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Formelsammlung HSD
% 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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\usepackage[naustrian]{babel}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage[fleqn]{amsmath}
\usepackage{amssymb,amsfonts,textcomp}
\usepackage{tikz,pgfplots}
\usepackage{pgfplotstable}
\usepackage{mathrsfs}
\usetikzlibrary{arrows}
\usepgfplotslibrary{dateplot}
\usetikzlibrary{shapes.symbols,shapes.geometric}
\usetikzlibrary{positioning}
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\usepackage[automark]{scrpage2}
\usepackage[
imagemodifier={-eu},type={CC},
modifier={by-nc-sa},
version={4.0}
]{doclicense}
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% \usepackage[pdftex]{graphicx}
\usepackage[%
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colorlinks=true, %% turn on colored links (true is better for on-screen reading, false is better for printout versions)
linkcolor=mygray,
urlcolor=blue,
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bookmarks=true, %% if true, generate PDF bookmarks (requires two passes of pdflatex)
bookmarksopen=false, %% if true, show all PDF bookmarks expanded
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}
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\DeclareGraphicsExtensions{.pdf}
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pdftitle={Formelsammlung HSD}, %%
pdfauthor={Paul Götzinger}, %%
pdfsubject={}, %%
pdfcreator={}, %%
pdfproducer={}, %%
pdfkeywords={} %%
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\parindent0pt
\pgfplotsset{compat=1.14}
\DeclareMathOperator{\sign}{Sign}
\DeclareMathOperator{\hevi}{H}
\DeclareMathOperator{\arcsinh}{arcsinh}
\DeclareMathOperator{\adj}{adj}
\DeclareMathOperator{\si}{si}
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\newcommand{\mygraphics}[3]{
\begin{center}
\includegraphics[width=#1, keepaspectratio=true]{#2} \\
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\title{Formelsammlung HSD}
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\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}}
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\begin{document}
% \pagenumbering{roman} %% small roman page numbers
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\thispagestyle{empty} %% no header/footer (only) on this page
\maketitle
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\newpage
\tableofcontents
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%%% begin main document
%%% 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]
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\draw (0.,0.)-- (6.,3.97);
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\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 ;
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\draw [line width=1.pt] (0.,0.) circle (1.2cm);
\draw [->,line width=1.pt] (0.,-5.) -- (0.,6.);
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\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]
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\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*}
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