x_n)^{\nu} &= (x_1 + (x_2 + \ldots + x_n))^{\nu} \\ + &\stackrel{\text{Bin. Formel}}{=} + \sum_{k=0}^{\nu} \binom{\nu}{k} x_1^{k} (x_2 + \ldots + x_n)^{\nu - k} \\ + &\stackrel{\text{I.V.}}{=} \sum_{k=0}^{\nu} x_1^{k} + (\nu - k)! \sum_{|\alpha|=\nu-k} \frac{\tilde{x}^{\alpha}}{\alpha!} \\ + &= \sum_{k=0}^{\nu} \frac{\nu!}{(\nu-k)!} (\nu-k)! \sum_{|\alpha| = \nu-k} + \frac{\tilde{x}^{\alpha}x_1^{k}}{\alpha! k!} \\ + &= \nu! \sum_{k=0}^{\nu} \sum_{|\alpha|=\nu-k} \frac{\tilde{x}^{\alpha} x_1^{k}}{\alpha! k!} \\ + &= \nu! \sum_{|\alpha|=\nu} \frac{x^{\alpha}}{\alpha!} + .\end{salign*} + \end{proof} + \item Beh.: Es ist mit $h = (h_1, h_2, h_3)^{T} \in \R^{3}$: + \[ + T_{2}^{f}(\hat{x} + h) = -e\left(-h_1 - h_2 -h_3 + \frac{1}{2} h_2^2 + h_1 h_2 + h_1h_3\right) + .\] + \begin{proof} + Mit $\hat{x} = (-1, -1, 0)^{T}$ folgt + der Gradient von $f$ bei $\hat{x}$ direkt + \begin{align*} + \nabla f(x) = \begin{pmatrix} e^{-x_2} - x_3e^{-x_1} \\ -x_1e^{-x_2} \\ e^{-x_1} \end{pmatrix} \implies \nabla f(\hat{x}) = \begin{pmatrix} e \\ e \\ e \end{pmatrix} + .\end{align*} + Die Hessematrix von $f$ ist gegeben als + \begin{align*} + H_f(x) = \begin{pmatrix} + x_3 e^{-x_1} & -e^{-x_2} & -e^{-x_1} \\ + -e^{-x_2} & x_1 e^{-x_2} & 0 \\ + -e^{-x_1} & 0 & 0 + \end{pmatrix} + \implies + H_f(\hat{x}) = \begin{pmatrix} 0 & -e & -e \\ + -e & -e & 0 \\ + -e & 0 & 0 + \end{pmatrix} + .\end{align*} + Damit folgt direkt + \begin{salign*} + T_2^{f}(\hat{x} + h) &= f(\hat{x}) + (\nabla f(\hat{x}), h)_2 + \frac{1}{2} (H_f(\hat{x})h, h)_2 \\ + &= -e + eh_1 + eh_2 + eh_3 - \frac{1}{2} eh_2^2 - eh_1h_2 - eh_1h_3 \\ + &= -e \left(-h_1 - h_2-h_3 + \frac{1}{2}h_2^2 + h_1h_2 + h_1 h_3\right) + .\end{salign*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe}[] + Beh.: $f\colon \R^2 \to \R$, $f(x,y) = (4x^2 + y^2) \exp(-x^2 - 4y^2)$ hat zwei strikte lokale Maxima + bei $P_1 = (1, 0)^{T}$ und $P_2 = (-1, 0)^{T}$ und ein striktes lokales Minimum bei $P_3 = (0,0)^{T}$. + \begin{proof} + Der Gradient ergibt sich als + \begin{align*} + \nabla f(x,y) = \begin{pmatrix} 8x - 8x^{3} - 2y^2x \\ 2y - 32x^2y - 8y^{3} \end{pmatrix} + \exp(-x^2 - 4y^2) + .\end{align*} + Daraus folgen die notwendigen Bedingungen + \begin{align} + 8x - 8x^{3} - 2y^2x &= 0 \\ + 2y - 32x^2y - 8y^{3} &= 0 + .\end{align} + Aus (1) folgt + \[ + 8x - 8x^{3} - 2y^2x = x(8 -8x^2 - 2y^2) = 0 \implies x = 0 \lor x^2 = 1 - \frac{1}{4}y^2 + .\] Für $x = 0$ folgt aus (2): + \begin{align*} + 2y - 8y^{3} \implies y (2 - 8y^2) \implies y = 0 \lor 2 = 8y^2 \implies y = 0 \lor y = \pm \frac{1}{2} + .\end{align*} + Damit folgen $P_3 = (0,0)^{T}$, $P_4 = \left( 0, \frac{1}{2} \right)^{T} $ und $P_5 = \left( 0, -\frac{1}{2} \right)^{T} $. + + Für $x = 1 - \frac{1}{4}y^2$ folgt aus (2): + \begin{align*} + 2y - 32\left( 1 - \frac{1}{4}y^2 \right) y - 8y^{3} = 0 \implies y - 16y = 0 \implies y = 0 + .\end{align*} + Damit folgt für $x$ mit $x = 1 - \frac{1}{4} y^2$: $x = \pm 1$. Also $P_1 = (1, 0)^{T}$ und + $P_2 = (-1, 0)^{T}$. + + Die Hessematrix von $f$ folgt direkt als + \begin{align*} + H_f(x) = \exp(-x^2 - 4y^2) + \begin{pmatrix} + 16 x^{4} + 4y^2x^2 - 40x^2 - 2y^2+8 & -68xy + 64x^{3}y + 16xy^{3} \\ + -68xy + 64 x^{3}y + 16xy^{3} & 2 - 32x^2 - 40y^2 + 256 x^2y^2 + 64y^{4} + \end{pmatrix} + .\end{align*} + Dann folgt durch Ablesen der Eigenwerte + \begin{align*} + H_f(P_1) &= H_f(1,0) = \frac{1}{e}\begin{pmatrix} -16 & 0\\ 0 & -30 \end{pmatrix} \quad + \text{negativ definit} \implies P_1 \text{ striktes lokales Min.} \\ + H_f(P_2) &= H_f(-1,0) = \frac{1}{e}\begin{pmatrix} -16 & 0\\ 0 & -30 \end{pmatrix} \quad + \text{negativ definit} \implies P_2 \text{ striktes lokales Min.} \\ + H_f(P_3) &= H_f(0,0) = \begin{pmatrix} 8 & 0 \\ 0 & 2 \end{pmatrix} \quad + \text{positiv definit} \implies P_3 \text{ striktes lokales Max.} \\ + H_f(P_4) &= H_f\left( 0, \frac{1}{2} \right) = \frac{1}{e}\begin{pmatrix} 7 \frac{1}{2} & 0 \\ 0 & -4 \end{pmatrix} \quad + \text{indefinit} \implies P_4 \text{ kein lokales Extremum} \\ + H_f(P_5) &= H_f\left( 0, - \frac{1}{2} \right) = \frac{1}{e}\begin{pmatrix} 7 \frac{1}{2} & 0 \\ 0 & -4 \end{pmatrix} \quad + \text{indefinit} \implies P_5 \text{ kein lokales Extremum} + .\end{align*} + Das zeigt die Behauptung. + \end{proof} +\end{aufgabe} + +\begin{aufgabe}[] + Beh.: Das Gleichungssystem + \begin{align*} + y_1 + \sin(y_1 y_2) &= y_1x_1 + 1 + \frac{\pi}{2} \\ + \cos(y_1) &= x_2 + y_2 + .\end{align*} + ist in einer Umgebung von $(x_1^{0}, x_2^{0}, y_1^{0}, y_2^{0})^{T} = (0, -1, \frac{\pi}{2}, 1)^{T} \in \R^{4}$ durch differenzierbare Funktionen $g_i \colon \R^2 \to \R$ mit + \begin{align*} + y_1 &= g_1(x_1, x_2) \\ + y_2 &= g_2(x_1, x_2) \\ + \end{align*} + eindeutig aufgelöst werden kann. + \begin{proof} + Das Gleichungssystem ist darstellbar als + \begin{align*} + \begin{cases} + y_1 + \sin(y_1 y_2) &= y_1x_1 + 1 + \frac{\pi}{2} \\ + \cos(y_1) &= x_2 + y_2 + \end{cases} + \implies + \begin{cases} + 0 = y_1 + \sin(y_1 y_2) - y_1 x_1 - 1 - \frac{\pi}{2} \\ + 0 = \cos(y_1) - x_2 - y_2 + \end{cases} + .\end{align*} + Mit $x = (x_1, x_2)^{T}$ und $y = (y_1, y_2)^{T}$ definiere + \[ + F(x,y) \coloneqq \begin{pmatrix} + y_1 + \sin(y_1 y_2) - y_1 x_1 - 1 - \frac{\pi}{2} \\ + \cos(y_1) - x_2 - y_2 + \end{pmatrix} + .\] + Damit folgt + \[ + D_yF(x,y) = \begin{pmatrix} 1 + y_2 \cos(y_1 y_2) -x_1 & y_1 \cos(y_1 y_2) \\ + - \sin(y_1) & -1 \end{pmatrix} \quad + D_xF(x,y) = \begin{pmatrix} -y_1 & 0 \\ 0 & -1 \end{pmatrix} + .\] und + \[ + D_yF(x^{0}, y^{0}) = \begin{pmatrix} 1 & 0 \\ -1 & -1 \end{pmatrix} + \quad + D_xF(x^{0}, y^{0}) = \begin{pmatrix} - \frac{\pi}{2} & 0 \\ 0 & -1 \end{pmatrix} + .\] Es ist + \[ + \begin{pmatrix} 1 & 0 \\ -1 & -1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ -1 & -1 \end{pmatrix} + = + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + .\] Also $D_yF(x^{0}, y^{0})^{-1} = D_yF(x^{0}, y^{0})$. Weiter ist + $F(x,y)$ stetig partiell differenzierbar, da alle partiellen Ableitungen stetig sind. + Außerdem gilt $F(x^{0}, y^{0}) = 0$. + + Damit folgt mit dem SIF: Es ex. eine eindeutige diff'bare Funktion $g\colon \R^2 \to \R^2$, für + die in einer Umgebung von $(x^{0}, y^{0})$ gilt: + \[ + F(x, g(x)) = 0 \implies y = g(x) + ,\] also das Gleichungssystem eindeutig auflöst. + Die Komponentenfunktionen $g_1$ und $g_2$ von $g$ zeigen die Behauptung. + \end{proof} + Beh.: $J_g(0, -1) = \begin{pmatrix} \frac{\pi}{2} & 0 \\ -\frac{\pi}{2} & -1 \end{pmatrix} $. + \begin{proof} + Es ist mit dem SIF: + \begin{align*} + J_g(0, -1) = J_g(x^{0}) = D_xg(x^{0}) &= - (D_yF(x^{0}, y^{0}))^{-1} D_xF(x^{0}, y^{0}) \\ + &= - \begin{pmatrix} 1 & 0 \\ -1 & -1 \end{pmatrix} \begin{pmatrix} - \frac{\pi}{2} & 0 \\ 0 & -1 \end{pmatrix} \\ + &= \begin{pmatrix} \frac{\pi}{2} & 0 \\ - \frac{\pi}{2} & -1 \end{pmatrix} + .\end{align*} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + Beh.: Die Gleichung + \[ + \ln(3 \epsilon + x) = \epsilon^2 x + .\] ist in einer Umgebung von $(\epsilon_0, x_0)^{T} = (0,1)^{T}$ eindeutig nach $x$ durch eine + Abbildung $x = g(\epsilon)$ auflösbar. + \begin{proof} + Definiere + \[ + F(\epsilon, x) = \epsilon^2x - \ln(3 \epsilon + x) + .\] Es gilt $F(\epsilon_0, x_0) = F(0,1) = -\ln(1) = 0$. Außerdem ist + \begin{align*} + D_xF(\epsilon, x) &= \frac{\partial F}{\partial x} = \epsilon^2 - \frac{1}{3 \epsilon + x} \\ + D_xF(\epsilon_0, x_0) &= -1 \implies (D_xF(\epsilon_0, x_0))^{-1} = -1 + .\end{align*} + Damit ex. nach dem SIF eine eindeutige Funktion $g$, die die gegebene Gleichung nach $x$ auflöst. + \end{proof} + Beh.: Das Taylor-Polynom 2. Ordnung von $g$ im Punkt $\epsilon_0 = 0$ ist gegeben als + \[ + T_2^{g}(\epsilon_0 + h) = 1 - 3h + 16h^2 + .\] + \begin{proof} + Es ist für $F$ aus der ersten Behauptung + \begin{align*} + \nabla F(\epsilon, x) = + \begin{pmatrix} + 2 \epsilon x - \frac{3}{3 \epsilon + x} \\ + \epsilon^2 - \frac{1}{3 \epsilon + x} + \end{pmatrix} + \quad + H_F(\epsilon, x) = + \begin{pmatrix} + 2 x + \frac{9}{(3 \epsilon + x)^2} & + 2 \epsilon - \frac{3}{(3 \epsilon +x)^2} \\ + 2 \epsilon - \frac{3}{(3 \epsilon +x)^2} & + \frac{1}{(3\epsilon + x)^2} + \end{pmatrix} + .\end{align*} + Ausgewertet an $(\epsilon_0, x_0) = (0, 1)$ folgt + \begin{align*} + \nabla F(\epsilon_0, x_0) = + \begin{pmatrix} + -3 \\ -1 + \end{pmatrix} + \quad + H_F(\epsilon_0, x_0) = + \begin{pmatrix} + 11 & -3 \\ + -3 & 1 + \end{pmatrix} + .\end{align*} + Durch implizites Differenzieren folgt + \[ + g'(\epsilon_0) = - \left( \frac{\partial F}{\partial x}(\epsilon_0, x_0) \right)^{-1} + \frac{\partial F}{\partial \epsilon}(\epsilon_0, x_0) = - (-1) \cdot (-3) = -3 + .\] Es gilt mit $F(\epsilon, g(\epsilon)) = 0$ und $x = g(\epsilon)$: + \[ + 0 = \frac{\d F}{\d x}(\epsilon, g(\epsilon)) = \left( \frac{\partial F}{\partial \epsilon} + \frac{\partial F}{\partial x} \cdot \frac{\partial F}{\partial \epsilon} \right) + .\] Angewendet auf $\frac{\d F(\epsilon, g(\epsilon)}{\d x} = 0$ ergibt + \begin{align*} + 0 &= \frac{\d}{\d x} \left( \frac{\partial F}{\partial \epsilon} + \frac{\partial F}{\partial x} f' \right) \\ + 0 &= \frac{\partial^2}{\partial \epsilon^2} + \frac{\partial^2F}{\partial x \partial \epsilon} + g' + + \left( \frac{\partial^2 F}{\partial \epsilon \partial x} + \frac{\partial^2 F}{\partial x^2}f' \right) f' + \frac{\partial F}{ \partial x} f'' + .\end{align*} + Damit folgt + \begin{align*} + g''(\epsilon_0) &= - \left( \frac{\partial F}{\partial x}(\epsilon_0, x_0) \right)^{-1} + \left( \frac{\partial^2 F}{\partial x^2} + 2 \frac{\partial^2 F}{\partial y \partial x}g' + + \frac{\partial^2 F}{\partial x^2} f'^2\right)\Big|_{(\epsilon,x) = (\epsilon_0, x_0)} \\ + &= - (-1) (11 + 2 (-3) (-3) + 1 \cdot 9) \\ + &= 32 + .\end{align*} + Es ist $F(\epsilon_0, x_0) = 0 \implies x_0 = g(\epsilon_0) \implies g(\epsilon_0) = 1$. Damit folgt + \[ + T_2^{g}(\epsilon_0 + h) = 1 - 3h + \frac{1}{2} \cdot 32 h^2 = 1 - 3h + 16h^2 + .\] + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + Sei $f\colon \R^{3} \to \R$ stetig diff'bar und in einem Punkt $x^{0} = (x_1^{0}, x_2^{0}, x_3^{0})^{T} \in \R^{3}$ gelte + \[ + \prod_{i=1}^{3} \frac{\partial f}{\partial x_i}(x^{0}) \neq 0 + .\] Weiter seien $g_i\colon \R^2 \to \R$ eine lokale Auflösung der Gleichung + \[ + f(x_1, x_2, x_3) = f(x^{0}) + \] nach $x_i$, $i = 1,2,3$. + Beh.: Es gilt + \[ + \frac{\partial g_1}{\partial x_2}(x_2^{0}, x_3^{0}) \cdot + \frac{\partial g_2}{\partial x_3}(x_1^{0}, x_3^{0}) \cdot + \frac{\partial g_3}{\partial x_1}(x_1^{0}, x_2^{0}) = -1 + .\] + \begin{proof} + Es gilt $\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \frac{\partial f}{\partial x_3} \neq 0$. Definiere + \[ + F(x_1, x_2, x_3) \coloneqq f(x_1, x_2, x_3) - f(x^{0}) + .\] $F$ stetig diff'bar, da $f$ stetig diff'bar und $D_xF(x) = D_xf(x)$. Es gilt außerdem + \[ + f(x_1, x_2, x_3) = f(x^{0}) \iff F(x_1, x_2, x_3) = 0 + .\] Also sind $g_i$ auch Auflösungen von $F$ nach $x_i$. Für diese gilt nach SIF: + \[ + D_{x_1 x_2} g_3(x_1^{0}, x_2^{0}) = - (D_{x_3}F(x_0))^{-1} D_{x_1, x_2}F(x^{0}) + .\] analog für $g_1$ und $g_2$. Damit folgt + \begin{align*} + \frac{\partial g_1}{\partial x_2}(x_2^{0}, x_3^{0}) &= - \left( \frac{\partial f(x^{0})}{\partial x_1} \right)^{-1} + \frac{\partial f(x^{0})}{\partial x_2} \\ + \frac{\partial g_2}{\partial x_3}(x_1^{0}, x_3^{0}) &= - \left( \frac{\partial f(x^{0})}{\partial x_2} \right)^{-1} + \frac{\partial f(x^{0})}{\partial x_3} \\ + \frac{\partial g_3}{\partial x_1} (x_1^{0}, x_2^{0}) &= + - \left( \frac{\partial f(x^{0})}{\partial x_3} \right)^{-1} \frac{\partial f(x_0)}{\partial x_1} + .\end{align*} + Insgesamt folgt damit + \begin{align*} + &\frac{\partial g_1}{\partial x_2}(x_2^{0}, x_3^{0}) \cdot + \frac{\partial g_2}{\partial x_3}(x_1^{0}, x_3^{0}) \cdot + \frac{\partial g_3}{\partial x_1} (x_1^{0}, x_2^{0}) \\ =& - +\left( \frac{\partial f(x^{0})}{\partial x_1} \right)^{-1} \frac{\partial f(x^{0})}{\partial x_2} +\cdot \left( \frac{\partial f(x^{0})}{\partial x_2} \right)^{-1} \frac{\partial f(x^{0})}{\partial x_3} +\cdot \left( \frac{\partial f(x^{0})}{\partial x_3} \right)^{-1} \frac{\partial f(x_0)}{\partial x_1} \\ +=& -1 + .\end{align*} + \end{proof} +\end{aufgabe} + +\end{document}