From c81e2200310504b3b4b174c21d8653c536170b6d Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Mon, 2 Aug 2021 09:47:15 +0200
Subject: Add appendix

---
 FuVar.tex | 132 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 131 insertions(+), 1 deletion(-)

(limited to 'FuVar.tex')

diff --git a/FuVar.tex b/FuVar.tex
index b607802..30e84d8 100644
--- a/FuVar.tex
+++ b/FuVar.tex
@@ -30,9 +30,14 @@
 
 %% Layout
 \usepackage{enumitem}
-\usepackage{booktabs}
 \usepackage{footmisc}
 
+%% Tables
+\usepackage{booktabs}
+\usepackage{multirow}
+\usepackage{tabularx}
+\usepackage{supertabular}
+
 %% Nice drwaings
 \usepackage{tikz}
 
@@ -74,6 +79,9 @@
 \newtheorem{note}{Note}
 
 \DeclareMathOperator{\tr}{\mathrm{tr}}
+\DeclareMathOperator{\arcsinh}{\mathrm{arcsinh}}
+\DeclareMathOperator{\arccosh}{\mathrm{arccosh}}
+\DeclareMathOperator{\arctanh}{\mathrm{arctanh}}
 
 \setlist[description]{
   format = { \normalfont\itshape }
@@ -821,6 +829,128 @@ Notice that the curl is a vector, normal to the enclosed surface \(A\).
   \end{itemize}
 \end{theorem}
 
+\appendix
+\section{Trigonometry}
+\begin{center}
+    \begin{tikzpicture}[scale=4]
+      \draw[gray,dashed] (0,0) --
+      node[pos=.7, sloped, above] {\(0\)}
+      node[pos=1, anchor=west, sloped] {\(\left(1,0,0\right)\)}
+      (1.1,0);
+
+      \draw[gray,dashed] (0,0) --
+      node[pos=.7, sloped, above] {\(\pi/2\)}
+      node[pos=1, anchor=west, sloped] {\(\left(0,1,\infty\right)\)}
+      (0,1.1);
+
+      \draw[gray,dashed] (0,0) --
+      node[pos=.7, sloped, above = -3pt] {\small \(\pi/12\)}
+      node[pos=1, anchor=west, sloped] {\(\left(\frac{1+ \sqrt3}{2\sqrt 2},\frac{\sqrt3 -1}{2\sqrt 2}\right)\)}
+      ({1.1 *cos(15)}, {1.1 * sin(15)});
+
+      \draw[gray,dashed] (0,0) --
+      node[pos=.7, sloped, above = -3pt] {\(\pi/8\)}
+      node[pos=1, anchor=west, sloped] {\(\scriptscriptstyle\left(\frac{\sqrt{2 + \sqrt{2}}}{2},\frac{\sqrt{2-\sqrt{2}}}{2}\right)\)}
+      ({1.1 *cos(pi/8 r)}, {1.1 * sin(pi/8 r)});
+
+      \draw[dashed] (0,0) --
+      node[pos=.7, sloped, above] {\(\pi/6\)}
+      node[pos=1, anchor=west, sloped] {\(\left(\frac{\sqrt 3}{2},\frac{1}{2},\frac{\sqrt3}{3}\right)\)}
+      ({1.1 *cos(30)}, {1.1 * sin(30)});
+
+      \draw[dashed] (0,0) --
+      node[pos=.7, sloped, above] {\(\pi/4\)}
+      node[pos=1, anchor=west, sloped] {\(\left(\frac{\sqrt 2}{2},\frac{\sqrt 2}{2}, 1\right)\)}
+      ({1.1 *cos(45)}, {1.1 * sin(45)});
+
+      \draw[dashed] (0,0) --
+      node[pos=.7, sloped, above] {\(\pi/3\)}
+      node[pos=1, anchor=west, sloped] {\(\left(\frac{1}{2},\frac{\sqrt 3}{2},\sqrt{3}\right)\)}
+      ({1.1 *cos(60)}, {1.1 * sin(60)});
+
+      \draw[black, thick] ({cos(-5)}, {sin(-5)}) arc (-5:100:1);
+    \end{tikzpicture}
+\end{center}
+\[
+\cos^2(x) + \sin^2(x) = 1 \quad \cosh^2(x) - \sinh^2(x) = 1
+\]
+\begin{center}
+  \begin{tabular}{>{\(}l<{\)} @{\(\;=\;\)} >{\(}r<{\)}   >{\(}l<{\)} @{\(\;=\;\)} >{\(}r<{\)} }
+    \toprule
+    \cos(\alpha + 2\pi) & \cos(\alpha) & \sin(\alpha + 2\pi) & \sin(\alpha) \\
+    \cos(-\alpha)                & \cos(\alpha)  & \sin(-\alpha)                & -\sin(\alpha) \\
+    \cos(\pi - \alpha)           & -\cos(\alpha) & \sin(\pi - \alpha)           & \sin(\alpha)  \\
+    \cos(\frac{\pi}{2} - \alpha) & \sin(\alpha)  & \sin(\frac{\pi}{2} - \alpha) & \cos(\alpha) \\
+    \midrule
+    \cos(\alpha + \beta) & \multicolumn{3}{l}{\(\cos\alpha\cos\beta - \sin\alpha\sin\beta\)} \\
+    \sin(\alpha + \beta) & \multicolumn{3}{l}{\(\sin\alpha\cos\beta - \cos\alpha\sin\beta\)} \\
+    \midrule
+    \cos(2\alpha) & \multicolumn{3}{l}{\(\cos^2{\alpha} - \sin^2{\alpha} \)} \\
+                  & \multicolumn{3}{l}{\(1 - 2\sin^2\alpha\)} \\
+                  & \multicolumn{3}{l}{\(2\cos^2\alpha - 1\)} \\
+    \sin(2\alpha) & \multicolumn{3}{l}{\(2\sin\alpha\cos\alpha\)} \\
+    \tan(2\alpha) & \multicolumn{3}{l}{\((2\tan\alpha)(1 + \tan^2\alpha)^{-1}\)} \\
+    \bottomrule
+  \end{tabular}
+\end{center}
+
+\section{Derivative}
+Let \(f, u, v\) be differentiable functions of \(x\).
+\begin{alignat*}{3}
+  (af)' &= af' &\quad&& (u(v))' &= u'(v)v' \\
+  (uv)' &= u'v + uv' &\quad&& \left(\frac{u}{v}\right)' &= \frac{u'v-uv'}{v^2} \\
+  \left(\sum u_i\right)' &= \sum u'_i &\quad&& (\ln u)' &= \frac{u'}{u} \\
+  (f^{-1})' &= \frac{1}{f'(f^{-1}(x))}
+\end{alignat*}
+
+\section{Integration}
+Let \(f, u, v\) be integrable functions of \(x\).
+\begin{center}
+  \setlength\extrarowheight{7pt}
+  \begin{tabularx}{\linewidth}{>{\itshape}p{.27\linewidth} >{\(\displaystyle}X<{\)}}
+    \toprule
+    Linearity & \int k(u + v) = k\left(\int u + \int v\right) \\
+    Partial fraction decomposition& \int \frac{Q}{P_n} \,dx = \sum_{k=1}^n \int \frac{A_k}{x-r_k}\,dx \\
+    Affine transformation & \int f(\lambda x + \ell) \,dx = \frac{1}{\lambda} F(\lambda x + \ell) + C \\
+    Integration by parts & \int u \,dv = uv - \int v \,du \\
+    Power rule \(n \neq -1\)& \int u^n \cdot u' = \frac{u^{n+1}}{n+1} + C \\
+    Logarithm rule & \int \frac{u'}{u} = \ln|u| + C \\
+    \multirow{2}{=}{General substitution \(x = g(u)\)} & \int f(x) \,dx = \int (f\circ g) ~ g' \,du \\
+    & = \int \frac{f \circ g}{(g^{-1})'\circ g} \,du \\
+    \multirow{2}{=}{Universal substitution} & t = \tan(x/2), dx = 2/(1+t^2) dt \\
+    & \sin(x) = \frac{2t}{1+t^2}, ~ \cos(t) = \frac{1-t^2}{1+t^2} \\
+    \bottomrule
+  \end{tabularx}
+\end{center}
+
+\section{Tables}
+Some useful derivatives and integrals:
+\begin{center}
+  \begin{tabularx}{\linewidth}{>{\(}l<{\)} >{\(}X<{\)} >{\(}l<{\)} >{\(}l<{\)}}
+    \toprule
+    f & f' & f & f'\\
+    \midrule
+    x^n & nx^{n-1} & a^x & a^x \ln a \\
+    \sqrt[n]{x} & 1/\left(x^n\sqrt[n]{x^{n-1}}\right) & \ln x & 1/x \\
+    \midrule
+    \sin x & \cos x &\cos x & -\sin x \\
+    \tan x & 1/\cos^2 x & 1/\tan x & -1/\sin^2 x \\
+    \arcsin x & 1/\sqrt{1-x^2} & \arccos x & -1/\sqrt{1-x^2} \\
+    \arctan x & 1/\left(1 + x^2\right) \\
+    \midrule
+    \sinh x & \cosh x & \tanh x & 1/\cosh^2 x \\
+    \arcsinh x & 1/\sqrt{1+x^2} & \arccosh x & 1/\sqrt{x^2 - 1} \\
+    \bottomrule
+  \end{tabularx}
+\end{center}
+\begin{align*}
+  \int \ln x \,dx &= x\ln x - x + C \\
+  \int \sin^2 ax \,dx &= \frac{x}{2} - \frac{\sin 2ax}{4a} +C\\
+  \int xe^{ax} \,dx &= \frac{e^{ax}}{a^2} (ax - 1) +C \\
+  \int x^2 e^{ax} \,dx &= e^{ax}\left(\frac{x^2}{a} - \frac{2x}{a^2} + \frac{2}{a^3}\right) +C \\
+  \int e^{ax} \sin bx \,dx &= \frac{e^{ax}}{a^2 + b^2} (a\sin bx - b\cos bx) +C
+\end{align*}
+
 \section*{License}
 \doclicenseText
 
-- 
cgit v1.2.1

