Add current work to version control

4 files changed, 998 insertions, 0 deletions

diff --git a/.gitignore b/.gitignore
new file mode 100644
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--- /dev/null
+++ b/.gitignore
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+
+# Created by https://www.gitignore.io/api/vim,tex,linux,macos,windows
+# Edit at https://www.gitignore.io/?templates=vim,tex,linux,macos,windows
+
+### Linux ###
+*~
+
+# temporary files which can be created if a process still has a handle open of a deleted file
+.fuse_hidden*
+
+# KDE directory preferences
+.directory
+
+# Linux trash folder which might appear on any partition or disk
+.Trash-*
+
+# .nfs files are created when an open file is removed but is still being accessed
+.nfs*
+
+### macOS ###
+# General
+.DS_Store
+.AppleDouble
+.LSOverride
+
+# Icon must end with two \r
+Icon

+
+# Thumbnails
+._*
+
+# Files that might appear in the root of a volume
+.DocumentRevisions-V100
+.fseventsd
+.Spotlight-V100
+.TemporaryItems
+.Trashes
+.VolumeIcon.icns
+.com.apple.timemachine.donotpresent
+
+# Directories potentially created on remote AFP share
+.AppleDB
+.AppleDesktop
+Network Trash Folder
+Temporary Items
+.apdisk
+
+### TeX ###
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Comment the next line if you want to keep your tikz graphics files
+*.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices
+*.xyc
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# KBibTeX
+*~[0-9]*
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+### TeX Patch ###
+# glossaries
+*.glstex
+
+### Vim ###
+# Swap
+[._]*.s[a-v][a-z]
+[._]*.sw[a-p]
+[._]s[a-rt-v][a-z]
+[._]ss[a-gi-z]
+[._]sw[a-p]
+
+# Session
+Session.vim
+Sessionx.vim
+
+# Temporary
+.netrwhist
+
+# Auto-generated tag files
+tags
+
+# Persistent undo
+[._]*.un~
+
+# Coc configuration directory
+.vim
+
+### Windows ###
+# Windows thumbnail cache files
+Thumbs.db
+Thumbs.db:encryptable
+ehthumbs.db
+ehthumbs_vista.db
+
+# Dump file
+*.stackdump
+
+# Folder config file
+[Dd]esktop.ini
+
+# Recycle Bin used on file shares
+$RECYCLE.BIN/
+
+# Windows Installer files
+*.cab
+*.msi
+*.msix
+*.msm
+*.msp
+
+# Windows shortcuts
+*.lnk
+
+# End of https://www.gitignore.io/api/vim,tex,linux,macos,windows
diff --git a/LICENSE.txt b/LICENSE.txt
new file mode 100644
index 0000000..a73481c
--- /dev/null
+++ b/LICENSE.txt
@@ -0,0 +1,428 @@
+Attribution-ShareAlike 4.0 International
+
+=======================================================================
+
+Creative Commons Corporation ("Creative Commons") is not a law firm and
+does not provide legal services or legal advice. Distribution of
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+
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+
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+Section 8 -- Interpretation.
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+
+  c. No term or condition of this Public License will be waived and no
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+
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+     as a limitation upon, or waiver of, any privileges and immunities
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+
+
+=======================================================================
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diff --git a/an2e_zf.pdf b/an2e_zf.pdf
new 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}
\ No newline at end of file