From bec744b47598e3420af0e67aa881733deb5702f8 Mon Sep 17 00:00:00 2001 From: henrydatei Date: Wed, 1 Aug 2018 10:42:00 +0200 Subject: [PATCH] added wichtige Sachen, die in der ANAG Klausur wichtig waren --- .../Wichtige Grundlagen der Analysis.pdf | Bin 171215 -> 176845 bytes .../Wichtige Grundlagen der Analysis.tex | 25 ++++++++++++++++++ 2 files changed, 25 insertions(+) diff --git a/2. Semester/Summray ANAG/Wichtige Grundlagen der Analysis.pdf b/2. 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Semester/Summray ANAG/Wichtige Grundlagen der Analysis.tex b/2. Semester/Summray ANAG/Wichtige Grundlagen der Analysis.tex index 2867a55..8cd243a 100644 --- a/2. Semester/Summray ANAG/Wichtige Grundlagen der Analysis.tex +++ b/2. Semester/Summray ANAG/Wichtige Grundlagen der Analysis.tex @@ -155,6 +155,31 @@ &= \arctan(x+1)\notag \end{align} +\subsection{Mehrfachintegrale} +\begin{align} + \int_{X\times Y\times Z} f(x,y,z)\,\mathrm{d}(x,y,z) = \int_X\int_Y\int_Z f\,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x\notag +\end{align} + +\subsection{Der Transformationssatz} +\begin{enumerate}[label=\textbf{\arabic*.}] + \item $f:V\subset \mathbb{R}^n\to\mathbb{R}$ integrierbar + \item $\phi:U\subset\mathbb{R}^n\to V$ Diffeomorphismus +\end{enumerate} +\begin{align} + \int_V f(x)\,\mathrm{d}x=\int_U f(\phi(y))\cdot\vert\phi(y)'\vert\,\mathrm{d}y\notag +\end{align} + +\subsection{parametrisierte Integrale} +\begin{enumerate}[label=\textbf{\arabic*.}] + \item $M\subset\mathbb{R}^n$ messbar, $P\subset\mathbb{R}^m$ Menge von Parametern offen, $f:M\times P\to\mathbb{R}$ + \item $f(\cdot, p)$ integrierbar auf $M$ $\forall p$ + \item $f(x,\cdot)$ stetig differenzierbar auf $P$ $\forall x$ + \item $\exists g:M\to\mathbb{R}$ integrierbar mit $\vert f_p(x,p)\vert\le g$ $\forall x,p$ +\end{enumerate} +\begin{align} + F(p)=\int_M f(x,p)\,\mathrm{d}x\Rightarrow F'(p)=\int_M f_p(x,p)\,\mathrm{d}x\notag +\end{align} + \section{Extremwerte} \subsection{ohne Nebenbedingung} \begin{enumerate}[label=\textbf{\arabic*.}]