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For +the avoidance of doubt, this paragraph does not form part of the +public licenses. + +Creative Commons may be contacted at creativecommons.org. + diff --git a/an2e_zf.pdf b/an2e_zf.pdf Binary files differnew file mode 100644 index 0000000..76358e9 --- /dev/null +++ b/an2e_zf.pdf diff --git a/an2e_zf.tex b/an2e_zf.tex new file mode 100644 index 0000000..16812c5 --- /dev/null +++ b/an2e_zf.tex @@ -0,0 +1,202 @@ +\documentclass[a4paper]{article} + +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{bm} + +\numberwithin{equation}{subsection} + +\usepackage{float} +\usepackage{array} +\usepackage{booktabs} +\usepackage{rotating} + +\usepackage[margin=2cm, bottom=2cm, top=2cm, marginpar=0pt]{geometry} +\usepackage{graphicx} +\usepackage{xcolor} + +%\usepackage{showframe} + +%\usepackage{tikz} +%\usepackage{tikz-3dplot} +%\usepackage{pgfplots} +%\pgfplotsset{compat=1.15} + +\usepackage{multicol} + +\usepackage[colorlinks = true, + linkcolor = red!50!black, + urlcolor = blue, + citecolor = black, + anchorcolor = blue]{hyperref} + + +\usepackage{polyglossia} +\setdefaultlanguage[variant=swiss]{german} + + +\title{Analysis 2 Zusammenfassung} +\author{Naoki Pross} +\date{Fr\"uhlingsstemester 2020} + + +\newcommand{\dd}[1]{\ensuremath{~\mathrm{d}#1}} +\newcommand{\deriv}[2]{\ensuremath{\frac{\dd{#1}}{\dd{#2}}}} +\newcommand{\pderiv}[2]{\ensuremath{\frac{\partial#1}{\partial#2}}} +\renewcommand{\vec}[1]{\ensuremath{\bm{#1}}} + +\newcommand{\brpage}[1]{\textcolor{red!70!black}{\small\texttt{S#1}}} + +\begin{document} + +\begin{multicols}{2} +\section{Integration \brpage{493,507}} +\subsection{Tricks \brpage{495}} +Linearit\"at \brpage{495} +\[ + \int k(u + v) = k\left(\int u + \int v\right) +\] +Partialbruchzerlegung \brpage{15,498} +\[ + \int \frac{f(x)}{P_n(x)} \dd{x} = \sum_{k=1}^n \int \frac{A_k}{x-r_k}\dd{x} +\] +Elementartransformation \brpage{496} +\[ + \int f(\lambda x + \ell) \dd{x} = \frac{1}{\lambda} F(\lambda x + \ell) + C +\] +Partielle Integration \brpage{497} +\[ + \int u \dd{v} = uv - \int v \dd{u} +\] +Potenzenregel \brpage{496} +\[ + \int u^n \cdot u' = \frac{u^{n+1}}{n+1} + C \qquad n \neq -1 +\] +Logaritmusregel \brpage{496} +\[ + \int \frac{u'}{u} = \ln|u| + C +\] +Allgemeine Substutution \brpage{497}\\ + \(x = g(u)\), und \(\dd{x} = g'(u)\dd{u}\) +\[ + \int f(x) \dd{x} = \int (f\circ g) ~ g' \dd{u} = \int \frac{f \circ g}{(g^{-1})'\circ g} \dd{u} +\] +Universalsubstitution \brpage{504} +\begin{align*} + t &= \tan(x/2) & \sin(x) &= \frac{2t}{1+t^2} \\ + \dd{x} &= \frac{2\dd{t}}{1+t^2} & \cos(t) &= \frac{1-t^2}{1+t^2} +\end{align*} +Womit +\[ + \int f(\sin(x), \cos(x), \tan(x)) \dd{x} = \int g(t) \dd{t} +\] + +\subsection{Uneigentliches Integral \brpage{520}} +\begin{align*} + \int\limits_a^\infty f \dd{x} &= \lim_{B \to \infty} \int\limits_a^B f \dd{x} \\ + \int\limits_{-\infty}^b f \dd{x} &= \lim_{A \to -\infty} \int\limits_A^b f \dd{x} \\ + \int\limits_{-\infty}^\infty f \dd{x} &= \lim_{\substack{A \to +\infty \\ B \to -\infty}} \int\limits_A^B f \dd{x} +\end{align*} +Wenn \(f\) im Punkt \(u \in (a,b)\) nicht definiert ist. +\begin{equation} \label{eqn:int-with-pole} + \int\limits_a^b f \dd{x} = + \lim_{\epsilon\to +0} \int\limits_a^{u-\epsilon} f \dd{x} + + \lim_{\delta\to +0} \int\limits_{u+\delta}^b f \dd{x} +\end{equation} + +\subsection{Cauchy Hauptwert \brpage{523}} +Der C.H. (oder PV f\"ur \emph{Principal Value} auf Englisch) eines uneigentlichen Integrals ist der Wert, wenn in einem Integral wie \eqref{eqn:int-with-pole} beide Grenzwerte mit der gleiche Geschwindigkeit gegen 0 sterben. +\[ + \text{C.H.} \int\limits_a^b f \dd{x} = + \lim_{\epsilon\to +0} \left( \int\limits_a^{u-\epsilon} f \dd{x} + + \int\limits_{u+\epsilon}^b f \dd{x} \right) +\] +Zum Beispiel \(x^{-1}\) ist nicht \"uber \(\mathbb{R}\) integrierbar, wegen des Poles bei 0. Aber intuitiv wie die Symmetrie vorschlagt +\[ + \text{C.H.} \int\limits^\infty_{-\infty} \frac{1}{x} \dd{x} = 0 +\] + +\subsection{Majorant-, Minorantenprinzip und Konvergenzkriterien \brpage{521,473,479,481}} + +Gilt f\"ur die Funktionen \(0 < f(x) \leq g(x)\) mit \(x \in [a,\infty)\) +\[ + \text{konvergiert } \int\limits_a^\infty g \dd{x} + \implies \text{ konvergiert } \int\limits_a^\infty f \dd{x} +\] +Die selbe gilt umgekehrt f\"ur Divergenz. Wenn \(0 < h(x) \leq f(x)\) +\[ + \text{divergiert } \int\limits_a^\infty h \dd{x} + \implies \text{ divergiert } \int\limits_a^\infty f \dd{x} +\] +\(g\) und \(h\) hei{\ss}en Majorant und Minorant bzw. + +\section{Implizite Ableitung \brpage{448}} +Alle normale differenziazionsregeln gelten. +\[ + \dd{y} = y'\dd{x} +\] +%Allgemeiner f\"ur die implizite Funktion \(F(x,y) = 0\) +%\[ +% \pderiv{F}{x} + \pderiv{F}{y} y' = 0 +%\] + +\end{multicols} + +\section{Ebene \brpage{250} und Raumkurven \brpage{263}} +\begin{sideways} +\centering +\renewcommand{\arraystretch}{3.5} +\begin{tabular}{l *{3}{>{\(\displaystyle}l<{\)}} } +\toprule +\textbf{Ebene Kurven} & \textbf{Explizit} & \textbf{Polar} & \textbf{Parameter} \\ +\midrule +Bogenl\"ange \brpage{251} + & \int\limits_a^b \sqrt{1 + (y')^2} \dd{x} + & \int\limits_\alpha^\beta \sqrt{(r')^2 + r^2} \dd{\varphi} + & \int\limits_{t_0}^{t_1} \sqrt{\dot{x}^2 + \dot{y}^2} \dd{t} = \int\limits_{t_0}^{t_1} |\vec{c}| \dd{t} +\\ +Fl\"ache + & \int\limits_a^b |f(x)| \dd{x} + & \frac{1}{2}\int\limits_\alpha^\beta r(\varphi)^2 \dd{\varphi} + & \frac{1}{2}\int\limits_{t_0}^{t_1} x\dot{y} - \dot{x}y \dd{t} = \frac{1}{2}\int\limits_{t_0}^{t_1}\det(\vec{c},\dot{\vec{c}}) \dd{t} +\\ +Rotationsvolumen um \(x\) + & \pi \left|\int\limits_a^b y^2 \dd{x} \right| + & \pi \left|\int\limits_{t_0}^{t_1} y \dot{x} \dd{t} \right| + & \pi \left|\int\limits_\alpha^\beta r^2 \sin^2 \varphi (r'\cos\varphi - r\sin\varphi) \dd{\varphi} \right| +\\ +Rotationsoberfl\"ache um \(x\) + & 2\pi \int\limits_a^b |y| \sqrt{1 + (y')^2} \dd{x} + & 2\pi \int\limits_\alpha^\beta |r\sin(\varphi)| \sqrt{(r')^2 + r^2} \dd{\varphi} + & 2\pi \int\limits_{t_0}^{t_1} |y| \sqrt{\dot{x}^2 + \dot{y}^2} \dd{t} +\\ +% Rotationsvolumen um \(y\) \\ +% Rotationsoberfl\"ache um \(y\) \\ +\bottomrule +\end{tabular} +\end{sideways} + +\subsection{Kr\"ummung} +\[ + \kappa = \deriv{\alpha}{s} = \frac{\ddot{y}}{(1+\dot{y}^2)^{3/2}} +\] + +\[ + \det(\vec{\dot{c}}, \vec{\ddot{c}})\,|\vec{c}|^{-3} + \quad \stackrel{\text{3D}}{=} \quad + |\vec{\dot{c}}\times\vec{\ddot{c}}|\,|\vec{c}|^{-3} +\] + +\section*{License} +{ \tt +An2E-ZF (c) by Naoki Pross +\\\\ +An2E-ZF is licensed under a Creative Commons Attribution-ShareAlike 4.0 Unported License. +\\\\ +You should have received a copy of the license along with this work. If not, see +\\\\ +\url{http://creativecommons.org/licenses/by-sa/4.0/} +} + + +\end{document}
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