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@@ -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 |