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Formelsammlung HSD
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| % Formelsammlung_HSD.tex -- Formelsammlung | |
| % | |
| % Copyright (C) 2017 Paul Goetzinger <paul70079@gmail.com> | |
| % | |
| % SPDX-License-Identifier: CC-BY-ND-4.0 | |
| \documentclass[ | |
| a4paper, %% defines the paper size: a4paper (default) | |
| twoside, | |
| titlepage, %% only the titlepage (using titlepage-environment) appears on the first page (alternatively: notitlepage) | |
| % parskip, %% insert an empty line between two paragraphs (alternatively: halfparskip, ...) | |
| 12pt %% set default font size to 12 point | |
| ]{scrartcl} %% article, see KOMA documentation (scrguide.dvi) | |
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| %%% use it only for multi-language documents or non-german ones | |
| \usepackage[naustrian]{babel} | |
| %%% inputenc: coding of german special characters | |
| \usepackage[utf8]{inputenc} | |
| %%% fontenc, ae, aecompl: coding of characters in PDF documents | |
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| %\usepackage{amsmath,amssymb,amstext} | |
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| \usepackage[fleqn]{amsmath} | |
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| \usepackage{tikz,pgfplots} | |
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| \usetikzlibrary{arrows} | |
| \usepgfplotslibrary{dateplot} | |
| \usetikzlibrary{shapes.symbols,shapes.geometric} | |
| \usetikzlibrary{positioning} | |
| %%% scrpage2: KOMA heading and footer | |
| %%% Note: if you don't use this package, please remove | |
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| \usepackage[automark]{scrpage2} | |
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| imagemodifier={-eu},type={CC}, | |
| modifier={by-nc-sa}, | |
| version={4.0} | |
| ]{doclicense} | |
| %%% graphicx: support for graphics | |
| % \usepackage[pdftex]{graphicx} | |
| \usepackage[% | |
| %%% general options | |
| % | |
| %%% extension options bibliography | |
| colorlinks=true, %% turn on colored links (true is better for on-screen reading, false is better for printout versions) | |
| linkcolor=mygray, | |
| urlcolor=blue, | |
| % | |
| %%% PDF-specific display options | |
| bookmarks=true, %% if true, generate PDF bookmarks (requires two passes of pdflatex) | |
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| bookmarksnumbered=false, %% if true, add the section numbers to the bookmarks | |
| %pdfstartpage={1}, %% determines, on which page the PDF file is opened (show thumbnails), FullScreen | |
| ]{hyperref} | |
| %%% provide all graphics (also) in this format, so you don't have | |
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| %%% and/or you don't have to distinguish between generating | |
| %%% dvi/ps (through latex) and pdf (through pdflatex) | |
| \DeclareGraphicsExtensions{.pdf} | |
| %%% sets the PDF-Information options | |
| %%% (see fields in Acrobat Reader: ``File -> Document properties -> Summary'') | |
| %%% Note: this method is better than as options of the hyperref-package (options are expanded correctly) | |
| \hypersetup{ | |
| pdftitle={Formelsammlung HSD}, %% | |
| pdfauthor={Paul Götzinger}, %% | |
| pdfsubject={}, %% | |
| pdfcreator={}, %% | |
| pdfproducer={}, %% | |
| pdfkeywords={} %% | |
| } | |
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| % \subject{} %% subject which appears above titlehead | |
| % \titlehead{} %% special heading for the titlepage | |
| %%% title | |
| \title{Formelsammlung HSD} | |
| %%% author(s) | |
| \author{Paul Götzinger % \and | |
| % Second Author (Matrikelnummer) \and | |
| % Third Author (Matrikelnummer) | |
| \thanks{Erik Kornfellner, Lukas Ebenstein, Stefan Wohlrab} %% use it instead of footnotes (only on titlepage) | |
| } | |
| %%% date | |
| \date{Hagenberg, am \today{}} | |
| \publishers{\small Lizenziert unter \doclicenseLongNameRef \\ \href{https://gist.github.com/paul70078/930925f7b38154c11fdff3567d599403}{Formelsammlung\_HSD.tex}} | |
| % \dedication{} %% generates a dedication-page after titlepage | |
| %%% uncomment following lines, if you want to: | |
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| \begin{document} | |
| % \pagenumbering{roman} %% small roman page numbers | |
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| %%% structure: \section \subsection \subsubsection \paragraph \subparagraph | |
| %%% | |
| \section{Allgemeines} | |
| $\mathbb{N}$ ... Menge der natürlichen Zahlen: Ganze Zahlen $>0$ | |
| $\mathbb{N}_0$ ... Menge der natürlichen Zahlen: Ganze Zahlen $\ge 0$ | |
| $\mathbb{Z}$ ... Menge der ganzen Zahlen | |
| $\mathbb{Q}$ ... Menge der rationalen Zahlen: Ganze Zahlen und Brüche | |
| $\mathbb{R}$ ... Menge der reellen Zahlen: Rationale Zahlen und alle weiteren Kommazahlen | |
| \subsection{Quadratische Lösung} | |
| \begin{align*} | |
| &x^2 + px + q = 0 \quad x_{1,2}=\frac{-p} 2 \pm \sqrt{(\frac p 2)^2-q} \\ | |
| &a x^2 + b x + c = 0 \quad x_{1,2}=\frac{-b\pm \sqrt{b^2-4 ac}}{2a} | |
| \end{align*} | |
| \subsubsection{Stefans magische Formel} | |
| \begin{align*} | |
| a x^2 \pm b x + c = \left(\sqrt{a}x \pm \frac{b}{2\sqrt{a}}\right)^2 + \left(c-\frac{b^2}{4a}\right) | |
| \end{align*} | |
| \subsection{Allgemeine Gerade} | |
| Allgemeine Gerade: | |
| \begin{align*} | |
| &y=k \cdot x+d \\ | |
| &k=\frac{\Delta y}{\Delta x} \\ | |
| \end{align*} | |
| \subsection{Kreisgleichung} | |
| \begin{align*} | |
| (x-a)^2+(y-b)^2=r^2 | |
| \end{align*} | |
| a ... Mittelpunkt auf der x-Achse | |
| b ... Mittelpunkt auf der y-Achse | |
| r ... Radius des Kreises | |
| %\subsection{Polynomdivision} | |
| %Still in development | |
| \subsection{Partialbruchzerlegung} | |
| \begin{align*} | |
| \frac{P(x)}{Q(x)} | |
| \end{align*} | |
| $Q(x)$ ist das Produkt von Faktoren der Form: | |
| \begin{align*} | |
| &ax+b \text{ ... einfache Linearfaktoren} \\ | |
| &(cx+d)^n \text{ ... n-fache Linearfaktoren} \\ | |
| & px^2+qx+r \text{ ... irreduzibles Polynom für } q^2 < 4pr | |
| \end{align*} | |
| $\frac{P(x)}{Q(x)}$ ist dann die Summe folgender Partialbrüche: | |
| \begin{align*} | |
| &\frac{k}{ax+b} \text{ für jeden einfachen Linearfaktor} \\ | |
| &\frac{L_1}{cx+d} + \frac{L_2}{(cx+d)^2} + \cdots + \frac{L_n}{(cx+d)^n} \text{ für jeden n-fachen Linearfaktor} \\ | |
| &\frac{Mx + N}{px^2+qx+r} \text{ für jedes irreduzibles Polynom} | |
| \end{align*} | |
| \begin{enumerate} | |
| \item Wenn der Grad von $P(x)$ kleiner als der von $Q(x)$: Polynomdivision | |
| \item Nullstellen ermitteln | |
| \item Zerlegung in Partialbrüche | |
| \item Gleichsetzen von Ursprünglicher Bruch und Partialbrüche | |
| \item Ausmultiplizieren der Nenner | |
| \item Bestimmung der Koeffizienten der Partialbrüche durch einsetzen der Nullstellen | |
| \item Integration der Summe der Partialbrüche | |
| \end{enumerate} | |
| \section{Absolutbetrag und Signumfunktion} | |
| \subsection{Absolutbetrag} | |
| \begin{align*} | |
| |x|= \begin{cases} | |
| x & \text{für } x>0 \\ | |
| -x & \text{für } x<0 | |
| \end{cases} | |
| \end{align*} | |
| \begin{tikzpicture} | |
| \begin{axis}[axis lines = middle,line width=1.2] | |
| \addplot[blue]{abs(x)}; | |
| \end{axis} | |
| \end{tikzpicture} | |
| \subsection{Signum} | |
| $\frac x{|x|} = \sign(x)=\begin{cases}1 & \text{für } x>0 \\ | |
| 0 & \text{für } x=0 \\ | |
| -1 & \text{für } x<0 | |
| \end{cases} | |
| $ | |
| \begin{tikzpicture} | |
| \begin{axis}[axis lines = middle,samples=1000,ymax=1.1,ymin=-1.1,line width=1.2] | |
| \addplot[blue]{sign(x)}; | |
| \end{axis} | |
| \end{tikzpicture} | |
| \subsection{Heviside} | |
| $\hevi(x)=\begin{cases} | |
| 0 & \text{für } x<0 \\ | |
| 1 & \text{für } x\ge 0 | |
| \end{cases}$ | |
| \begin{tikzpicture} | |
| \begin{axis}[axis lines = middle,samples=1000,ymax=1.1,line width=1.2] | |
| \addplot[blue]{(sign(x)+1)/2}; | |
| \end{axis} | |
| \end{tikzpicture} | |
| \section{Winkelfunktionen} | |
| \begin{tabular}{c c c c} | |
| {} & Sinus & Kosinus & Tangens \\ | |
| \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.6cm,y=0.6cm] | |
| \draw [color=eqeqeq,fill=eqeqeq] (0,0) -- plot[domain=0.:1.57,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)}) -- cycle ; | |
| \fill[color=ffzzcc] (0.,0.) -- (5.,0.) -- (5.,3.31) -- cycle; | |
| \draw [color=wwccff,fill=wwccff] (0,0) -- plot[domain=0.:0.58,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)}) -- cycle ; | |
| \draw [shift={(0.,0.)}] (0,0) -- (0.:1.2) arc (0.:33.7:1.2) -- cycle; | |
| \draw [shift={(4.15,0.)},fill opacity=0.] (0,0) -- (90.:0.6) arc (90.:180.:0.6) -- cycle; | |
| \draw [shift={(0.,0.)}] plot[domain=-0.2:1.8,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)}); | |
| \draw (0.,0.)-- (6.,3.97); | |
| \draw [->] (5.,0.) -- (5.,3.3); | |
| \draw [->] (4.15,0.) -- (4.15,2.75); | |
| \draw [->] (0.:1.2) arc (0.:33.7:1.2); | |
| \draw [->] (0.,0.) -- (4.15,0.); | |
| \draw [->] (0.,-1.) -- (0.,6.); | |
| \draw [->] (-1.,0.) -- (6.,0.); | |
| \draw [->] (0.,0.) -- (4.15,2.75); | |
| \fill[] (3.95,0.25) circle (0.05); | |
| \filldraw[fill=white] (4.15,0) circle (0.1); | |
| \filldraw[fill=white] (4.15,2.75) circle (0.1); | |
| \draw (1.5,2.8) node[anchor=north west] {r=1}; | |
| \draw (1.2,0) node[anchor=north west] {cos $\alpha$}; | |
| \draw (2.3,1.5) node[anchor=north west] {sin $\alpha$}; | |
| \draw (5.,2.) node[anchor=north west] {tan $\alpha$}; | |
| \draw (3.6,0) node[anchor=north west] {\small M}; | |
| \draw (3.7,4.) node[anchor=north west] {\small P}; | |
| \draw[] (1.7,0.5) node {$\alpha$}; | |
| \end{tikzpicture}& | |
| % | |
| \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.3cm,y=0.3cm] | |
| \draw [shift={(0.,0.)},color=ffzzcc,fill=ffzzcc,fill opacity=1.0] (0,0) -- plot[domain=1.57:3.14,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [shift={(0.,0.)},color=wwccff,fill=wwccff,fill opacity=1.0] (0,0) -- plot[domain=3.14:4.7,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [shift={(0.,0.)},color=wwccff,fill=wwccff,fill opacity=1.0] (0,0) -- plot[domain=-1.57:0.,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [shift={(0.,0.)},color=ffzzcc,fill=ffzzcc,fill opacity=1.0] (0,0) -- plot[domain=0.:1.57,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [line width=1.pt] (0.,0.) circle (1.2cm); | |
| \draw [->,line width=1.pt] (0.,-5.) -- (0.,6.); | |
| \draw [->,line width=1.pt] (-5.,0.) -- (6.,0.); | |
| \draw (0.5,0.5) node[anchor=south west] {\Large +}; | |
| \draw (-0.5,0.5) node[anchor=south east] {\Large +}; | |
| \draw (-1.,-0.5) node[anchor=north east] {\Large -}; | |
| \draw (1.,-0.5) node[anchor=north west] {\Large -}; | |
| \end{tikzpicture}& | |
| % | |
| \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.3cm,y=0.3cm] | |
| \draw [shift={(0.,0.)},color=wwccff,fill=wwccff,fill opacity=1.0] (0,0) -- plot[domain=1.57:3.14,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [shift={(0.,0.)},color=wwccff,fill=wwccff,fill opacity=1.0] (0,0) -- plot[domain=3.14:4.7,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [shift={(0.,0.)},color=ffzzcc,fill=ffzzcc,fill opacity=1.0] (0,0) -- plot[domain=-1.57:0.,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [shift={(0.,0.)},color=ffzzcc,fill=ffzzcc,fill opacity=1.0] (0,0) -- plot[domain=0.:1.57,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [line width=1.pt] (0.,0.) circle (1.2cm); | |
| \draw [->,line width=1.pt] (0.,-5.) -- (0.,6.); | |
| \draw [->,line width=1.pt] (-5.,0.) -- (6.,0.); | |
| \draw (0.5,0.5) node[anchor=south west] {\Large +}; | |
| \draw (0.5,-0.5) node[anchor=north west] {\Large +}; | |
| \draw (-1.,-0.5) node[anchor=north east] {\Large -}; | |
| \draw (-1.,0.5) node[anchor=south east] {\Large -}; | |
| \end{tikzpicture}& | |
| % | |
| \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.3cm,y=0.3cm] | |
| \draw [shift={(0.,0.)},color=wwccff,fill=wwccff,fill opacity=1.0] (0,0) -- plot[domain=1.57:3.14,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [shift={(0.,0.)},color=ffzzcc,fill=ffzzcc,fill opacity=1.0] (0,0) -- plot[domain=3.14:4.7,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [shift={(0.,0.)},color=wwccff,fill=wwccff,fill opacity=1.0] (0,0) -- plot[domain=-1.57:0.,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [shift={(0.,0.)},color=ffzzcc,fill=ffzzcc,fill opacity=1.0] (0,0) -- plot[domain=0.:1.57,variable=\t]({1.*4.*cos(\t r)+0.*4.*sin(\t r)},{0.*4.*cos(\t r)+1.*4.*sin(\t r)}) -- cycle ; | |
| \draw [line width=1.pt] (0.,0.) circle (1.2cm); | |
| \draw [->,line width=1.pt] (0.,-5.) -- (0.,6.); | |
| \draw [->,line width=1.pt] (-5.,0.) -- (6.,0.); | |
| \draw (0.5,0.5) node[anchor=south west] {\Large +}; | |
| \draw (-0.5,-0.5) node[anchor=north east] {\Large +}; | |
| \draw (-1.,0.5) node[anchor=south east] {\Large -}; | |
| \draw (1.,-0.5) node[anchor=north west] {\Large -}; | |
| \end{tikzpicture} | |
| \end{tabular} | |
| \subsection{Trigonometrische Formeln} | |
| \subsubsection{Additionstheoreme} | |
| \begin{align*} | |
| &\sin (x\pm y)=\sin x\cos y\pm \cos x\sin y \\ | |
| &\cos (x\pm y)=\mp \sin x\sin y+\cos x\cos y \\ | |
| &\tan (x\pm y)=\frac{\tan x\pm \tan y}{1\mp \tan x\tan y} \\ | |
| \end{align*} | |
| \subsubsection{Produktformel} | |
| \begin{align*} | |
| \cos x\cos y=\frac 1 2[\cos (x+y)+\cos (x-y)] \\ | |
| \sin x\sin y=\frac 1 2[\cos (x-y)-\cos (x+y)] \\ | |
| \cos x\sin y=\frac 1 2[\sin (x+y)-\sin (x-y)] \\ | |
| \end{align*} | |
| \subsubsection{Quadratische Formeln} | |
| \begin{align*} | |
| \cos ^2x=\frac 1 2[1+\cos (2x)] \\ | |
| \sin ^2x=\frac 1 2[1-\cos (2x)] \\ | |
| \end{align*} | |
| \subsubsection{Goniometrische Beziehungen} | |
| \begin{align*} | |
| &\sin ^2\alpha +\cos ^2\alpha = 1 \\ | |
| &\tan \alpha =\frac{\sin \alpha }{\cos \alpha } \\ | |
| &1+\tan^2\alpha =\frac 1{\cos ^2\alpha }\hfill \\ | |
| &\cos (2\alpha )=\cos ^2\alpha -\sin ^2\alpha | |
| \end{align*} | |
| \subsection{sin, cos zu arctan Transformation} | |
| \begin{align*} | |
| &u=2 \arctan z \\ | |
| &\sin u = \frac{2z}{1+z^2} \\ | |
| &\cos u = \frac{1-z^2}{1+z^2} \\ | |
| &\frac{du}{dz}=\frac{2}{1+z^2} | |
| \end{align*} | |
| \subsection{Weitere Funktionen} | |
| \subsubsection{Transformation zwischen $e^{ix} \pm e^{-ix}$ zu $\sin x$, $\cos x$} | |
| \begin{align*} | |
| e^{i x} &= \left(\cos(x) + i \sin(x)\right) \\ | |
| e^{-i x} &= \left(\cos(x) - i \sin(x)\right) \\ | |
| e^{i x} \pm e^{-i x} &= \left(\cos(x) + i \sin(x)\right) \pm \left(\cos(x) - i \sin(x)\right) \\ | |
| \rightarrow ~ e^{i x} + e^{-i x} &= 2 \cos(x) \\ | |
| \rightarrow ~ e^{i x} - e^{-i x} &= j 2 \sin(x) | |
| \end{align*} | |
| \subsubsection{$\si(x)$ Funktion} | |
| \begin{align*} | |
| \si(x) &= \frac{\sin(x)}{x} | |
| \end{align*} | |
| \subsubsection{$\arg(x+iy)$ Funktion} | |
| \begin{align*} | |
| \arg(x+i y) &= \begin{cases} | |
| \arctan(\frac{y}{x}) & \text{für } x > 0, \\ | |
| \arctan(\frac{y}{x}) + \pi & \text{für } x < 0 \text{ und } y \ge 0, \\ | |
| \arctan(\frac{y}{x}) - \pi & \text{für } x < 0 \text{ und } y < 0, \\ | |
| +\frac{\pi}{2} & \text{für } x = 0 \text{ und } y > 0, \\ | |
| -\frac{\pi}{2} & \text{für } x = 0 \text{ und } y < 0, \\ | |
| \text{undefiniert} & \text{für } x = 0 \text{ und } y = 0. \\ | |
| \end{cases} | |
| \end{align*} | |
| \section{Geometrische Reihe/Summe} | |
| \begin{align*} | |
| &1+x+x^2+x^3+...+x^n \\ | |
| &\sum _{n=0}^Nx^n=\frac{1-x^{N+1}}{1-x} \\ | |
| &\sum _{n=0}^{\infty}x^n=\frac 1{1-x} | |
| \end{align*} | |
| \section{Inverse Funktionen} | |
| \begin{tabular}{l l} | |
| Funktion: & $f(x)=y$ \\ | |
| Inverse Funktion: & $f(y)=x$ | |
| \end{tabular} | |
| \section{Logarithmus} | |
| \begin{align*} | |
| &\log(x)=y \Leftrightarrow x=10^y \\ | |
| &\ln (x)=y \Leftrightarrow x=e^y \\ | |
| &e^{\ln x}=x \\ | |
| &\ln (e^y)=y \\ | |
| &\ln (1)=0 \\ | |
| &\ln (e)=1 | |
| \end{align*} | |
| \subsection{Logarithmusgesetze} | |
| \begin{align*} | |
| &\ln (a \; \cdot b)=\ln a + \ln b \\ | |
| &\ln \left(\frac a b\right)=\ln a-\ln b \\ | |
| &\ln (a^b)=b\ln a \\ | |
| &\ln a=\ln b \Leftrightarrow \; a=b | |
| \end{align*} | |
| \subsection{Hyperbelfunktion} | |
| Definition: | |
| \begin{align*} | |
| &\cosh x = \frac{1}{2} (e^x + e^{-x}) \\ | |
| &\sinh x = \frac{1}{2} (e^x - e^{-x}) | |
| \end{align*} | |
| Regeln: | |
| \begin{align*} | |
| &\cosh x=\cosh (-x) \\ | |
| &\sinh x=-\sinh (-x) \\ \\ | |
| &\sinh (x\pm y)=\sinh a\;\cdot \cosh b\pm \cosh a\;\cdot \sinh b \\ | |
| &\cosh (x\pm y)=\cosh a\;\cdot \cosh b\pm \sinh a\;\cdot \sinh b \\ \\ | |
| &\cosh^2x+\sinh ^2x=\cosh (2x) \\ | |
| &\cosh ^2x-\sinh ^2x=1 | |
| \end{align*} | |
| \section{Differentiation} | |
| \begin{align*} | |
| f'(x)=\lim_{x \to 0} \frac{f(x+h)-f(x)}{h} | |
| \end{align*} | |
| \subsection{Linearität der Differentation} | |
| \begin{align*} | |
| (a \; f(x) + b \; g(x))' = a \; f'(x) + b \; g'(x) | |
| \end{align*} | |
| \subsection{Ableitung spezieller Funktionen} | |
| \begin{align*} | |
| &(e^x)' = e^x \\ | |
| &(\sin x)' = \cos x \\ | |
| &(\cos x)' = -\sin x \\ | |
| &(\tan x)' = 1 + \tan^2 x = \frac{1}{\cos^2 x} \\ | |
| &(\ln x)' = \frac{1}{x} \\ | |
| &(\arcsin x)' = \frac{1}{\sqrt{1-x^2}} \\ | |
| &(\arccos x)' = -\frac{1}{\sqrt{1-x^2}} \\ | |
| &(\arctan x)' = \frac{1}{1+x^2} | |
| \end{align*} | |
| \subsection{Differentationstechniken} | |
| \subsubsection{Produktregel} | |
| \begin{align*} | |
| (f(x) \; g(x))' = f'(x) \; g(x) + f(x) \; g'(x) | |
| \end{align*} | |
| \subsubsection{Quotientenregel} | |
| \begin{align*} | |
| \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x) \; g(x) - f(x) \; g'(x)}{g^2(x)} | |
| \end{align*} | |
| \subsubsection{Kettenregel} | |
| \begin{align*} | |
| \left(f\left(g(x)\right)\right)' = f'\left(g(x)\right) \cdot g'(x) | |
| \end{align*} | |
| \subsubsection{Differenzieren über Umkehrfunktion} | |
| % umkehr von sin x = | |
| \begin{align*} | |
| (f^{(-1)})'(x) = \frac{1}{f'\left(f^{-1}(x)\right)} | |
| \end{align*} | |
| \subsubsection{Logarithmische Ableitung} | |
| Variante der Produktregel | |
| \begin{align*} | |
| &y(x) = f(x) \cdot g(x) \cdot h(x) \\ | |
| &y'(x) = y'''(x) \left[\frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)} + \frac{h'(x)}{h(x)}\right] | |
| \end{align*} | |
| \subsection{Kurvendiskussion} | |
| \subsubsection{Symmetrie} | |
| \begin{alignat*}{2} | |
| &f(x) = f(-x) \quad &&\rightarrow \text{gerade Funktion}\\ | |
| &f(x) = -f(-x) \quad &&\rightarrow \text{ungerade Funktion} | |
| \end{alignat*} | |
| \subsubsection{Extremwerte} | |
| \begin{alignat*}{4} | |
| &f'(x) = 0 \quad &&\& \quad f''(x) > 0 \quad &&\to \text{lok. min} \\ | |
| &f'(x) = 0 \quad &&\& \quad f''(x) < 0 \quad &&\to \text{lok. max} \\ | |
| &f''(x) = 0 \quad &&\& \quad f'''(x) \neq 0 \quad &&\to \text{Wendepunkt} | |
| \end{alignat*} | |
| \subsection{Taylorreihe und Taylorpolynom} | |
| Taylorpolynom mit Grad $N$: | |
| \begin{align*} | |
| P(f)(x) = f(0) + f'(0) \; x + f''(0) \frac{x^2}{2!} + \cdots + f^{(N)}(0) \; \frac{x^N}{N!} | |
| \end{align*} | |
| Taylorreihe: | |
| \begin{align*} | |
| f(x) = \sum_{i = 0}^{n} f^{(n)}(0) \; \frac{x^n}{n!} | |
| \end{align*} | |
| \subsubsection{Taylorreihe und Taylorpolynom im Punkt C} | |
| Taylorpolynom mit Grad $N$: | |
| \begin{align*} | |
| P(f)(x) = f(C) + f'(C) \; (x - C) + f''(C) \frac{(x - C)^2}{2!} + \cdots + f^{(N)}(C) \; \frac{(x - C)^N}{N!} | |
| \end{align*} | |
| Taylorreihe: | |
| \begin{align*} | |
| \sum_{k=0}^{\infty} f^{(k)}(C) \; \frac{(x-C)^k}{k!} | |
| \end{align*} | |
| \section{Stammfunktion und Integral} | |
| \subsection{Das unbestimmte Integral} | |
| Für $F' = f$ | |
| \begin{align*} | |
| &\int_{a}^{b} f(x) dx := F(b) - F(a) \quad \text{für} \; a < b \\ | |
| &\int_{a}^{b} f(x) dx = -\int_{b}^{a} f(x) dx \quad \text{für} \; a > b \\ | |
| &\int f(x) := F(x) | |
| \end{align*} | |
| \subsection{Grundintegrale} | |
| \begin{tabular}{l | l} | |
| Funktion $f(x)$ & Stammfunktion $F(x)$ \\ \hline | |
| $a$ & $ax + C$ \\ \\ | |
| $x^m$ für $m \neq -1$ & $x^{m+1} \; \frac{1}{m+1} + C$ \\ \\ | |
| $x^{-1}$ & $\ln|x|+C$ \\ \\ | |
| $e^{ax}$ & $\frac{1}{a} \; e^{ax} + C$ \\ \\ | |
| $\cos(ax)$ & $\frac{1}{a} \; \sin(ax) + C$ \\ \\ | |
| $\sin(ax)$ & $-\frac{1}{a} \; \cos(ax) + C$ \\ \\ | |
| $\frac{h'(x)}{h(x)}$ & $\ln|h(x)| + C$ | |
| \end{tabular} | |
| \subsection{Weitere Integrale} | |
| \begin{tabular}{l | l} | |
| $f(x)$ & $F(x)$ \\ \hline \\ | |
| $\frac{1}{x^2+a^2}$ & $\frac{1}{a} \; \arctan \left(\frac{x}{a}\right) + C$ \\ \\ | |
| $\frac{1}{u^2 + 1}$ & $\arctan(u)$ \\ \\ | |
| $\sqrt{\frac{1}{x^2+1}}$ & $\ln(x + \sqrt{x^2 + 1}) + C = \arcsinh(x)$ \\ \\ | |
| $\tan(ax)$ & $-\frac{1}{a} \ln|\cos(ax)| + C$ \\ | |
| \end{tabular} | |
| \subsection{Transformationsregeln} | |
| \subsection{Substitution} | |
| \begin{align*} | |
| \int f[g(x)] \; dx &= \int f(u) \frac{du}{g'(x)} \\ | |
| &\uparrow \\ | |
| u &= g(x)& \\ | |
| \frac{du}{dx} &= g'(x) \\ | |
| dx &= \frac{du}{g'(x)} | |
| \end{align*} | |
| \subsubsection{Partielle Integration} | |
| \begin{align*} | |
| &\int f'(x) \; g(x) \; dx = f(x) \; g(x) - \int f(x) \; g'(x) \; dx + C \\ | |
| &\int_{a}^{b} f'(x) \; g(x) \; dx = f(b) \; g(b) - f(a) \; g(a) - \int_{a}^{b} f(x) \; g'(x) \; dx + C | |
| \end{align*} | |
| \section{Matrizen} | |
| \begin{align*} | |
| & m \times n \text{ Matrix } A = \begin{bmatrix} | |
| a_{1,1} & \dots & a_{1,n} \\ | |
| \vdots & \ddots & \vdots \\ | |
| a_{m,1} & \dots & a_{m,n} | |
| \end{bmatrix} \\ | |
| & A = (a_{i,j})\begin{matrix}m & n \\ i = 1 & j = 1 \end{matrix} | |
| \end{align*} | |
| \subsubsection{Einheitsmatrix} | |
| \begin{align*} | |
| & I_n \quad n \times n \text{ Matrix} \\ | |
| & I_n = \begin{bmatrix} | |
| 1 & 0 & \dots & 0 \\ | |
| 0 & 1 & & \vdots \\ | |
| \vdots & & \ddots & \vdots \\ | |
| 0 & \dots & \dots & 1 | |
| \end{bmatrix} | |
| \end{align*} | |
| für $A \quad m \times n$ Matrix | |
| \begin{align*} | |
| & A \cdot I_n = A | |
| \end{align*} | |
| \subsection{Vektor} | |
| $n\times 1$ oder $1 \times m$ Matrix | |
| \subsection{Matrixoperationen} | |
| \subsubsection{Summe von Matrizen} | |
| \begin{align*} | |
| & A, B \text{ Matrizen vom selben Format } m \times n \\ | |
| & C = A + B \\ | |
| & c_{i,j} = a_{i,j} + b_{i,j} \\ | |
| & C = (c_{i,j})\begin{matrix}m & n \\ i = 1 & j = 1 \end{matrix} | |
| \end{align*} | |
| \subsubsection{Skalarprodukt von Vektoren} | |
| Für $a, b$ Vektoren der Länge $n$ | |
| \begin{align*} | |
| & \langle a, b \rangle = a_1 b_1 + \dots + a_n b_n | |
| \end{align*} | |
| \subsubsection{Kreuzprodukt von Vektoren} | |
| Für $a, b$ Vektoren der Länge 3 | |
| \begin{align*} | |
| & a \times b = \begin{pmatrix} | |
| a_2 b_3 - a_3 b_2 \\ | |
| -(a_1 b_3 - a_3 b_1) \\ | |
| a_1 b_2 - a_2 b_1 | |
| \end{pmatrix} | |
| \end{align*} | |
| \subsection{Multiplikation von Matrizen} | |
| Für $A$ ist eine $m \times k$ Matrix und $B$ ist eine $k \times n$ Matrix. | |
| Die Ergebnismatrix ist eine $m \times n$ Matrix. | |
| \begin{tabular}{c | c} | |
| $A \cdot B$ & $\begin{bmatrix} \\ \text{Spalte } j \text{ von } B \\ \\ \end{bmatrix}$ \\ \\ \hline \\ | |
| $\begin{bmatrix} & \text{Zeile} & \\ & i & \\ & \text{von } A & \end{bmatrix}$ & $\begin{bmatrix} \\ & \langle a_i, b_j \rangle & \\ & & \end{bmatrix}$ | |
| \end{tabular} | |
| \subsection{Transposition} | |
| Es werden Spalten mit den Zeilen getauscht. | |
| \begin{align*} | |
| A & = (a_{i,j})\begin{matrix} m & n \\ i = 1 & j = 1 \end{matrix} \\ | |
| A^T & = (a_{j,i})\begin{matrix} n & m \\ i = 1 & j = 1 \end{matrix} \\ | |
| A & = \begin{bmatrix} | |
| a_{1,1} & \dots & a_{1,n} \\ | |
| \vdots & \ddots & \vdots \\ | |
| a_{m,1} & \dots & a_{m,n} | |
| \end{bmatrix} \\ | |
| A^T & = \begin{bmatrix} | |
| a_{1,1} & \dots & a_{m,1} \\ | |
| \vdots & \ddots & \vdots \\ | |
| a_{1,n} & \dots & a_{m,n} | |
| \end{bmatrix} \\ | |
| \\ | |
| \left(A^T\right)^T & =A \\ | |
| \left(A \cdot B\right)^T &= B^T \cdot A^T | |
| \end{align*} | |
| \subsection{Inversion} | |
| Für $A, B \quad n \times n$ Matrix | |
| $A$ ist invers zu $B$, wenn $A \cdot B = B \cdot A = I_n$ | |
| \subsubsection{$2 \times 2$ Matrix} | |
| Für $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ | |
| $A^{-1}$ existiert wenn $\det A \neq 0$ | |
| \begin{align*} | |
| \adj A &= \begin{bmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{bmatrix} \\ | |
| A^{-1} & = \frac{\adj A}{\det A} \\ | |
| \det A & = a_{11} \cdot a_{22} - a_{21} \cdot a_{12} | |
| \end{align*} | |
| \subsubsection{$3 \times 3$ Matrix} | |
| $A^{-1}$ existiert wenn $\det A \neq 0$ | |
| \begin{align*} | |
| A^{-1} & = \frac{\adj A}{\det A} \\ | |
| \det A & = +a_{11} U_{11} - a_{12} U_{12} + a_{13} U_{13}\\ | |
| \adj A & = \begin{bmatrix} | |
| +U_{11} & -U_{12} & +U_{13} \\ | |
| -U_{21} & +U_{22} & -U_{23} \\ | |
| +U_{31} & -U_{32} & +U_{33} | |
| \end{bmatrix}^T | |
| \end{align*} | |
| Eine Unterdeterminante $U_{ij}$ ist die Determinante einer Matrix ohne der Zeile $i$ und der Spalte $j$. | |
| Alternative für die Determinante über die Summe der Diagonalen: | |
| \begin{align*} | |
| \det A = a_{11} \; a_{22} \; a_{33} + a_{21} \; a_{32} \; a_{13} + a_{31} \; a_{12} \; a_{23} - a_{13} \; a_{22} \; a_{31} - a_{23} \; a_{32} \; a_{11} - a_{33} \; a_{12} \; a_{21} | |
| \end{align*} | |
| \subsection{Eigenschaften einer Matrix} | |
| Für $A \quad n \times n$ Matrix | |
| \subsubsection{Symmetrie} | |
| Wenn $A = A^T$ | |
| \subsubsection{Asymmetrie} | |
| Wenn $A = -A^T$ | |
| \subsubsection{Orthogonal} | |
| Wenn $A \cdot A^T = I_n, \quad A^T = A^{-1}$ | |
| \subsubsection{Dreiecksmatrix} | |
| Wenn ein Dreieck der Matrix (ohne Diagonale) den Wert 0 beinhaltet. | |
| \subsection{Gauß-Algorithmus} | |
| Zum Lösen eines Gleichungssystems der Form $A x = b$ | |
| für $A \; n \times n$ Matrix und $x, \; b$ Vektoren der Länge $n$. | |
| \begin{alignat*}{3} | |
| a_1 & x_1 + \cdots + a_1 && x_n =\; && b_1 \\ | |
| & \vdots && \vdots && \vdots \\ | |
| a_n & x_1 + \cdots + a_n && x_n =\; && b_n | |
| \end{alignat*} | |
| Anschreiben als: | |
| \begin{align*} | |
| \left(\begin{matrix} | |
| a_{1,1} & \dots & a_{1,n} \\ | |
| \vdots & \ddots & \vdots \\ | |
| a_{n,1} & \dots & a_{n,n} | |
| \end{matrix} \; \middle\vert \; \begin{matrix} | |
| b_1 \\ \vdots \\ b_n | |
| \end{matrix}\right) | |
| \end{align*} | |
| Ziel ist die Rechte obere Dreiecksmatrix durch Äquivalenzumformung. | |
| Dies wird erreicht indem Vielfache der ersten Gleichung zu den folgenden Gleichungen addiert wird. | |
| Die Reihenfolge der Gleichungen kann geändert werden. | |
| \section{Vektoren im Raum} | |
| \begin{align*} | |
| \vec{a} = \begin{pmatrix}x\\y\\z\end{pmatrix} = (x, y, z) | |
| \end{align*} | |
| \subsection{Rechenoperationen} | |
| für: | |
| \begin{align*} | |
| \vec{a} &= \begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix} & | |
| \vec{b} &= \begin{pmatrix}b_1\\b_2\\b_3\end{pmatrix} | |
| \end{align*} | |
| \subsubsection{Summe} | |
| \begin{align*} | |
| \vec{a} + \vec{b} = \begin{pmatrix}a_1+b_1\\a_2+b_2\\a_3+b_3\end{pmatrix} | |
| \end{align*} | |
| \subsubsection{Betrag} | |
| \begin{align*} | |
| |\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} | |
| \end{align*} | |
| \subsubsection{Skalarprodukt} | |
| \begin{align*} | |
| \vec{a} \cdot \vec{b} &= \langle \vec{a}, \vec{b} \rangle = a_1 b_1 + a_2 b_2 + a_3 b_3 \\ | |
| \end{align*} | |
| \subsubsection{Kreuzprodukt (Vektorprodukt)} | |
| \begin{align*} | |
| \vec{a} \times \vec{b} &= \begin{pmatrix} a_2 b_3 - a_3 b_2 \\ -(a_1 b_3 - a_3 b_1) \\ a_1 b_2 - a_2 b_1 \end{pmatrix} \\ | |
| \vec{a} \times \vec{b} &\perp \vec{a}, \vec{b} | |
| \end{align*} | |
| \subsubsection{Winkel zwischen zwei Vektoren} | |
| \label{sec:vec-calc-angle} | |
| \begin{align*} | |
| &\Theta \dots \text{Winkel zwischen zwei Vektoren} \\ | |
| \\ | |
| &\cos{\measuredangle(\vec{a}, \vec{b})} = \frac{ |\langle \vec{a}, \vec{b} \rangle|}{|\vec{a}| |\vec{b}|} | |
| \end{align*} | |
| \subsection{Geradengleichung} | |
| \begin{align*} | |
| A,B &\dots \text{Punkte im Raum} \\ | |
| \overrightarrow{AP} &\dots \text{Vektor von $A$ nach $B$} \\ | |
| g &\dots \text{Gerade durch $A$, $B$} \\ | |
| \overrightarrow{AP} &= P - A \\ | |
| \vec{g} : \begin{pmatrix} x \\ y \\ z \end{pmatrix} &= A + t \; \overrightarrow{AP} | |
| \end{align*} | |
| \subsection{Ebenen} | |
| für | |
| \begin{align*} | |
| &E \dots \text{Ebene} \\ | |
| &\vec{N} \dots \text{Normalvektor der Ebene} | |
| &A \dots \text{Punkt auf der Ebene} \\ | |
| \\ | |
| &E: n_1 \; x + n_2 \; y + n_3 \; z = a_1 n_1 + a_2 n_2 + a_3 n_3 | |
| \end{align*} | |
| \subsubsection{Ebenengleichung} | |
| für | |
| \begin{align*} | |
| \vec{N} &\dots \text{Normalvektor auf die Ebene} \\ | |
| A &\dots \text{Punkt in der Ebene} \\ | |
| \\ | |
| &\left(\begin{pmatrix}x \\ y \\ z\end{pmatrix} - A\right) \perp \vec{N} \\ | |
| \rightarrow &\left(\begin{pmatrix} x \\ y \\ z \end{pmatrix} - A\right) \times \vec{N} | |
| \end{align*} | |
| \subsubsection{Umformen auf Parameterfreie Form} | |
| \begin{align*} | |
| &E: \vec{x} = A + s \begin{pmatrix}u_1 \\ u_2 \\ u_3\end{pmatrix} + t \begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix} \dots \text{Parameterbehaftete Form}\\ | |
| &\begin{pmatrix}n_1 \\ n_2 \\ n_3\end{pmatrix}^T \cdot \begin{pmatrix}u_1 \\ u_2 \\ u_3\end{pmatrix} = 0 | |
| &\Rightarrow n_1 u_1 + n_2 u_2 + n_3 u_3 = 0 \\ | |
| &\begin{pmatrix}n_1 \\ n_2 \\ n_3\end{pmatrix}^T \cdot \begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix} = 0 | |
| &\Rightarrow n_1 v_1 + n_2 v_2 + n_3 v_3 = 0 \\ | |
| \end{align*} | |
| \subsubsection{Winkel zwischen zwei Ebenen} | |
| Siehe: \ref{sec:vec-calc-angle} \nameref{sec:vec-calc-angle} \\ | |
| (mittels den Normalvektoren der Ebenen) | |
| \subsubsection{Schnittpunkt mit Gerade} | |
| In $n_1 \; x + n_2 \; y + n_3 \; z = a_1 n_1 + a_2 n_2 + a_3 n_3$ Gerade für $\begin{pmatrix}x\\y\\z\end{pmatrix}$ einsetzen | |
| \subsubsection{Schnittpunkt zweier Ebenen} | |
| \begin{align*} | |
| A &\dots \text{Punkt auf beiden Ebenen} \\ | |
| \\ | |
| g &: \vec{x} = A + t \vec{N_1} \times \vec{N_2} | |
| \end{align*} | |
| \subsection{Eigenwert und Eigenvektor von Matrizen} | |
| \begin{align*} | |
| A &\dots \text{Matrix} & n &\times n \; \text{Matrix}\\ | |
| \lambda &\dots \text{Eigenwerte} & \lambda &\in \mathbb{C}\\ | |
| s &\dots \text{Eigenvektor} & s &\in \mathbb{R}^n | |
| \end{align*} | |
| $s$ ist ein Eigenvektor von $A$, wenn | |
| \begin{align*} | |
| &As = \lambda s = \lambda I_n s \\ | |
| &s (A - \lambda I_n) = 0 \\ | |
| &\det( A - \lambda I_n) = 0 | |
| \end{align*} | |
| \begin{enumerate} | |
| \item Berechnen von Eigenwerten | |
| \item Berechnen von Eigenvektor zu Eigenwerten | |
| \end{enumerate} | |
| \section{Komplexe Zahlen} | |
| für | |
| \begin{align*} | |
| z &\dots \text{Komplexe Zahl} \\ | |
| x &\dots \text{Realteil von } z \\ | |
| y &\dots \text{Imaginärteil von } z \\ | |
| z &\in \mathbb{C} \;\; x,y \in \mathbb{R} \\ | |
| z &= x + i y | |
| \end{align*} | |
| \subsection{Rechenregeln} | |
| \subsubsection{Addition} | |
| \begin{align*} | |
| z_1 + z_2 = (x_1 + i y_1) + (x_2 + i y_2) = (x_1 + x_2) + i (y_1 + y_2) | |
| \end{align*} | |
| \subsubsection{Multiplikation} | |
| \begin{align*} | |
| z_1 \cdot z_2 = (x_1 + i y_1) \cdot (x_2 + i y_2) = (x_1 x_2 - y_1 y_2) + i (x_1 y_2 + x_2 y_1) | |
| \end{align*} | |
| \subsubsection{Inverse: Division} | |
| $\conj{z} \dots$ Konjungiert komplexe Zahl | |
| \begin{align*} | |
| \conj{z} &= x - i y \\ | |
| \frac{1}{z} &= \frac{1}{z} \cdot \frac{\conj{z}}{\conj{z}} = \frac{\conj{z}}{x^2 + y^2} = \frac{x}{x^2 + y^2} - i \frac{y}{x^2 + y^2} \\ | |
| \end{align*} | |
| Es gilt: | |
| \begin{align*} | |
| \conj{z_1 \cdot z_2} &= \conj{z_1} \cdot \conj{z_2} \\ | |
| \conj{\conj{z}} &= z \\ | |
| x &= \frac{1}{2} (z + \conj{z}) \\ | |
| y &= \frac{1}{2} (z - \conj{z}) | |
| \end{align*} | |
| \subsection{Polarkoordinaten} | |
| \begin{align*} | |
| r &\dots \text{Radius} \\ | |
| \varphi &\dots \text{Winkel} \\ | |
| z &= r \cdot e^{i \varphi} \\ | |
| x &= r \cos \varphi \\ | |
| y &= r \sin \varphi \\ | |
| r^2 &= x^2 + y^2 \\ | |
| \frac{y}{x} &= \tan \varphi | |
| \end{align*} | |
| \subsubsection{Formel von Moivre} | |
| \begin{align*} | |
| e^{i\varphi_1} e^{i\varphi_2} &= e^{i (\varphi_1 + \varphi_2)} = \cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2) \\ | |
| \left(e^{i\varphi}\right)^k &= e^{i k \varphi} | |
| \end{align*} | |
| \subsection{Wurzel einer komplexen Zahl} | |
| \begin{align*} | |
| \sqrt[n]{z} = \sqrt[n]{|z|} \; e^{\frac{\varphi + 2 \pi k}{n}} \text{ für } k = 0 \; ... \; n-1 | |
| \end{align*} | |
| \section{Differentialgleichungen} | |
| für | |
| \begin{align*} | |
| x(t) & \dots \text{Funktion der Variable } t \\ | |
| x'(t) = \frac{d x(t)}{dt} & \dots \text{1. Ableitung der Funktion } x(t) \\ | |
| x^{(n)}(t) = \frac{d^n x(t)}{d t^n} &\dots \text{n-te Ableitung der Funktion } x(t) | |
| \end{align*} | |
| \subsection{Lineare Differentialgleichung der Ordnung 1} | |
| für | |
| \begin{align*} | |
| &x'(t) + a \; x(t) = 0 & a \in \mathbb{R} | |
| \end{align*} | |
| Allgemeine Lösung: | |
| \begin{align*} | |
| &x(t) = C \; e^{-at} & C \in \mathbb{C} | |
| \end{align*} | |
| \subsection{Lineare Differentialgleichung der Ordnung 2} | |
| für | |
| \begin{align*} | |
| &x''(t) + b \; x'(t) + c \; x(t) = 0 & b,c \in \mathbb{C} | |
| \end{align*} | |
| Allgemeine Lösung: | |
| \begin{align*} | |
| &x(t) = C \; e^{\lambda t} & \lambda \in \mathbb{C} | |
| \end{align*} | |
| \subsubsection{Charakteristische Gleichung} | |
| \begin{align*} | |
| &\lambda^2 + b \; \lambda + c = 0 \\ | |
| &\lambda_{1,2} = -\frac{b}{2} \pm \sqrt{\frac{b^2}{4} - c} | |
| \end{align*} | |
| Fall bestimmen: | |
| \begin{enumerate} | |
| \item Fall: $\lambda_1 \neq \lambda_2$ (reell) | |
| \begin{align*} | |
| &x(t) = C_1 \; e^{\lambda_1 t} + C_2 \; e^{\lambda_2 t} \\ | |
| &C_1, C_2 \in \mathbb{R} | |
| \end{align*} | |
| \item Fall: $\lambda = \lambda_1 = \lambda_2$ (reell) | |
| \begin{align*} | |
| &x(t) = (C_1 + t \; C_2) e^{\lambda t} \\ | |
| &C_1, C_2 \in \mathbb{R} | |
| \end{align*} | |
| \item Fall: $\lambda_{1,2} = \alpha \pm i \; \beta$ (konjungiert Komplex) \\ | |
| Reelle Lösungen: | |
| \begin{align*} | |
| x(t) &= C_1 \; e^{\alpha t} \cos(\beta t) + C_2 \; e^{\alpha t} \sin(\beta t) & C_1, C_2 &\in \mathbb{R} \\ | |
| x(t) &= C \; e^{\alpha t} \cos(\beta t - \varphi) & C &\in \mathbb{R} & \varphi \in [0, 2\pi] | |
| \end{align*} | |
| \end{enumerate} | |
| \subsubsection{Inhomogene Differentialgleichung 2. Ordnung} | |
| für | |
| \begin{align*} | |
| &x''(t) + b \; x'(t) + c \; x(t) = f(x) & b,c \in \mathbb{C} | |
| \end{align*} | |
| \begin{align*} | |
| x_h &\dots \text{Homogene Lösung} \rightarrow \text{Lösung für } x_h''(t) + b \; x_h'(t) + c \; x_h(t) = 0 \\ | |
| x_p &\dots \text{Partikuläre Lösung} \\ | |
| x &\dots \text{Lösung der DGL} \\ | |
| x &= x_h + x_p | |
| \end{align*} | |
| Partikuläre Lösung: | |
| \begin{enumerate} | |
| \item Ansatz aus Tabelle wählen: | |
| \begin{tabular}{l|l} | |
| $f(t)$ & $x_p$ \\ \hline | |
| $f(t) = k \; e^{\alpha t}$ & $x_p = p \; e^{\alpha t}$ \\ | |
| $f(t) = k \cos(\beta t)$ & $x_p = p \cos(\beta t) + q \sin(\beta t)$ \\ | |
| $f(t) = k \sin(\beta t)$ & $x_p = p \cos(\beta t) + q \sin(\beta t)$ \\ | |
| $f(t) \dots$ Polynom der Ordnung N & $x_p \dots$ Polynom der Ordnung N | |
| \end{tabular} | |
| \item $x_p$ aus Ansatz in DGL als $x$ einsetzen | |
| \item Koeffizienten bestimmen | |
| \end{enumerate} | |
| \subsection{Differentialgleichungen für Funktionen in mehreren Variablen} | |
| für | |
| \begin{align*} | |
| f(x,y) \dots \text{Funktion in Abhängigkeit der Variablen x,y} | |
| \end{align*} | |
| \begin{align*} | |
| \frac{\partial}{\partial x} f(x,y) &= f_x(x,y) &&\dots \; \text{Partielle Ableitung von $f$ nach $x$} \\ | |
| \frac{\partial}{\partial y} f(x,y) &= f_y(x,y) &&\dots \; \text{Partielle Ableitung von $f$ nach $y$} \\ | |
| \nabla f(x,y) &= \begin{pmatrix}f_x(x,y) \\ f_y(x,y)\end{pmatrix} &&\dots \; \text{Gradient} \\ | |
| H_f &= \begin{bmatrix} | |
| f_{xx} & f_{xy} \\ | |
| f_{yx} & f_{yy} | |
| \end{bmatrix} &&\dots \; \text{Hessematrix} \\ | |
| \triangle f(x,y) &= sp(H_f (x,y)) = f_{xx} + f_{yy} &&\dots \; \text{Laplace (operator)} | |
| \end{align*} | |
| %\section{Anhang} | |
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