diff options
| -rw-r--r-- | .gitignore | 368 | ||||
| -rw-r--r-- | LICENSE.txt | 428 | ||||
| -rw-r--r-- | an2e_zf.pdf | bin | 0 -> 53843 bytes | |||
| -rw-r--r-- | an2e_zf.tex | 202 |
4 files changed, 998 insertions, 0 deletions
diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..daa0042 --- /dev/null +++ b/.gitignore @@ -0,0 +1,368 @@ + +# 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 +Creative Commons public licenses does not create a lawyer-client or +other relationship. Creative Commons makes its licenses and related +information available on an "as-is" basis. Creative Commons gives no +warranties regarding its licenses, any material licensed under their +terms and conditions, or any related information. Creative Commons +disclaims all liability for damages resulting from their use to the +fullest extent possible. + +Using Creative Commons Public Licenses + +Creative Commons public licenses provide a standard set of terms and +conditions that creators and other rights holders may use to share +original works of authorship and other material subject to copyright +and certain other rights specified in the public license below. The +following considerations are for informational purposes only, are not +exhaustive, and do not form part of our licenses. + + Considerations for licensors: Our public licenses are + intended for use by those authorized to give the public + permission to use material in ways otherwise restricted by + copyright and certain other rights. Our licenses are + irrevocable. Licensors should read and understand the terms + and conditions of the license they choose before applying it. + Licensors should also secure all rights necessary before + applying our licenses so that the public can reuse the + material as expected. Licensors should clearly mark any + material not subject to the license. This includes other CC- + licensed material, or material used under an exception or + limitation to copyright. More considerations for licensors: + wiki.creativecommons.org/Considerations_for_licensors + + Considerations for the public: By using one of our public + licenses, a licensor grants the public permission to use the + licensed material under specified terms and conditions. If + the licensor's permission is not necessary for any reason--for + example, because of any applicable exception or limitation to + copyright--then that use is not regulated by the license. Our + licenses grant only permissions under copyright and certain + other rights that a licensor has authority to grant. Use of + the licensed material may still be restricted for other + reasons, including because others have copyright or other + rights in the material. A licensor may make special requests, + such as asking that all changes be marked or described. + Although not required by our licenses, you are encouraged to + respect those requests where reasonable. More considerations + for the public: + wiki.creativecommons.org/Considerations_for_licensees + +======================================================================= + +Creative Commons Attribution-ShareAlike 4.0 International Public +License + +By exercising the Licensed Rights (defined below), You accept and agree +to be bound by the terms and conditions of this Creative Commons +Attribution-ShareAlike 4.0 International Public License ("Public +License"). To the extent this Public License may be interpreted as a +contract, You are granted the Licensed Rights in consideration of Your +acceptance of these terms and conditions, and the Licensor grants You +such rights in consideration of benefits the Licensor receives from +making the Licensed Material available under these terms and +conditions. + + +Section 1 -- Definitions. + + a. Adapted Material means material subject to Copyright and Similar + Rights that is derived from or based upon the Licensed Material + and in which the Licensed Material is translated, altered, + arranged, transformed, or otherwise modified in a manner requiring + permission under the Copyright and Similar Rights held by the + Licensor. For purposes of this Public License, where the Licensed + Material is a musical work, performance, or sound recording, + Adapted Material is always produced where the Licensed Material is + synched in timed relation with a moving image. + + b. Adapter's License means the license You apply to Your Copyright + and Similar Rights in Your contributions to Adapted Material in + accordance with the terms and conditions of this Public License. + + c. BY-SA Compatible License means a license listed at + creativecommons.org/compatiblelicenses, approved by Creative + Commons as essentially the equivalent of this Public License. + + d. Copyright and Similar Rights means copyright and/or similar rights + closely related to copyright including, without limitation, + performance, broadcast, sound recording, and Sui Generis Database + Rights, without regard to how the rights are labeled or + categorized. For purposes of this Public License, the rights + specified in Section 2(b)(1)-(2) are not Copyright and Similar + Rights. + + e. Effective Technological Measures means those measures that, in the + absence of proper authority, may not be circumvented under laws + fulfilling obligations under Article 11 of the WIPO Copyright + Treaty adopted on December 20, 1996, and/or similar international + agreements. + + f. Exceptions and Limitations means fair use, fair dealing, and/or + any other exception or limitation to Copyright and Similar Rights + that applies to Your use of the Licensed Material. + + g. License Elements means the license attributes listed in the name + of a Creative Commons Public License. The License Elements of this + Public License are Attribution and ShareAlike. + + h. Licensed Material means the artistic or literary work, database, + or other material to which the Licensor applied this Public + License. + + i. Licensed Rights means the rights granted to You subject to the + terms and conditions of this Public License, which are limited to + all Copyright and Similar Rights that apply to Your use of the + Licensed Material and that the Licensor has authority to license. + + j. Licensor means the individual(s) or entity(ies) granting rights + under this Public License. + + k. Share means to provide material to the public by any means or + process that requires permission under the Licensed Rights, such + as reproduction, public display, public performance, distribution, + dissemination, communication, or importation, and to make material + available to the public including in ways that members of the + public may access the material from a place and at a time + individually chosen by them. + + l. Sui Generis Database Rights means rights other than copyright + resulting from Directive 96/9/EC of the European Parliament and of + the Council of 11 March 1996 on the legal protection of databases, + as amended and/or succeeded, as well as other essentially + equivalent rights anywhere in the world. + + m. You means the individual or entity exercising the Licensed Rights + under this Public License. Your has a corresponding meaning. + + +Section 2 -- Scope. + + a. License grant. + + 1. Subject to the terms and conditions of this Public License, + the Licensor hereby grants You a worldwide, royalty-free, + non-sublicensable, non-exclusive, irrevocable license to + exercise the Licensed Rights in the Licensed Material to: + + a. reproduce and Share the Licensed Material, in whole or + in part; and + + b. produce, reproduce, and Share Adapted Material. + + 2. Exceptions and Limitations. For the avoidance of doubt, where + Exceptions and Limitations apply to Your use, this Public + License does not apply, and You do not need to comply with + its terms and conditions. + + 3. Term. The term of this Public License is specified in Section + 6(a). + + 4. Media and formats; technical modifications allowed. The + Licensor authorizes You to exercise the Licensed Rights in + all media and formats whether now known or hereafter created, + and to make technical modifications necessary to do so. The + Licensor waives and/or agrees not to assert any right or + authority to forbid You from making technical modifications + necessary to exercise the Licensed Rights, including + technical modifications necessary to circumvent Effective + Technological Measures. For purposes of this Public License, + simply making modifications authorized by this Section 2(a) + (4) never produces Adapted Material. + + 5. Downstream recipients. + + a. Offer from the Licensor -- Licensed Material. Every + recipient of the Licensed Material automatically + receives an offer from the Licensor to exercise the + Licensed Rights under the terms and conditions of this + Public License. + + b. Additional offer from the Licensor -- Adapted Material. + Every recipient of Adapted Material from You + automatically receives an offer from the Licensor to + exercise the Licensed Rights in the Adapted Material + under the conditions of the Adapter's License You apply. + + c. No downstream restrictions. You may not offer or impose + any additional or different terms or conditions on, or + apply any Effective Technological Measures to, the + Licensed Material if doing so restricts exercise of the + Licensed Rights by any recipient of the Licensed + Material. + + 6. No endorsement. Nothing in this Public License constitutes or + may be construed as permission to assert or imply that You + are, or that Your use of the Licensed Material is, connected + with, or sponsored, endorsed, or granted official status by, + the Licensor or others designated to receive attribution as + provided in Section 3(a)(1)(A)(i). + + b. Other rights. + + 1. Moral rights, such as the right of integrity, are not + licensed under this Public License, nor are publicity, + privacy, and/or other similar personality rights; however, to + the extent possible, the Licensor waives and/or agrees not to + assert any such rights held by the Licensor to the limited + extent necessary to allow You to exercise the Licensed + Rights, but not otherwise. + + 2. Patent and trademark rights are not licensed under this + Public License. + + 3. To the extent possible, the Licensor waives any right to + collect royalties from You for the exercise of the Licensed + Rights, whether directly or through a collecting society + under any voluntary or waivable statutory or compulsory + licensing scheme. In all other cases the Licensor expressly + reserves any right to collect such royalties. + + +Section 3 -- License Conditions. + +Your exercise of the Licensed Rights is expressly made subject to the +following conditions. + + a. Attribution. + + 1. If You Share the Licensed Material (including in modified + form), You must: + + a. retain the following if it is supplied by the Licensor + with the Licensed Material: + + i. identification of the creator(s) of the Licensed + Material and any others designated to receive + attribution, in any reasonable manner requested by + the Licensor (including by pseudonym if + designated); + + ii. a copyright notice; + + iii. a notice that refers to this Public License; + + iv. a notice that refers to the disclaimer of + warranties; + + v. a URI or hyperlink to the Licensed Material to the + extent reasonably practicable; + + b. indicate if You modified the Licensed Material and + retain an indication of any previous modifications; and + + c. indicate the Licensed Material is licensed under this + Public License, and include the text of, or the URI or + hyperlink to, this Public License. + + 2. You may satisfy the conditions in Section 3(a)(1) in any + reasonable manner based on the medium, means, and context in + which You Share the Licensed Material. For example, it may be + reasonable to satisfy the conditions by providing a URI or + hyperlink to a resource that includes the required + information. + + 3. If requested by the Licensor, You must remove any of the + information required by Section 3(a)(1)(A) to the extent + reasonably practicable. + + b. ShareAlike. + + In addition to the conditions in Section 3(a), if You Share + Adapted Material You produce, the following conditions also apply. + + 1. The Adapter's License You apply must be a Creative Commons + license with the same License Elements, this version or + later, or a BY-SA Compatible License. + + 2. You must include the text of, or the URI or hyperlink to, the + Adapter's License You apply. You may satisfy this condition + in any reasonable manner based on the medium, means, and + context in which You Share Adapted Material. + + 3. You may not offer or impose any additional or different terms + or conditions on, or apply any Effective Technological + Measures to, Adapted Material that restrict exercise of the + rights granted under the Adapter's License You apply. + + +Section 4 -- Sui Generis Database Rights. + +Where the Licensed Rights include Sui Generis Database Rights that +apply to Your use of the Licensed Material: + + a. for the avoidance of doubt, Section 2(a)(1) grants You the right + to extract, reuse, reproduce, and Share all or a substantial + portion of the contents of the database; + + b. if You include all or a substantial portion of the database + contents in a database in which You have Sui Generis Database + Rights, then the database in which You have Sui Generis Database + Rights (but not its individual contents) is Adapted Material, + + including for purposes of Section 3(b); and + c. You must comply with the conditions in Section 3(a) if You Share + all or a substantial portion of the contents of the database. + +For the avoidance of doubt, this Section 4 supplements and does not +replace Your obligations under this Public License where the Licensed +Rights include other Copyright and Similar Rights. + + +Section 5 -- Disclaimer of Warranties and Limitation of Liability. + + a. UNLESS OTHERWISE SEPARATELY UNDERTAKEN BY THE LICENSOR, TO THE + EXTENT POSSIBLE, THE LICENSOR OFFERS THE LICENSED MATERIAL AS-IS + AND AS-AVAILABLE, AND MAKES NO REPRESENTATIONS OR WARRANTIES OF + ANY KIND CONCERNING THE LICENSED MATERIAL, WHETHER EXPRESS, + IMPLIED, STATUTORY, OR OTHER. THIS INCLUDES, WITHOUT LIMITATION, + WARRANTIES OF TITLE, MERCHANTABILITY, FITNESS FOR A PARTICULAR + PURPOSE, NON-INFRINGEMENT, ABSENCE OF LATENT OR OTHER DEFECTS, + ACCURACY, OR THE PRESENCE OR ABSENCE OF ERRORS, WHETHER OR NOT + KNOWN OR DISCOVERABLE. WHERE DISCLAIMERS OF WARRANTIES ARE NOT + ALLOWED IN FULL OR IN PART, THIS DISCLAIMER MAY NOT APPLY TO YOU. + + b. TO THE EXTENT POSSIBLE, IN NO EVENT WILL THE LICENSOR BE LIABLE + TO YOU ON ANY LEGAL THEORY (INCLUDING, WITHOUT LIMITATION, + NEGLIGENCE) OR OTHERWISE FOR ANY DIRECT, SPECIAL, INDIRECT, + INCIDENTAL, CONSEQUENTIAL, PUNITIVE, EXEMPLARY, OR OTHER LOSSES, + COSTS, EXPENSES, OR DAMAGES ARISING OUT OF THIS PUBLIC LICENSE OR + USE OF THE LICENSED MATERIAL, EVEN IF THE LICENSOR HAS BEEN + ADVISED OF THE POSSIBILITY OF SUCH LOSSES, COSTS, EXPENSES, OR + DAMAGES. WHERE A LIMITATION OF LIABILITY IS NOT ALLOWED IN FULL OR + IN PART, THIS LIMITATION MAY NOT APPLY TO YOU. + + c. The disclaimer of warranties and limitation of liability provided + above shall be interpreted in a manner that, to the extent + possible, most closely approximates an absolute disclaimer and + waiver of all liability. + + +Section 6 -- Term and Termination. + + a. This Public License applies for the term of the Copyright and + Similar Rights licensed here. However, if You fail to comply with + this Public License, then Your rights under this Public License + terminate automatically. + + b. Where Your right to use the Licensed Material has terminated under + Section 6(a), it reinstates: + + 1. automatically as of the date the violation is cured, provided + it is cured within 30 days of Your discovery of the + violation; or + + 2. upon express reinstatement by the Licensor. + + For the avoidance of doubt, this Section 6(b) does not affect any + right the Licensor may have to seek remedies for Your violations + of this Public License. + + c. For the avoidance of doubt, the Licensor may also offer the + Licensed Material under separate terms or conditions or stop + distributing the Licensed Material at any time; however, doing so + will not terminate this Public License. + + d. Sections 1, 5, 6, 7, and 8 survive termination of this Public + License. + + +Section 7 -- Other Terms and Conditions. + + a. The Licensor shall not be bound by any additional or different + terms or conditions communicated by You unless expressly agreed. + + b. Any arrangements, understandings, or agreements regarding the + Licensed Material not stated herein are separate from and + independent of the terms and conditions of this Public License. + + +Section 8 -- Interpretation. + + a. For the avoidance of doubt, this Public License does not, and + shall not be interpreted to, reduce, limit, restrict, or impose + conditions on any use of the Licensed Material that could lawfully + be made without permission under this Public License. + + b. To the extent possible, if any provision of this Public License is + deemed unenforceable, it shall be automatically reformed to the + minimum extent necessary to make it enforceable. If the provision + cannot be reformed, it shall be severed from this Public License + without affecting the enforceability of the remaining terms and + conditions. + + c. No term or condition of this Public License will be waived and no + failure to comply consented to unless expressly agreed to by the + Licensor. + + d. Nothing in this Public License constitutes or may be interpreted + as a limitation upon, or waiver of, any privileges and immunities + that apply to the Licensor or You, including from the legal + processes of any jurisdiction or authority. + + +======================================================================= + +Creative Commons is not a party to its public +licenses. Notwithstanding, Creative Commons may elect to apply one of +its public licenses to material it publishes and in those instances +will be considered the “Licensor.” The text of the Creative Commons +public licenses is dedicated to the public domain under the CC0 Public +Domain Dedication. Except for the limited purpose of indicating that +material is shared under a Creative Commons public license or as +otherwise permitted by the Creative Commons policies published at +creativecommons.org/policies, Creative Commons does not authorize the +use of the trademark "Creative Commons" or any other trademark or logo +of Creative Commons without its prior written consent including, +without limitation, in connection with any unauthorized modifications +to any of its public licenses or any other arrangements, +understandings, or agreements concerning use of licensed material. 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}
\ No newline at end of file |