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b/ana14.tex index 4c595f0..51b7a0a 100644 --- a/ana14.tex +++ b/ana14.tex @@ -294,11 +294,31 @@ Fragestellung: Sei $f \colon D\subset \R^n \to \R^n$. Existiert die Umkehrabbild \end{satz} \begin{proof} Sei $\hat x \in D$ und $\hat y \coloneqq f(\hat x)\in f(D)$. + \begin{figure} + \centering + \begin{tikzpicture} + \coordinate (x) at (-.5,-.5); + \coordinate (y) at (5.5,-.5); + \draw[color = black] (0,0) circle (2cm); + \node at (0,1.5) {$D$}; + \draw[thick, color = red] (x) circle (1cm); + \draw[color = blue, fill = blue!20!white] (-.5,-.5) circle (.7cm); + \node[color = red] at (1,0) {$U(\hat x)$}; + \draw[->] (2,0) -- node[pos = 0.6, above] {$f$} (4,0); + \draw[color = black] (6,0) circle (2cm); + \node at (6,1.5) {$f(D)$}; + \draw[thick, color = red, fill = blue!20!white] (y) circle (1cm); + \node[color = red] at (7,0) {$U(\hat y)$}; + \node[color = blue] at (-.6,-.2) {$V(\hat x)$}; + \node [fill=black,inner sep=1pt,circle,label=-45:$\hat x$] at (x) {}; + \node [fill=black,inner sep=1pt,circle,label=0:$\hat y$] at (y) {}; + \end{tikzpicture} + \end{figure} Betrachte $F \colon \R^n \times D \to \R^n, F(y,x) = y-f(x)$. Für $(\hat x, \hat y)$ gilt $F(\hat y, \hat x) = 0$. Die Jacobimatrix $D_xF(y,x) = -J_f(x)$ ist regulär in $\hat x$. Mit Vertauschung von $x$ und $y$ folgt aus dem Satz über implizite Funktionen, dass Umgebungen $U(\hat y)$, $U(\hat x)$ und genau eine stetig differenzierbare Abbildung $g:U(\hat y) \to U(\hat x)$ existieren, sodass $\forall y \in U(\hat y)$ \[ 0 = F(y,g(y)) = y-f(g(y)), \implies \exists! x = g(y)\in U(\hat x) \text{ mit } y= f(x) \] - Setze dann \[V(\hat x) \coloneqq U(\hat x) \cap f^{-1}(U(\hat y)) = \{ x\in U(\hat x)\mid f(x) \in U(\hat)y\}.\] + Setze dann \[V(\hat x) \coloneqq U(\hat x) \cap f^{-1}(U(\hat y)) = \{ x\in U(\hat x)\mid f(x) \in U(\hat y)\}.\] Da $U(\hat x)$ und $f^{-1}(U(\hat y))$ offen sind, ist auch $V(\hat x)$ offen. Daher ist $f\colon V(\hat x)\to U(\hat y)$ bijektiv und die Umkehrabbildung $f^{-1}\colon U(\hat y) \to V(\hat x)$ ist gerade $g$. \begin{align*} J_{f\circ f^{-1}}(\cdot) = J_{\mathrm{id}}(\cdot) &= \mathbb{I} diff --git a/analysisII.pdf b/analysisII.pdf index 65a9816a6ed86799ea3f2d961a1a4b67ee95042d..63ba4c5c69d8d6c910a8e365f97ade945e7b3dbb 100644 GIT binary patch delta 23904 zcmX`Sb8uk6(*GUX+1O5YW83z|wrwY8V|!z4Y}>YNZfslc{_cJ5TXp`J>Y49nrcTw& z^y%(1_Q1IP$@nV>GzcsRJP4uzQd0?ng^PtF1(Xz$2Ebo;T5r1D)bG8coDQPCOmTai ziAu~Qms6s0Ba)wEAzN?A6Hn0Av|ID}(ENo3LM|!YbZ&AuF@OpY7&*A*UVHCOWea*CUuuBN6_=_jyvOn;<4Dosg$z5koQORTd_rvvhNc@BHIHK5Ix{n8^#crY4Ru z2nc*l19yp3pv1qcA(=}^Ryn&+;N;hDf>{$Jh9awSN&Yd0 zvV;_o$S4d}v|N5F7V{txbB9{IrwVeqVz2r)-PJe|*r5HN#eBA%bo7`o zz-Yq658{Ysn3oS0H`D;v!!n!p}agyTHp^uc3Hl;y&t} zD^Q1^@B!2kv+sGOf9!G~@&`oxrueo%f$g1{K`WlW{8ix@2@}~Tdy-P&!b;jFg7_Xh zRRuI*(QQg#2Gj^3#A%ukJ;e$_#0zapXh|z=i9v~bc*r&f`og;kH6n9v>Jln;0Q0aA 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