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10) * 1}; \end{axis} \end{tikzpicture} \caption{$f_n(x) = x^{n}$ und ihre Grenzfunktion} @@ -125,7 +126,7 @@ ymin=-0.4, ] \addplot[domain=0:1.1,samples=50,smooth,dashed, blue] {0.3}; - \addplot[domain=0:1.001,samples=1000,smooth,red] {(and(x>=0 , x<1) * 0 + (and(x>=1, x< 10) * 1}; + \addplot[domain=0:1.001,samples=1000,red] {(and(x>=0 , x<1) * 0 + (and(x>=1, x< 10) * 1}; \addplot[domain=0:1,samples=50,smooth,blue] {x^8}; \addplot[domain=0:1.1,samples=50,smooth,dashed, blue] {-0.3}; \end{axis} diff --git a/ana2.pdf b/ana2.pdf index c6a6df467c4f611b89018fad1d543aa99ff94d1f..644a892b2ee5502daec64b10f96d07552f087d99 100644 GIT binary patch delta 76182 zcmV)5K*_)P_zjT33XmlMG?Q@{D1X&jS(Dqw5q{ra0l%ckvM`$aP|6Q-B)KG)6lb+c zsv_xuYht;`l31G{Ek(aRJqL~fhafjg9ez+Pgc3<8P0w> z`|;xJ;WKVBgN&pyyST~}L7eeS35~R**~L72JHd#cX)dLlJb8LP%^Bm9XMb;A{e1EK z@2{S|%FiE9sh+%fIemZe;^8wwdcug{h-n5(IXUNkitaCmBqz=I+k^;;l+?|H$A6vX zM5@V;yMhQj3fWA2?d)YU9uMyew9;s>;jCX8%pGu=((Ew)Ug5ZQ@9CDt`>3LS&HXAX$sgV}>~A 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z%I~~J0e3ompBj}XAt9A-jnXN>r>9Sa+#2-`@NN@%Vf0O<30YXt3t6h)ovH@twXX|5 zD66HU>w)-`#e%r^F|-Uy1rzDn8o+b~T7AMZ#C{fH%29bSAnAKFXdvhoxk3iL+dl*) z6B)Xp42;1PQ67Sz-n87jK5fWsdV^s(x_$DcLCQ|A$~ST5dUQ%d`3-_aQtp!%Hdz{_ zJ1adV76f~^ZgFC#(8j0wP@Zv{^GHv63`eBeE3XD1?mgoBA>mF?fv$rRf{%j>9}o$N zg+}|q^84Y#38BaoA=EpY*m%9XF+Hqep&%~|VA8}u;A(*x@loHsnN*irV6n=H7jZ@cx3(QKI;ZyByziZ5S*VKQ}RlF zT~}#5J3Y1bCLkS&JZ1he@HMkYZ`m|38MzCxP>&$=NtZnnfT0 oF75Dt52YMFJDS?yxVo9RxOuslTj22V2@3InaG06p)f90417pyiZvX%Q diff --git a/ana2.tex b/ana2.tex index 76be396..e9bcc82 100644 --- a/ana2.tex +++ b/ana2.tex @@ -62,7 +62,7 @@ Jetzt: Fourier Analysis! \end{definition} \begin{definition}[Skalarprodukt] - Sei $V$ Vektorraum über $\mathbb{K}$. Die Abbildung $<\cdot, \cdot >\colon V \times V \to \mathbb{K}$ + Sei $V$ Vektorraum über $\mathbb{K}$. Die Abbildung $\langle\cdot, \cdot \rangle\colon V \times V \to \mathbb{K}$ heißt Skalarprodukt auf $V$, falls $\forall u, v, w \in V$ und $\alpha \in \mathbb{K}$ gilt: \begin{enumerate}[(S1)] \item $\langle v, u\rangle = \overline{\langle u, v\rangle}$ (Symmetrie, @@ -150,7 +150,7 @@ Jetzt: Fourier Analysis! \begin{definition}[Konvergenz im Quadratischen Mittel ($L^2$-Konvergenz)] Seien $f_n \in R[a,b], n \in \N, f \in R[a,b]$. $f_n$ konvergiert gegen $f$ im Quadratischen - Mittel $f_n \xrightarrow[L^2]{n \to \infty}$, wenn gilt + Mittel $f_n \xrightarrow[L^2]{n \to \infty}f$, wenn gilt \[ \Vert f_n - f \Vert_{L^2} \xrightarrow{n \to \infty} 0 .\] Das heißt, dass die quadratische Abweichung zwischen $f_n$ und $f$ gegen Null konvergiert: diff --git a/ana20.tex b/ana20.tex index e5c9b33..b2feedc 100644 --- a/ana20.tex +++ b/ana20.tex @@ -199,7 +199,7 @@ \left.\gamma\right|_{[s_{i-1}, s_i]} \in C^1([s_{i-1},s_i], \R^n),\quad \forall i = 1, \dots, M. \] Dann ist $\gamma$ rektifizierbar und hat die Länge \[S(\gamma) = \int_a^b \norm{\gamma'(t)}_2\d t = \sum_{i = 1}^{M}\int_{s_{i-1}}^{s_i} \norm{\gamma'(t)}_2 \d t.\] - Insbesondere hat der Graph einer $C_1$-Funktion $f\in C^1([a,b], \R), \ \gamma_f(t)\coloneqq (t, f(t))^T$ die Länge \[S(\gamma_f) = \int_a^b \sqrt{1 + f'(t)^2} \d t.\] + Insbesondere hat der Graph einer $C^1$-Funktion $f\in C^1([a,b], \R), \ \gamma_f(t)\coloneqq (t, f(t))^T$ die Länge \[S(\gamma_f) = \int_a^b \sqrt{1 + f'(t)^2} \d t.\] \end{satz} \begin{proof} Sei $\mathcal{Z} = \{t_0, \dots, t_N\}$ eine Partition, betrachte $\mathcal{Z}^* = \mathcal{Z} \cup \{s_0,\dots, s_M\} = \{x_0,\dots, x_K\}$. Dann gilt diff --git a/ana21.tex b/ana21.tex index 0421cc8..48756c5 100644 --- a/ana21.tex +++ b/ana21.tex @@ -219,8 +219,8 @@ Dann ist $\gamma$ ein geschlossener Integrationsweg, also folgt \begin{salign*} 0 &= \int_{\gamma}^{} F \\ - &= \int_{-1}^{0} (F(\gamma_1(t)), \gamma_1'(t)) \d t + \int_{0}^{1} (F(\gamma_2(-t), - \gamma_2'(-t)) \d t \\ - &\stackrel{s \coloneqq -t}{=} \int_{\gamma_1}^{} F + \int_{0}^{-1} (F(\gamma_2(s)), \gamma_2'(s) \d s \\ + &= \int_{-1}^{0} (F(\gamma_1(t)), \gamma_1'(t)) \d t + \int_{0}^{1} (F(\gamma_2(-t)), - \gamma_2'(-t)) \d t \\ + &\stackrel{s \coloneqq -t}{=} \int_{\gamma_1}^{} F + \int_{0}^{-1} (F(\gamma_2(s)), \gamma_2'(s)) \d s \\ &= \int_{\gamma_1}^{} F - \int_{\gamma_2}^{} F .\end{salign*} @@ -256,7 +256,7 @@ \int_{\gamma} F = \varphi(x) - \varphi(x_0) .\] Damit folgt \[ - \varphi(x) - \varphi(x_0) = \varphi_0(x) \implies \varphi(x) = \varphi_0(x) + \underbrace{\varphi(x_0)}_{\text{konst}} = \varphi_0(x) + c + \varphi(x) - \varphi(x_0) = \varphi_0(x) \implies \varphi(x) = \varphi_0(x) + \underbrace{\varphi(x_0)}_{\text{konst.}} = \varphi_0(x) + c .\] \end{proof} diff --git a/ana22.tex b/ana22.tex index 523b2d4..8438227 100644 --- a/ana22.tex +++ b/ana22.tex @@ -31,7 +31,7 @@ .\] mit z.B. $\gamma(t) \coloneqq t \begin{pmatrix} x \\ y \end{pmatrix} $, $t \in [0,1]$. Dann gilt \begin{salign*} - \int_{\gamma} F &= \int_{0}^{1} \left( F(\gamma(t), \gamma'(t) \right) \d t \\ + \int_{\gamma} F &= \int_{0}^{1} \left( F(\gamma(t)), \gamma'(t) \right) \d t \\ &= \int_{0}^{1} \left( \begin{pmatrix} ty \\ tx \end{pmatrix} , \begin{pmatrix} x \\ y \end{pmatrix} \right) \d t \\ &= \int_{0}^{1} \left( t y x + tx y \right) \d t \\ &= \int_{0}^{1} 2 t x y \d t \\ @@ -87,24 +87,24 @@ Die Integrabilitätsbedingungen sind nicht hinreichend. Kurven. \begin{enumerate}[(i)] \item Es gelte $\gamma_0(a) = A = \gamma_1(a)$ und - $\gamma_0(b) = B = \gamma_1(b)$. $y_0$ und $y_1$ heißen \underline{homotop} in $D$, falls + $\gamma_0(b) = B = \gamma_1(b)$. $\gamma_0$ und $\gamma_1$ heißen \underline{homotop} in $D$, falls eine stetige Abbildung $H\colon [a,b] \times [0,1] \to D$ (Homotopie) existiert, s.d. - $H(t, 0) = \gamma_0(t)$ und $H(t,1) = \gamma_1(t)$, $\forall t \in [a,b]$ und + $H(t, 0) = \gamma_0(t)$ und $H(t,1) = \gamma_1(t)$, $\forall t \in [a,b]$ sowie $H(a, s) = A$ und $H(b,s) = B$, $\forall s \in [0,1]$. Für $s \in [0,1]$ sind - $\gamma_s^{(t)} \coloneqq H(t,s)$, $t \in [a,b]$ mit $\gamma_s(a) = A$ und $\gamma_s(b) = B$ + $\gamma_s(t) \coloneqq H(t,s)$, $t \in [a,b]$ mit $\gamma_s(a) = A$ und $\gamma_s(b) = B$ stetige Kurven von $A$ nach $B$ in $D$. $H$ heißt stetige Deformation von $\gamma_0$ nach $\gamma_1$. \item $\gamma_0$ und $\gamma_1$ seien geschlossen. $\gamma_0$ und $\gamma_1$ heißen - \underline{frei homotop} in $D$ falls eine stetige Abbildung - $H\colon [a,b] \times [0,1] \to D$ mit $H(t,0) = \gamma_0(t)$, $\forall t \in [a,b]$ und - $H(a,s) = H(b,s)$, $\forall x \in [0,1]$ d.h. für $s \in [0,1]$ sind $\gamma_s(t) \coloneqq H(t,s)$ - geschlossene Kurven in $D$. + \underline{frei homotop} in $D$, falls eine stetige Abbildung + $H\colon [a,b] \times [0,1] \to D$ existiert mit $H(t,0) = \gamma_0(t)$ und $H(t,1) = \gamma_1(t)$, $\forall t \in [a,b]$ und + $H(a,s) = H(b,s)$, $\forall s \in [0,1]$, d.h. für $s \in [0,1]$ ist $\gamma_s(t) \coloneqq H(t,s)$ + eine geschlossene Kurve in $D$. $H$ heißt stetige Deformation innerhalb von $D$ der geschlossenen Kurve $\gamma_0$ nach der geschlossenen Kurve $\gamma_1$. - \item Eine geschlossene Kurve heißt zusammenziehbar in $D$, wenn sie frei homotop zu + \item Eine geschlossene Kurve heißt \underline{zusammenziehbar} in $D$, wenn sie frei homotop zu einer konstanten Kurve ist, d.h. sie sich in $D$ zu einem Punkt zusammenziehen lässt. \end{enumerate} \end{definition} @@ -134,7 +134,7 @@ Die Integrabilitätsbedingungen sind nicht hinreichend. \begin{satz}[Zweiter Hauptsatz der Kurvenintegrale: Homotopieinvarianz] Sei $D \subseteq \R^{n}$ offen, $F \in C^{1}(D, \R^{n})$ erfülle - die Integrabilitätsbedingungen und $\gamma_0, \gamma_1 \colon [a,b] \to D$ Integrationswege. + die Integrabilitätsbedingungen und seien $\gamma_0, \gamma_1 \colon [a,b] \to D$ Integrationswege. 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