\begin
Opdag hvorfor 18 millioner mennesker verden rundt stoler på Overleaf med deres arbejde.
\title {formelsamling}
\documentclass{article}
\pagenumbering{gobble} %inga bladnummer
%\renewcommand{\familydefault}{\sfdefault} %will switch to using sans-serif for everything except mathematics. The sans-serif will be computer modern sans unless you also put
%\usepackage{helvet} %san serif Helvetica-kopia (typsnitt)
%\documentclass{article}
%\usepackage{mathtools} % \abs{F} absoluta värden? clashes with what?
\usepackage{cancel} %att kunna stryka ut variabler
\usepackage{amssymb}
\usepackage[fleqn]{amsmath} % flush left equations
\mathindent=0pt % ingen indentering
\setlength{\columnsep}{30pt} % så att det skalll finnas "luft" mellan spalterna
\linespread{1.5} %ökar avstånd mellan rader till 1,8
\usepackage{textcomp} %för att kunna skriva \textmu{}
\usepackage [swedish]{babel}
\usepackage[utf8x]{inputenc} %krävs för att å ä ö skall skrivas ut rakt av
\usepackage{multicol} % så att jag kan ha flera spalter
\usepackage[a4paper,landscape]{geometry}
\usepackage{ziffer} %för korrekt avstånd mellan siffror och kommatecken
\usepackage{siunitx} %för att enhetsbeteckningar inte skrivs kursivt & för att kunna skriva Å för ångström i ekvationer
\sisetup{per-mode=fraction} %SI set-up ger horisontellt bråkstreck på alla enheter
%siunitx.pdf page 38
\sisetup{fraction-function=\dfrac} %dfrac skall göra att bråken blir större, luftigare
\sisetup{locale = FR} % SI set-up som gör allt FRanskt (kommatecken, t.ex.)
\let\times\cdot %gör att kommandot "\times" blir en prick varje gång (multiplikation)
\usepackage{tikz}
\usetikzlibrary{calc, intersections, decorations.markings, arrows, positioning}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usetikzlibrary{scopes} %krävs för exemplet med frikroppsdiagrammet
\setlength{\textheight}{450pt}
\setlength{\textwidth}{760pt}
\setlength{\topmargin}{-0.87in}
\setlength{\oddsidemargin}{-0.4in}
\setlength{\evensidemargin}{-0.4in}
\setlength{\parindent}{0pt}
\begin{document}
\begin{multicols}3
%\noindent A \hfill Z
%\setlength{\mathindent}{0pt}
\section*{TRIGONOMETRI}
\begin{minipage}[t]{0.27\textwidth}
\begin{multicols}2
\begin{equation*}
a^2 + b^2 = c^2
\end{equation*}
\begin{tikzpicture}[thick]
\coordinate (O) at (0,0);
\coordinate (A) at (3,2);
\coordinate (R) at (3,0);
\draw (O)--(A)--(R)--cycle;
\tkzLabelSegment[above left=2pt](O,A){$c$};
\tkzLabelSegment[below=2pt](O,R){$b$};
\tkzLabelSegment[right=2pt](A,R){$a$};
\tkzMarkRightAngle[size=0.5](O,R,A)% square angle here
\tkzMarkAngle[size=1cm](R,O,A)% sequence of points describes angle
\tkzLabelAngle[pos = 0.7](R,O,A){$\theta$}
\end{tikzpicture}
%\vfill
\columnbreak
\begin{eqnarray*}
\\ \sin\theta & = & \frac{a}{c}
\\ \cos\theta & = & \frac{b}{c}
\\ \tan\theta & = & \frac{a}{b}
% \tan\theta & = & \frac {\sin\theta}{\cos\theta}
\end{eqnarray*}
\end{multicols}
\end{minipage}
\section*{VEKTORER}
\begin{multicols}2
\begin{minipage}[t]{0.27\textwidth}
\begin{tikzpicture}
\tkzDefPoint(0,0){O} \tkzDefPoint(3,0){X} \tkzDefPoint(0,2){Y} \tkzDefPoint(3,2){P}
\draw [->][thin] (0,0)--(0,2);
\tkzLabelSegment[above=3pt](O,P){$\mathbf{\vec{F}_{res}}$};
\tkzLabelSegment[left=3pt](O,Y){$\vec{F}_y$};
\tkzLabelSegment[below=3pt](O,X){$\vec{F}_x$};
\draw [->][thin] (0,0)--(3,0);
\draw [->][very thick, black] (0,0)--(3,2);
\draw [thin, loosely dotted] (3,0)--(3,2);
\draw [thin, loosely dotted] (0,2)--(3,2);
\end{tikzpicture}
\end{minipage}
Summan av krafter i jämviktat system lika med noll (se kraftdiagram till höger)\\
$\vec{F}_N + \vec{F}_f + \vec{F}_g = 0$
\columnbreak
%\vfill
\begin{eqnarray*}
\mathbf{\vec{F}_{res}} &= \vec{F}_x + \vec{F}_y
\\ F_{res}^2 &= F_x^2 + F_y^2
\\ \Rightarrow F_{res} &= \sqrt{F_x^2 + F_y^2}
\end{eqnarray*}
\\
\begin{tikzpicture}[scale=1]%[dot/.style={draw,circle,minimum size=10mm,inner sep=0pt,outer sep=0pt,fill=black}
% Har inte fått det ifyllda cirkeln i mitten av kraftdiagrammet OMG !!!
%{coordinate [dot] (mitt) at (0,0)};
\tkzDefPoint(0,0){O} \tkzDefPoint(1,0.577){f} \tkzDefPoint(-1,1.73){N} \tkzDefPoint(0,-2.31){g}
\tkzLabelSegment[right=9pt](O,f){$\vec{F}_{f}$};
\tkzLabelSegment[left=1pt](O,N){$\vec{F}_N$};
\tkzLabelSegment[left=1pt](O,g){$\vec{F}_g$};
\filldraw [black] (0,0) circle (4pt);
\draw [->][thick] (0,0)--(1,0.577);
\draw [->][thick] (0,0)--(-1,1.73);
\draw [->][thick] (0,0)--(0,-2.31);
\end{tikzpicture}
\end{multicols}
\section*{POTENSER}
%\noindent A \hfill Z
%\\
\begin{align*}
a^x \times a^y &= a^{x+y} & 2^3 \times 2^2 &= 2^{(3+2)}
\\ \frac{a^x}{a^y} &= a^{x-y} & \frac{2^3}{2^2} &= 2^{(3-2)}
\\ \left(a^x \right)^y &= a^{xy} & (2^3)^2 &= 2^{(3 \times 2)}
\\ a^{-x} &= \frac{1}{a^x} & 2^{-3} &= \frac{1}{2^3}
\\ a^x \times b^x &= \left(a \times b \right)^x & 2^3 \times 5^3 &= (2 \times 5)^3
\\ \frac {a^x}{b^x} &= \left(\frac{a}{b}\right)^x & \frac {2^2}{5^2} &= \left(\frac{2}{5}\right)^2
\\ a^{\frac{1}{n}} &= \sqrt[n]{a} & 8^{\frac{1}{3}} &= \sqrt[3]{8}
\\ a^0 &= 1 & 2^0 &= 1
\end{align*}
\begin{tabular}{ >{$}l<{$} >{$}r<{$}@{,}>{$}l<{$} r l} % >{$}l<{$} definierar vilka spalter som är i math mode; @ tillåter alignering runt kommatecknet
%\text{tiopotens}& \text{siffra} & &&prefix \\
%\hline
10^{-12}& 0 & 000 000 000 001 & p & pico- \\
10^{-9} &0 & 000 000 001 & n & nano- \\
10^{ -6} &0 & 000 001 & \textmu{} & micro-\\
10^{ -3} &0 &001& m & milli-\\
10^{-2} &0 &01 & c & centi-\\
10^{ -1} &0 & 1 & d & deci-\\
10^{ 0} &1 & & & \\
10^{ 2} &100 & & h & hekto-\\
10^{ 3} &1000 & & k & kilo-\\
10^{ 6} &1000 000 & & M & mega-\\
10^{ 9} &1000 000 000 & & G & giga-\\
10^{ 12} &1000 000 000 000& & T & tera-\\
%\hline
\end{tabular}
%\begin{table*}
%\tabular[l l l r]
%\begin{tabular}{r@{.}l}
%\begin{math}
% 3 & 14159 \\
% 16 & 2 \\
% 123 & 456 \\
%\end{math}
%\end{tabular}
%\end{table*}
%
\section*{AREA, VOLYM \& DENSITET}
\begin{align*}
& A_{kvadrat} = x \times y &&A_{cirkel} = \pi \times r^2
%\\ &= \SI{2}{\m} \times \{SI}{2}{\m} &
%\\ &= \SI{4}{\m\squared}
% kvadrat
\\
&\begin{tikzpicture}
\draw[step=1cm][dashed, color=gray] (0,0) grid (2,2);
\draw [thick] (0,0)--(0,2)--(2,2)--(2,0)--(0,0);
\draw [|-|][thin] (2.5,0)--(2.5,2) node[midway, right] {y};
\draw [|-|][thin] (0,-0.5)--(2,-0.5) node[midway, below] {x};
\end{tikzpicture}
% cirkel
&&\begin{tikzpicture}
\draw[step=1cm] [dashed, color=gray](1,0) grid (3,2);
\draw [thick] (2,1)--(3,1) node[midway, above]{$r$};
\draw[thick](2,1) circle [radius=1cm] ;
\draw (0,-0.8) circle [radius=0pt]; %osynlig cirkel för att tvinga fram positionering
\end{tikzpicture}
\\
% rectangular tower !!!!!!!!!!!!!!!!
&\begin{tikzpicture} [scale=0.5]
%\fill[top color=gray!50!black,bottom color=gray!10,middle color=gray,shading=axis,opacity=0.2] (0,0) circle (2cm and 0.5cm);
%\fill[left color=gray!50!black,right color=gray!50!black,middle color=gray!50,shading=axis,opacity=0.2] (2,0) -- (2,6) arc (360:180:2cm and 0.5cm) -- (-2,0) arc (180:360:2cm and 0.5cm);
%\fill[top color=gray!90!,bottom color=gray!2,middle color=gray!30,shading=axis,opacity=0.2] (0,6) circle (2cm and 0.5cm);
%\draw (-2.5,6) -- (-2.5,0) arc (180:360:2.5cm and 0.5cm) -- (2.5,6) ++ (-2.5,0) circle (2.5cm and 0.5cm);
\draw (-2.5,0) -- (-2.5,6) -- (-0.3,5.6) -- (2.5,6) -- (2.5,0); % sidor & fram top
\draw (-2.5,0) -- (-0.3,-0.4) -- (2.5,0); %fram botten
\draw (-2.5,6) -- (0.3,6.4) -- (2.5,6); %bak top
\draw[densely dashed] (-2.5,0) -- (0.3,0.4) -- (2.5,0); %bak botten
\draw[densely dashed] (0.3,6.4) -- (0.3,0.4); %bak vertikalen
\draw (-0.3,5.6) -- (-0.3,-0.4); %framvertikalen
\fill[left color=gray!50!black,right color=gray!50!black,middle color=gray!50,shading=axis,opacity=0.2] (2.5,0) -- (2.5,6);
\draw[|-|][thin] (3.5,0) -- (3.5,6) node[midway, right] {$h$};
%\draw[|-|][densely dashed] (0,0) -- (2.5,0) node[midway, above] {$r$};
\end{tikzpicture}
%shaded cylinder **********
&&\begin{tikzpicture} [scale=0.5]
\fill[top color=gray!50!black,bottom color=gray!10,middle color=gray,shading=axis,opacity=0.2] (0,0) circle (2cm and 0.5cm);
\fill[left color=gray!50!black,right color=gray!50!black,middle color=gray!50,shading=axis,opacity=0.2] (2,0) -- (2,6) arc (360:180:2cm and 0.5cm) -- (-2,0) arc (180:360:2cm and 0.5cm);
\fill[top color=gray!90!,bottom color=gray!2,middle color=gray!30,shading=axis,opacity=0.2] (0,6) circle (2cm and 0.5cm);
\draw (-2,6) -- (-2,0) arc (180:360:2cm and 0.5cm) -- (2,6) ++ (-2,0) circle (2cm and 0.5cm);
\draw[densely dashed] (-2,0) arc (180:0:2cm and 0.5cm);
\draw[|-|][thin] (-3,0) -- (-3,6) node[midway, left] {$h$};
\draw[|-|][densely dashed] (0,0) -- (2,0) node[midway, above] {$r$};
\end{tikzpicture}
% *******************
\\ V &= A \times h & V &= A \times h
\\ &= \SI{4}{\m\squared} \times \SI{6}{\m}
\\ &= \SI{24}{\m\cubed}
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Densitet}
\begin{align*}
\\ \rho &= \frac {m}{V} &
\\ \rho_{vatten} &=\SI{1000}{\kg\per\m\cubed}= \frac{\SI{1000}{\kg}} {\SI{1}{\m\cubed}} \tag{vattnets densitet}
\\ &= \SI{1,00}{\kg\per\liter}
\end{align*}
\\
\subsection*{Omräkning kubikmeter till liter}
\begin{align*}
\SI{1}{m\cubed} &= (\SI{1}{\m})^3 \times 1
\\&= (\SI{1}{\m})^3 \times \left(\frac{\SI{10}{\dm}}{\SI{1}{\m}}\right)^3
\\&= \SI{1}{\cancel\m\cubed} \times \frac{\SI{e3}{\dm\cubed}}{\SI{1}{\cancel\m\cubed}}
%\\&= \SI{1}{\cancelto{\m}{\m\cubed}} \times \frac{\SI{e3}{\dm\cubed}}{\SI{1}{\cancel\m\cubed}}
\\ &= \SI{e3}{\dm\cubed}
\\ &= \SI{e3}{\litre}
\end{align*}
\section*{RÖRELSE}
\begin{align*}
v &= \frac {\Delta s}{\Delta t} & \SI{1}{\m\per\s}&= \frac{\SI{1}{\m}} {\SI{1}{\s}} \tag{hastighet}
\\
\\a &= \frac {\Delta v}{\Delta t} & \SI{1}{\m\per\s\squared}&= \frac{\frac{\SI{1}{\m}} {\SI{1}{\s}}}{\SI{1}{\s}} \tag{acceleration}
\\F &= m \times a & \SI{1}{\N} &= \SI{1}{\kg} \times \SI{1}{\m\per\s\squared} \tag{kraft}
\\W &= F \times \Delta s & \SI{1}{\J} &= \SI{1}{\N} \times \SI{1}{\m} \tag{arbete: Joule}
\\ &= E & \SI{1}{\J} \tag{energi: Joule}
\\P &= \frac{\Delta W}{\Delta t} & \SI{1}{\W} &= \frac{\SI{1}{\J}}{\SI{1}{\s}} \tag{effekt: Watt}
\end{align*}
\begin{align*} %alignera två ggr så att förskjutningen inte blir för stor för hela spalten
\\p=m \times v & &\SI{1}{\kg\m\per\s} &= \SI{1}{\kg} \times \SI{1}{\m\per\s} \tag{rörelsemängd}
\\E_{kinetisk} &= \frac{1}{2} \times m \times v^2 & \SI{0,5}{\J} &= \frac{1}{2} \times \SI{1}{\kg} \times \left(\SI{1}{\m\per\s}\right)^2 \tag{rörelseenergi}
\\E_{potential} &= m \times g \times h & \SI{10}{\J} &= \SI{1}{\kg} \times \SI{10}{\m\per\s\squared} \times \SI{1}{\m} \tag{lägesenergi}
\end{align*}
\section*{TRYCK \& VÄRME}
\begin{align*}
p &= \frac {F}{A} & \SI{1}{\pascal} &= \frac{\SI{1}{\newton}} {\SI{1}{\m\squared}} \tag{tryck: Pascal}
\\ p &= {\rho} \times g \times h & \SI{10000}{\Pa} &= \SI{1000}{\kg\per\m\cubed} \times \SI{10}{\m\per\s\squared} \times \SI{1}{\m} \tag{vätsketryck}
\\ F &= {\rho} \times g \times V & \SI{10000}{\N} &= \SI{1000}{\kg\per\m\cubed} \times \SI{10}{\m\per\s\squared} \times \SI{1}{\m\cubed} \tag{Arkimedes princip: lyftkraft i Newton}
\end{align*}
\begin{align*}
\\p \times V &= n \times R \times T \tag{Allmänna gaslagen}
\\ \SI{2e5}{\Pa} \times \SI{1}{\m\cubed} &= \SI{24,0}{\mol} \times \SI{8,314}{\Pa\m\cubed\per\mol\per\K} \times \SI{1002}{\K}
\end{align*}
\begin{align*}
\\ E &= c \times m \times T & \SI{800}{\kJ} &= \SI{4}{\kJ\per\kg\per\K} \times \SI{4}{\kg} \times \SI{50}{\kelvin} \tag{c = specifik värmekapacitet}
\\ \Delta E &= c \times m \times \Delta T & \SI{1000}{J} &= \SI{1000}{\joule\per\kg\per\K} \times \SI{1}{\kg} \times \SI{1}{\kelvin} \tag{c = specifik värmekapacitet}
\\ \Delta E &= l_s \times m & \SI{1000}{J} &= \SI{1000}{\joule\per\kg} \times \SI{1}{\kg} \tag{smältvärme}
\\ \Delta E &= l_a \times m & \SI{1000}{J} &= \SI{1000}{\joule\per\kg} \times \SI{1}{\kg} \tag{ångbildningsvärme} %OBS: subscript funkar inte att ha med i tags
\end{align*} % no empty line allowed before end
\section*{ELEKTRICITET}
\begin{align*}
F &= k_e \times \frac{Q_1 \times Q_2}{r^2} \tag{Coulombs lag, där $k_e = \SI{8,99e9}{\N\m\squared\per\coulomb\squared}$}
%\\ k_e &= \SI{8,99e9}{\N\m\squared\per\coulomb\squared}
\\ I &= \frac{Q}{t} & \SI{1}{A} &= \frac{\SI{1}{C}}{\SI{1}{s}} \tag{ström: Ampere}
\\ U &= \frac{E}{Q} & \SI{1}{\V} &= \frac{\SI{1}{J}}{\SI{1}{C}} \tag{spänning: Volt}
\\ U &= R \times I & \SI{1}{V} &= \SI{1}{\ohm} \times \SI{1}{A}\tag{Ohms lag, där R=Resistans=motstånd: Ohm}
\\ P &= U \times I & \SI{1}{W} &= \SI{1}{V} \times \SI{1}{A} \tag{Effekt: Watt}
\\ &= R \times I^2 & &= \SI{1}{\ohm} \times \SI{1}{A\squared}
\\ &= R \times I^2 & \SI{1}{W} &= \SI{1}{\ohm} \times (\SI{1}{A})^2)
\\ R_{tot} &= R_1 + R_2 \tag {för två seriekopplade motstånd}
\\ \frac{1}{R_{tot}} &= \frac{1}{R_1} + \frac{1}{R_2} \tag {för två parallelkopplade motstånd}
\end{align*}
\end{multicols}
\end{document}