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109
BA1 - Analyse I/BA1 - Analyse I.tex Normal file
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\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}
\input{../Base.tex}
\title{Formulaire d'Analyse I}
\begin{document}
\begin{tabu}to \textwidth{ |X|X| }
\hline
\textbf{Polynômes de Taylor} \newline
$ \begin{aligned}
\e^x & = \sum\limits_{k = 0}^\infty \frac{x^k}{k!}, &&x \in \symbb{R} \\
\sinh \left( x \right) & = \sum\limits_{k = 0}^\infty \frac{x^{2k+1}}{\left( 2k+1 \right)!}, &&x \in \symbb{R} \\
\cosh \left( x \right) & = \sum\limits_{k = 0}^\infty \frac{x^{2k}}{\left( 2k \right)!}, &&x \in \symbb{R} \\
\sin \left( x \right) & = \sum\limits_{k = 0}^\infty \left( -1 \right)^k \cdot \frac{x^{2k+1}}{\left( 2k+1 \right)!}, &&x \in \symbb{R} \\
\cos \left( x \right) & = \sum\limits_{k = 0}^\infty \left( -1 \right)^k \cdot \frac{x^{2k}}{\left( 2k \right)!}, &&x \in \symbb{R} \\
\ln \left( 1+x \right) & = \sum\limits_{k = 0}^\infty \left( -1 \right)^{k+1} \cdot \frac{x^k}{k}, &&x \in \left] -1, 1 \right[ \\
\frac{1}{1+x} & = \sum\limits_{k = 0}^\infty \left( -1 \right)^{k} \cdot x^k, &&x \in \left] -1, 1 \right[ \\
\arctan \left( x \right) & = \sum\limits_{k = 0}^\infty \left( -1 \right)^k \cdot \frac{x^{2k+1}}{2k+1}, &&x \in \left] -1, 1 \right[ \\
\end{aligned} $
&
\textbf{Intégrales} \newline
$ \begin{aligned}
&\int \frac{f' \left( x \right)}{f \left( x \right)} \cdot \dif x && = \ln \abs{f \left( x \right)} + C \\
&\int \frac{f' \left( x \right)}{1+f^2 \left( x \right)} \cdot \dif x && = \arctan \left[ f \left( x \right) \right] + C \\
&\int \left[ f \left( x \right) \right]^\alpha \cdot f' \left( x \right) \cdot \dif x && = \frac{\left[ f \left( x \right) \right]^{\alpha+1}}{\alpha + 1} + C, &\forall \alpha \neq -1 \\
&\int \e^{f \left( x \right)} \cdot f' \left( x \right) \cdot \dif x && = \e^{f \left( x \right)} + C \\
&\int \frac{f' \left( x \right)}{\sqrt{1-f^2 \left( x \right)}} \cdot \dif x && = \arcsin \left[ f \left( x \right) \right] + C \\
\end{aligned} $
\\
\end{tabu}
\nointerlineskip
\begin{tabu}to \textwidth{ |X|X|X| }
\hline
\textbf{Racine carrée complexe} \newline
$ \begin{aligned}
w = u + v \cdot \im, z = a + b \cdot \im, z^2 = w \\
\begin{cases}
a^2 - b^2 & = u \\
2 \cdot a \cdot b & = v \\
a^2 + b^2 & = \sqrt{u^2 + v^2} \\
\end{cases}
\end{aligned} $
&
\textbf{Somme géométrique} \newline
$ \begin{aligned}
\sum\limits_{k = 0}^n q^k & = \frac{1-q^{n+1}}{1-q} \\
\sum\limits_{k = 0}^\infty q^k & = \frac{1}{1-q} \\
\end{aligned} $
&
\\\hline
\textbf{Exponentielle} \newline
$ \begin{aligned}
\cos \left( \theta \right) & = \frac{\e^{\im \cdot \theta} + \e^{-\im \cdot \theta}}{2} \\
\sin \left( \theta \right) & = \frac{\e^{\im \cdot \theta} - \e^{-\im \cdot \theta}}{2 \cdot \im} \\
\cosh \left( \theta \right) & = \frac{\e^{\theta} + \e^{-\theta}}{2} \\
\sinh \left( \theta \right) & = \frac{\e^{\theta} - \e^{-\theta}}{2} \\
\end{aligned} $
&
\textbf{Exponentielle} \newline
$ \begin{aligned}
\lim_{n \to \infty} \left( 1 + \frac{L}{n} \right)^n & = \e^L \\
\text{De manière\ générale~:} \\
\lim_{x \to \infty} f \left( x \right) & = +\infty \\
\lim_{x \to \infty} f \left( x \right) \cdot h \left( x \right) & = L \\
\lim_{x \to \infty} \left[ 1 + h \left( x \right) \right]^{f \left( x \right)} & = \e^L \\
\end{aligned} $
&
\textbf{Trigonométrie} \newline
$ \begin{aligned}
\cosh^2 \left( x \right) - \sinh^2 \left( x \right) = 1 \\
\cos^2 \left( x \right) + \sin^2 \left( x \right) = 1 \\
\sin \left( x+y \right) = \sin \left( x \right) \cdot \cos \left( y \right) + \cos \left( x \right) \cdot \sin \left( y \right) \\
\cos \left( x+y \right) = \cos \left( x \right) \cdot \cos \left( y \right) + \sin \left( x \right) \cdot \sin \left( y \right) \\
\sin \left( x \right) + \sin \left( y \right) = 2 \cdot \sin \left( \frac{x+y}{2} \right) \cdot \cos \left( \frac{x-y}{2} \right) \\
\sin \left( x \right) - \sin \left( y \right) = 2 \cdot \sin \left( \frac{x-y}{2} \right) \cdot \cos \left( \frac{x+y}{2} \right) \\
\cos \left( x \right) + \cos \left( y \right) = 2 \cdot \cos \left( \frac{x+y}{2} \right) \cdot \cos \left( \frac{x-y}{2} \right) \\
\cos \left( x \right) - \cos \left( y \right) = -2 \cdot \sin \left( \frac{x+y}{2} \right) \cdot \sin \left( \frac{x-y}{2} \right) \\
\end{aligned} $
\\\hline
\end{tabu}
\nointerlineskip
\begin{tabu}to \textwidth{ |X| }
\textbf{Angles particuliers} \newline
$ \begin{aligned}
\cos \left( 0 \right) = 1 \qquad &\cos \left( \frac{\pi}{6} \right) = \frac{1}{2} \qquad &\cos \left( \frac{\pi}{4} \right) = \frac{1}{2} \cdot \sqrt{2} \qquad &\cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \cdot \sqrt{3} \qquad &\cos \left( \frac{\pi}{2} \right) = 0 \\
\sin \left( 0 \right) = 0 \qquad &\sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \cdot \sqrt{3} \qquad &\sin \left( \frac{\pi}{4} \right) = \frac{1}{2} \cdot \sqrt{2} \qquad &\sin \left( \frac{\pi}{3} \right) = \frac{1}{2} \qquad &\sin \left( \frac{\pi}{2} \right) = 1 \\
\end{aligned} $
\\\hline
\textbf{Convergence} \newline
$ \begin{aligned}
&\int_M^\infty x^a \cdot \e^{-bx} \cdot \dif x &\text{ converge pour tout } a \in \symbb{R} \text{ et tout } b > 0 \\
&\int_a^\infty \frac{1}{x^p} \cdot \dif x &\text{ converge si et seulement si } p > 1 \quad \left( a > 0 \right) \\
&\int_0^b \frac{1}{x^p} \cdot \dif x &\text{ converge si et seulement si } p < 1 \\
\end{aligned} $
\\\hline
\end{tabu}
\end{document}

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BA1 - Analyse/BA1 - Analyse I.tex
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\documentclass[a4paper,10pt]{article}
%\documentclass[a4paper,10pt]{scrartcl}
\input{../Common.tex}
\begin{document}
\begin{tabularx}{\textwidth}{ |X|X| }
\hline
\textbf{Polynômes de Taylor} \newline
$\begin{aligned}
e^x &= \sum\limits_{k=0}^\infty \frac{x^k}{k!}, &x \in \mathbb{R} \\
\sinh(x) &= \sum\limits_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}, &x \in \mathbb{R} \\
\cosh(x) &= \sum\limits_{k=0}^\infty \frac{x^{2k}}{(2k)!}, &x \in \mathbb{R} \\
\sin(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!}, &x \in \mathbb{R} \\
\cos(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!}, &x \in \mathbb{R} \\
\ln(1+x) &= \sum\limits_{k=0}^\infty (-1)^{k+1} \frac{x^k}{k}, &x \in {]-1,1[} \\
\frac{1}{1+x} &= \sum\limits_{k=0}^\infty (-1)^{k} x^k, &x \in {]-1,1[} \\
\arctan(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{2k+1}, &x \in {]-1,1[} \\
\end{aligned}$ \newline
&
\textbf{Intégrales} \newline
$\begin{aligned}
&\int \frac{f'(x)}{f(x)}\mathrm{d}x &&= \ln \left|f(x)\right| + C \\
&\int \frac{f'(x)}{1+f^2(x)}\mathrm{d}x &&= \arctan \left[f(x)\right] + C \\
&\int \left[f(x)\right]^\alpha f'(x) \mathrm{d}x &&= \frac{\left[f(x)\right]^{\alpha+1}}{\alpha + 1} + C, &\forall \alpha \neq -1 \\
&\int e^{f(x)} f'(x) \mathrm{d}x &&= e^{f(x)} + C \\
&\int \frac{f'(x)}{\sqrt{1-f^2(x)}}\mathrm{d}x &&= \arcsin \left[f(x)\right] + C \\
\end{aligned}$ \newline
\\
\end{tabularx}
\offinterlineskip
\begin{tabularx}{\textwidth}{ |X|X|X| }
\hline
\textbf{Racine carrée complexe} \newline
$\begin{aligned}
w = u + vi, z = a + bi, z^2 = w \\
\begin{cases}
a^2 - b^2 &= u \\
2ab &= v \\
a^2 + b^2 &= \sqrt{u^2 + v^2} \\
\end{cases}
\end{aligned}$
&
\textbf{Somme géométrique} \newline
$\begin{aligned}
\sum\limits_{k=0}^n q^k &= \frac{1-q^{n+1}}{1-q} \\
\sum\limits_{k=0}^\infty q^k &= \frac{1}{1-q} \\
\end{aligned}$ \newline
&
\\ \hline
\textbf{Exponentielle} \newline
$\begin{aligned}
\cos(\theta) &= \frac{e^{i\theta} + e^{-i\theta}}{2} \\
\sin(\theta) &= \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
\cosh(\theta) &= \frac{e^{\theta} + e^{-\theta}}{2} \\
\sinh(\theta) &= \frac{e^{\theta} - e^{-\theta}}{2} \\
\end{aligned}$ \newline
&
\textbf{Exponentielle} \newline
$\begin{aligned}
\lim_{n \to \infty} \left(1 + \frac{L}{n}\right)^n &= e^L \\
\text{De manière\ générale :} \\
\lim_{x \to \infty} f(x) &= +\infty \\
\lim_{x \to \infty} f(x)h(x) &= L \\
\lim_{x \to \infty} \left[1 + h(x)\right]^{f(x)} &= e^L \\
\end{aligned}$ \newline
&
\textbf{Trigonométrie} \newline
$\begin{aligned}
\cosh^2(x) - \sinh^2(x) = 1 \\
\cos^2(x) + \sin^2(x) = 1 \\
\sin(x+y) = \sin x \cos y + \cos x \sin y \\
\cos(x+y) = \cos x \cos y + \sin x \sin y \\
\sin x + \sin y = 2 \sin(\frac{x+y}{2})\cos(\frac{x-y}{2}) \\
\sin x - \sin y = 2 \sin(\frac{x-y}{2})\cos(\frac{x+y}{2}) \\
\cos x + \cos y = 2 \cos(\frac{x+y}{2})\cos(\frac{x-y}{2}) \\
\cos x - \cos y = -2 \sin(\frac{x+y}{2})\sin(\frac{x-y}{2}) \\
\end{aligned}$ \newline
\\ \hline
\multicolumn{2}{|X|}{
\textbf{Angles particuliers} \newline
$\begin{aligned}
\cos(0) = 1 \quad &\cos(\frac{\pi}{6}) = \frac{1}{2} \quad &\cos(\frac{\pi}{4}) = \frac{1}{2}\sqrt{2} \quad &\cos(\frac{\pi}{3}) = \frac{1}{2}\sqrt{3} \quad &\cos(\frac{\pi}{2}) = 0 \\
\sin(0) = 0 \quad &\sin(\frac{\pi}{6}) = \frac{1}{2}\sqrt{3} \quad &\sin(\frac{\pi}{4}) = \frac{1}{2}\sqrt{2} \quad &\sin(\frac{\pi}{3}) = \frac{1}{2} \quad &\sin(\frac{\pi}{2}) = 1 \\
\end{aligned}$ \newline
}
&
\\ \hline
\multicolumn{2}{|X|}{
\textbf{Convergence} \newline
$\begin{aligned}
&\int_M^\infty x^a e^{-bx} \mathrm{d}x &\text{ converge pour tout } a \in \mathbb{R} \text{ et tout } b > 0 \\
&\int_a^\infty \frac{1}{x^p} \mathrm{d}x &\text{ converge si et seulement si } p > 1 \quad (a > 0) \\
&\int_0^b \frac{1}{x^p} \mathrm{d}x &\text{ converge si et seulement si } p < 1 \\
\end{aligned}$ \newline
}
&
\\ \hline
\end{tabularx}
\end{document}

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BA1 - Physique I/BA1 - Physique I.tex
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\documentclass[a4paper,10pt]{article}
%\documentclass[a4paper,10pt]{scrartcl}
\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}
\input{../Common.tex}
\input{../Base.tex}
\title{Formulaire de Physique I}
\begin{document}
\begin{tabularx}{\textwidth}{ |X|X|X| }
\begin{tabu}to \textwidth{ |X|X|X| }
\hline
\textbf{Produits vectoriels} \newline
$ \vec{e}_x \times \vec{e}_y = -\vec{e}_y \times \vec{e}_x = \vec{e}_z $ \newline
$ \vec{e}_y \times \vec{e}_z = -\vec{e}_z \times \vec{e}_y = \vec{e}_x $ \newline
$ \vec{e}_z \times \vec{e}_x = -\vec{e}_x \times \vec{e}_z = \vec{e}_y $ \newline
$ \vec{e}_x \times \vec{e}_x = \vec{e}_y \times \vec{e}_y = \vec{e}_z \times \vec{e}_z = \vec{0} $ \newline
$ \vec{e}_x \times \vec{e}_x = \vec{e}_y \times \vec{e}_y = \vec{e}_z \times \vec{e}_z = \vec{0} $
&
\textbf{MRUA} \newline
$ r = \frac{1}{2} \cdot a_0 \cdot t^2 + v_0 \cdot t + r_0 $ \newline
$ v = a_0 \cdot t + v_0 $ \newline
$ a = a_0 $ \newline
$ a = a_0 $
&
\textbf{MCU} \newline
$ a = \frac{v^2}{r} = \omega^2 \cdot r$ \newline
$ a = \frac{v^2}{r} = \omega^2 \cdot r $ \newline
$ \vec{v} = \vec{\omega} \times \vec{r} $ \newline
$ \vec{a} = \vec{\alpha} \times \vec{r} $ \newline
$ \omega \cdot T = 2 \cdot \pi $ \newline
\\ \hline
$ \omega \cdot T = 2 \cdot \pi $
\\\hline
\textbf{Moments / Centre de masse} \newline
$ \vec{L}_O = \vec{r} \times \vec{p} = m \cdot \vec{r} \times \vec{v} $ \newline
$ \vec{M}_O = \vec{r} \times \vec{F} = \frac{\mathrm{d}\vec{L}_O}{\mathrm{d}t} $ \newline
$ \vec{r}_{cm} = \frac{1}{M} \int_{M} \vec{r} \cdot \mathrm{d}m = \frac{1}{M} \int_{V} \vec{r} \cdot \rho(\vec{r}) \cdot \mathrm{d}V $ \newline
$ I_{cm,\Delta} = \int_{M} r_\bot^2 \cdot \mathrm{d}m $ \newline
$ \vec{L}_{cm,\Delta} = I_{cm,\Delta} \cdot \vec{\omega} $ \newline
$ \vec{M}_{cm,\Delta} = I_{cm,\Delta} \cdot \vec{\alpha} $ \newline
$ \vec{M}_O = \vec{r} \times \vec{F} = \frac{\dif\vec{L}_O}{\dif t} $ \newline
$ \vec{r}_{cm} = \frac{1}{M} \int_{M} \vec{r} \cdot \dif m = \frac{1}{M} \int_{V} \vec{r} \cdot \rho \left( \vec{r} \right) \cdot \dif V $ \newline
$ I_{cm, \Delta} = \int_{M} r_\bot^2 \cdot \dif m $ \newline
$ \vec{L}_{cm, \Delta} = I_{cm, \Delta} \cdot \vec{\omega} $ \newline
$ \vec{M}_{cm, \Delta} = I_{cm, \Delta} \cdot \vec{\alpha} $ \newline
$ I = I_{cm} + M \cdot r^2 $ \newline
$ \vec{r}_{cm} = \frac{1}{M} \sum m_i \cdot \vec{r}_i $ \newline
$ \vec{r}_{cm} = \frac{1}{M} \sum m_i \cdot \vec{r}_i $
&
\textbf{Forces} \newline
$ \vec{p} = m \cdot \vec{v} $ \newline
$ \vec{F} = m \cdot \vec{a} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t} $ \newline
$ \vec{F} = m \cdot \vec{a} = \frac{\dif\vec{p}}{\dif t} $ \newline
$ \vec{F}_f = \mu \cdot \vec{N} $ \newline
$ \vec{F}_f = -K \cdot \eta \cdot \vec{v} $ \newline
$ W = \int \vec{F} \bullet \mathrm{d}\vec{r} $ \newline
$ P_{inst} = \frac{\mathrm{d}W}{\mathrm{d}t} = \vec{F} \bullet \vec{v} $ \newline
$ P_{moy} = \frac{W}{\Delta t} $ \newline
$ W = \int \vec{F} \bullet \dif\vec{r} $ \newline
$ P_{inst} = \frac{\dif W}{\dif t} = \vec{F} \bullet \vec{v} $ \newline
$ P_{moy} = \frac{W}{\Delta t} $
&
\textbf{Énergie} \newline
$ W = \Delta E $ \newline
$ E_{mec} = E_{cin} + E_{pot} $ \newline
$ E_{mec,sat} = - \frac{G \cdot M \cdot m}{2 \cdot r} $ \newline
$ E_{mec, sat} = - \frac{G \cdot M \cdot m}{2 \cdot r} $ \newline
$ E_{cin} = \frac{1}{2} \cdot m \cdot v^2 $ \newline
$ E_{cin} = \frac{1}{2} \cdot m \cdot \omega_0^2 \cdot (A^2 - x^2) $ \newline
$ E_{cin} = \frac{1}{2} \cdot I_{cm,\Delta} \cdot \omega^2 $ \newline
$ E_{cin} = \frac{1}{2} \cdot m \cdot \omega_0^2 \cdot \left( A^2 - x^2 \right) $ \newline
$ E_{cin} = \frac{1}{2} \cdot I_{cm, \Delta} \cdot \omega^2 $ \newline
$ E_{pot} = m \cdot g \cdot h $ \newline
$ E_{pot} = \frac{1}{2} \cdot k \cdot x^2 = \frac{1}{2} \cdot m \cdot \omega_0^2 \cdot x^2 $ \newline
$ E_{pot} = - \frac{G \cdot M \cdot m}{r} $ \newline
\\ \hline
$ E_{pot} = - \frac{G \cdot M \cdot m}{r} $
\\\hline
\textbf{Référentiel non-galiléen} \newline
$ m \cdot \vec{a}' = \sum \vec{F}_{ext} - m \cdot \vec{a}_e - m \cdot \vec{a}_{Cor} $ \newline
$ - m \cdot \vec{a}_e = - m \cdot \vec{\omega} \times (\vec{\omega} \times \vec{r})$ \newline
$ - m \cdot \vec{a}_{Cor} = - 2 \cdot m \cdot \vec{\omega} \times \vec{v}' $ \newline
$ - m \cdot \vec{a}_e = - m \cdot \vec{\omega} \times \left( \vec{\omega} \times \vec{r} \right) $ \newline
$ - m \cdot \vec{a}_{Cor} = - 2 \cdot m \cdot \vec{\omega} \times \vec{v}' $
&
\textbf{Balistique} \newline
$ h_{max} = \frac{(v_0 \cdot \sin(\alpha))^2}{2 \cdot g} $ \newline
$ p = \frac{v_0^2 \cdot \sin(2 \cdot \alpha)}{g} $ \newline
$ h_{max} = \frac{\left( v_0 \cdot \sin \left( \alpha \right) \right)^2}{2 \cdot g} $ \newline
$ p = \frac{v_0^2 \cdot \sin \left( 2 \cdot \alpha \right)}{g} $
&
\textbf{Intégrales volumiques} \newline
$ V = \iiint\limits_{cube} \mathrm{d}V = \iiint \mathrm{d}x \cdot \mathrm{d}y \cdot \mathrm{d}z $ \newline
$ V = \iiint\limits_{cylindre} \mathrm{d}V = \iiint \rho \cdot \mathrm{d}\rho \cdot \mathrm{d}\varphi \cdot \mathrm{d}z $ \newline
$ V = \iiint\limits_{boule} \mathrm{d}V = \iiint r^2 \cdot \sin(\theta) \cdot \mathrm{d}r \cdot \mathrm{d}\theta \cdot \mathrm{d}\varphi $ \newline
\\ \hline
$ V = \iiint\limits_{cube} \dif V = \iiint \dif x \cdot \dif y \cdot \dif z $ \newline
$ V = \iiint\limits_{cylindre} \dif V = \iiint \rho \cdot \dif\rho \cdot \dif\varphi \cdot \dif z $ \newline
$ V = \iiint\limits_{boule} \dif V = \iiint r^2 \cdot \sin \left( \theta \right) \cdot \dif r \cdot \dif\theta \cdot \dif\varphi $
\\\hline
\textbf{Kepler} \newline
$ \frac{a^3}{T^2} = \frac{G \cdot M}{4 \cdot \pi^2} $ \hfill 1\textsuperscript{ère} loi \newline
$ \frac{\mathrm{d}\vec{A}}{\mathrm{d}t} = \frac{1}{2} \cdot \vec{r} \times \vec{v} = \frac{\vec{L}_O}{2 \cdot m} $ \hfill 2\textsuperscript{ème} loi \newline
$ \frac{\dif\vec{A}}{\dif t} = \frac{1}{2} \cdot \vec{r} \times \vec{v} = \frac{\vec{L}_O}{2 \cdot m} $ \hfill 2\textsuperscript{ème} loi \newline
$ \vec{F} = - \frac{G \cdot M \cdot m}{r^2} \cdot \vec{u_r} $ \hfill 3\textsuperscript{ème} loi \newline
$ T = 2 \cdot \pi \sqrt{\frac{R^3}{G \cdot M}} $ \newline
$ T = 2 \cdot \pi \cdot \sqrt{\frac{R^3}{G \cdot M}} $
&
\textbf{Dérivées usuelles} \newline
$ v = \dot{r} $ \newline
@ -85,99 +86,85 @@
$ \alpha = \dot{\omega} = \ddot{\varphi} $ \newline
$ F = \dot{p} $ \newline
$ P = \dot{W} $ \newline
$ M = \dot{L} $ \newline
$ M = \dot{L} $
&
\textbf{} \newline
\includegraphics[width=0.25\textwidth,keepaspectratio=true]{./Systèmes de coordonnées.png} \newline
\\ \hline
\textbf{Systèmes de coordonnées} \newline
\includegraphics[width=0.25\textwidth, keepaspectratio=true]{./Systèmes de coordonnées.png}
\\\hline
\textbf{Ressort / Pendule} \newline
$ \vec{F} = -k \cdot \vec{r} = -k \cdot (\vec{l} - \vec{l}_0) $ \hfill (ressort) \newline
$ \vec{F} = -k \cdot \vec{r} = -k \cdot \left( \vec{l} - \vec{l}_0 \right) $ \hfill (ressort) \newline
$ T_0 = \frac{2 \cdot \pi}{\omega_0} $ \newline
$ f_0 = \frac{1}{T_0} = \frac{\omega_0}{2 \cdot \pi} $ \newline
$ \omega_0 = \sqrt{\frac{k}{m}} \text{ ou } \omega_0 = \sqrt{\frac{g}{l}} $ \newline
$ \ddot{x} + \omega_0^2 \cdot x = 0 $ \newline
$ x(t) = A_1 \cdot \cos(\omega_0 \cdot t + \Phi) $ \newline
$ x \left( t \right) = A_1 \cdot \cos \left( \omega_0 \cdot t + \Phi \right) $
&
\textbf{Oscillateurs} \newline
$ \ddot{x} + 2 \cdot \lambda \cdot \dot{x} + \omega_0^2 \cdot x = 0 \mid x = C \cdot e^{\gamma \cdot t} $ \newline
$ \ddot{x} + 2 \cdot \lambda \cdot \dot{x} + \omega_0^2 \cdot x = 0 \mid x = C \cdot \e^{\gamma \cdot t} $ \newline
$ \gamma = - \lambda \pm \sqrt{\lambda^2 - \omega_0^2} $ \newline
$ \omega = \sqrt{| \omega_0^2 - \lambda^2 |} $ \newline
$ x(t) = A \cdot e^{- \lambda \cdot t} \cdot \cos(\omega \cdot t + \Phi), $ \hfill $ \lambda^2 < \omega_0^2 $ \newline
$ x(t) = e^{- \lambda \cdot t} \cdot (A_1 \cdot e^{\omega \cdot t} + A_2 \cdot e^{-\omega \cdot t}), $ \hfill $ \lambda^2 > \omega_0^2 $ \newline
$ x(t) = (A + B \cdot t) \cdot e^{- \lambda \cdot t}, $ \hfill $ \lambda^2 = \omega_0^2 $ \newline
$ \omega = \sqrt{\abs{\omega_0^2 - \lambda^2}} $ \newline
$ x \left( t \right) = A \cdot \e^{- \lambda \cdot t} \cdot \cos \left( \omega \cdot t + \Phi \right), $ \hfill $ \lambda^2 < \omega_0^2 $ \newline
$ x \left( t \right) = \e^{- \lambda \cdot t} \cdot \left( A_1 \cdot \e^{\omega \cdot t} + A_2 \cdot \e^{-\omega \cdot t} \right), $ \hfill $ \lambda^2 > \omega_0^2 $ \newline
$ x \left( t \right) = \left( A + B \cdot t \right) \cdot \e^{- \lambda \cdot t}, $ \hfill $ \lambda^2 = \omega_0^2 $
&
\textbf{Oscillateurs forcés} \newline
$ \ddot{x} + 2 \cdot \lambda \cdot \dot{x} + \omega_0^2 \cdot x = f \cdot \cos(\Omega \cdot t) $ \newline
$ x = A(\Omega) \cdot \cos(\Omega \cdot t + \psi) $ \newline
$ \underline{x} = A(\Omega) \cdot e^{i \cdot \psi(\Omega)} \cdot e^{i \cdot \Omega \cdot t} = x_0 \cdot e^{i \cdot \Omega \cdot t} $ \newline
$ \ddot{x} + 2 \cdot \lambda \cdot \dot{x} + \omega_0^2 \cdot x = f \cdot \cos \left( \Omega \cdot t \right) $ \newline
$ x = A \left( \Omega \right) \cdot \cos \left( \Omega \cdot t + \psi \right) $ \newline
$ \underline{x} = A \left( \Omega \right) \cdot \e^{\im \cdot \psi \left( \Omega \right)} \cdot \e^{\im \cdot \Omega \cdot t} = x_0 \cdot \e^{\im \cdot \Omega \cdot t} $ \newline
$ \omega_0 = \sqrt{\frac{k}{m}}, \lambda = \frac{\chi}{2 \cdot m}, f = \frac{F_e}{m} $ \newline
$ \omega = \sqrt{w_0^2 - \lambda^2}$ \newline
$ x_0 = A(\Omega) \cdot e^{i \cdot \psi(\Omega)} = \frac{f}{\omega_0^2 - \Omega^2 + i \cdot 2 \cdot \lambda \cdot \Omega} $ \newline
$ A(\Omega) = \|x_0\| = \frac{f}{\sqrt{(\omega_0^2 - \Omega^2)^2 + (2 \cdot \lambda \cdot \Omega)^2}} $ \newline
$ \psi(\Omega) = \arctan(\frac{\Im(x_0)}{\Re(x_0)}) = \arctan(\frac{-2 \cdot \lambda \cdot \Omega}{\omega_0^2 - \Omega^2}) $ \newline
$ \Omega_r = \sqrt{w_0^2 - 2 \cdot \lambda^2} $ \hfill $ \frac{\mathrm{d}A(\Omega)}{\mathrm{d}\Omega} = 0 $ \newline
$ Q = \frac{\Omega_r}{\Delta \Omega} = \frac{\Omega_r^2}{2 \cdot \lambda \cdot \omega} $ \newline
\\ \hline
$ \omega = \sqrt{w_0^2 - \lambda^2} $ \newline
$ x_0 = A \left( \Omega \right) \cdot \e^{\im \cdot \psi \left( \Omega \right)} = \frac{f}{\omega_0^2 - \Omega^2 + \im \cdot 2 \cdot \lambda \cdot \Omega} $ \newline
$ A \left( \Omega \right) = \abs{x_0} = \frac{f}{\sqrt{\left( \omega_0^2 - \Omega^2 \right)^2 + \left( 2 \cdot \lambda \cdot \Omega \right)^2}} $ \newline
$ \psi \left( \Omega \right) = \arctan \left( \frac{\Im \left( x_0 \right)}{\Re \left( x_0 \right)} \right) = \arctan \left( \frac{-2 \cdot \lambda \cdot \Omega}{\omega_0^2 - \Omega^2} \right) $ \newline
$ \Omega_r = \sqrt{w_0^2 - 2 \cdot \lambda^2} $ \hfill $ \frac{\dif A \left( \Omega \right)}{\dif\Omega} = 0 $ \newline
$ Q = \frac{\Omega_r}{\Delta \Omega} = \frac{\Omega_r^2}{2 \cdot \lambda \cdot \omega} $
\\\hline
\textbf{Coordonnées polaires $ (O,\vec{e_r},\vec{e}_{\varphi}) $} \newline
$ \vec{r} = r \cdotbis \vec{e_r} $ \newline
$ \vec{v} = \dot{r} \cdotbis \vec{e_r} + r \cdotbis \dot{\varphi} \cdotbis \vec{e}_{\varphi} $ \newline
$ \vec{a} = (\ddot{r} - r \cdotbis \dot{\varphi}^2) \cdotbis \vec{e_r} + (r \cdotbis \ddot{\varphi} + 2 \cdotbis \dot{r} \cdotbis \dot{\varphi}) \cdotbis \vec{e}_{\varphi} $ \newline
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e_r} = \dot{\varphi} \cdotbis \vec{e}_{\varphi} $ \newline
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\varphi} = -\dot{\varphi} \cdotbis \vec{e_r} $ \newline
\textbf{Coordonnées polaires $ \symbf{\left( O, \vec{e_r}, \vec{e}_{\varphi} \right)} $} \newline
$ \vec{r} = r \nocdot \vec{e_r} $ \newline
$ \vec{v} = \dot{r} \nocdot \vec{e_r} + r \nocdot \dot{\varphi} \nocdot \vec{e}_{\varphi} $ \newline
$ \vec{a} = \left( \ddot{r} - r \nocdot \dot{\varphi}^2 \right) \nocdot \vec{e_r} + \left( r \nocdot \ddot{\varphi} + 2 \nocdot \dot{r} \nocdot \dot{\varphi} \right) \nocdot \vec{e}_{\varphi} $ \newline
$ \frac{\dif}{\dif t} \nocdot \vec{e_r} = \dot{\varphi} \nocdot \vec{e}_{\varphi} $ \newline
$ \frac{\dif}{\dif t} \nocdot \vec{e}_{\varphi} = -\dot{\varphi} \nocdot \vec{e_r} $
&
\textbf{Coord. cylindriques $ (O,\vec{e}_{\rho},\vec{e}_{\varphi},\vec{e}_z) $} \newline
$ \vec{r} = \rho \cdotbis \vec{e}_{\rho} + z \cdotbis \vec{e}_z $ \newline
$ \vec{v} = \dot{\rho} \cdotbis \vec{e}_{\rho} + \rho \cdotbis \dot{\varphi} \cdotbis \vec{e}_{\varphi} + \dot{z} \cdotbis \vec{e}_z $ \newline
$ \vec{a} = (\ddot{\rho} - \rho \cdotbis \dot{\varphi}^2) \cdotbis \vec{e}_{\rho} + (\rho \cdotbis \ddot{\varphi} + 2 \cdotbis \dot{\rho} \cdotbis \dot{\varphi}) \cdotbis \vec{e}_{\varphi} + \ddot{z} \cdotbis \vec{e}_z $ \newline
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\rho} = \dot{\varphi} \cdotbis \vec{e}_{\varphi} $ \newline
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\varphi} = -\dot{\varphi} \cdotbis \vec{e}_{\rho} $ \newline
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_z = 0 $ \newline
\textbf{Coord. cylindriques $ \symbf{\left( O, \vec{e}_{\rho}, \vec{e}_{\varphi}, \vec{e}_z \right)} $} \newline
$ \vec{r} = \rho \nocdot \vec{e}_{\rho} + z \nocdot \vec{e}_z $ \newline
$ \vec{v} = \dot{\rho} \nocdot \vec{e}_{\rho} + \rho \nocdot \dot{\varphi} \nocdot \vec{e}_{\varphi} + \dot{z} \nocdot \vec{e}_z $ \newline
$ \vec{a} = \left( \ddot{\rho} - \rho \nocdot \dot{\varphi}^2 \right) \nocdot \vec{e}_{\rho} + \left( \rho \nocdot \ddot{\varphi} + 2 \nocdot \dot{\rho} \nocdot \dot{\varphi} \right) \nocdot \vec{e}_{\varphi} + \ddot{z} \nocdot \vec{e}_z $ \newline
$ \frac{\dif}{\dif t} \nocdot \vec{e}_{\rho} = \dot{\varphi} \nocdot \vec{e}_{\varphi} $ \newline
$ \frac{\dif}{\dif t} \nocdot \vec{e}_{\varphi} = -\dot{\varphi} \nocdot \vec{e}_{\rho} $ \newline
$ \frac{\dif}{\dif t} \nocdot \vec{e}_z = 0 $
&
\textbf{Coord. sphériques $ (O,\vec{e}_{r},\vec{e}_{\theta},\vec{e}_{\varphi}) $} \newline
$ \vec{r} = r \cdotbis \vec{e_r}$ \newline
$ \vec{v} = \dot{r} \cdotbis \vec{e_r} + r \cdotbis \dot{\theta} \cdotbis \vec{e}_{\theta} + r \cdotbis \dot{\varphi} \cdotbis \sin(\theta) \cdotbis \vec{e}_{\varphi} $ \newline
\textbf{Coord. sphériques $ \symbf{\left( O, \vec{e}_{r}, \vec{e}_{\theta}, \vec{e}_{\varphi} \right)} $} \newline
$ \vec{r} = r \nocdot \vec{e_r} $ \newline
$ \vec{v} = \dot{r} \nocdot \vec{e_r} + r \nocdot \dot{\theta} \nocdot \vec{e}_{\theta} + r \nocdot \dot{\varphi} \nocdot \sin \left( \theta \right) \nocdot \vec{e}_{\varphi} $ \newline
$ \vec{a} = \begin{pmatrix}
\ddot{r} - \dot{r} \cdotbis \dot{\theta}^2 - r \cdotbis \dot{\varphi}^2 \cdotbis \sin^2(\theta) \\
2 \cdotbis \dot{r} \cdotbis \dot{\theta} + r \cdotbis \ddot{\theta} - r \cdotbis \dot{\varphi}^2 \cdotbis \sin(\theta) \cdotbis \cos(\theta) \\
2 \cdotbis \dot{r} \cdotbis \dot{\varphi} \cdotbis \sin(\theta) + r \cdotbis \ddot{\varphi} \cdotbis \sin(\theta) + 2 \cdotbis r \cdotbis \dot{\varphi} \cdotbis \dot{\theta} \cdotbis \cos(\theta) \\
\ddot{r} - \dot{r} \nocdot \dot{\theta}^2 - r \nocdot \dot{\varphi}^2 \nocdot \sin^2 \left( \theta \right) \\
2 \nocdot \dot{r} \nocdot \dot{\theta} + r \nocdot \ddot{\theta} - r \nocdot \dot{\varphi}^2 \nocdot \sin \left( \theta \right) \nocdot \cos \left( \theta \right) \\
2 \nocdot \dot{r} \nocdot \dot{\varphi} \nocdot \sin \left( \theta \right) + r \nocdot \ddot{\varphi} \nocdot \sin \left( \theta \right) + 2 \nocdot r \nocdot \dot{\varphi} \nocdot \dot{\theta} \nocdot \cos \left( \theta \right) \\
\end{pmatrix} $ \newline
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e_r} = \dot{\theta} \cdotbis \vec{e}_{\theta} + \dot{\varphi} \cdotbis \sin(\theta) \cdotbis \vec{e}_{\varphi} $ \newline
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\theta} = -\dot{\theta} \cdotbis \vec{e_r} + \dot{\varphi} \cdotbis \cos(\theta) \cdotbis \vec{e}_{\varphi} $ \newline
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\varphi} = -\dot{\varphi} \cdotbis \sin(\theta) \cdotbis \vec{e_r} - \dot{\varphi} \cdotbis \cos(\theta) \cdotbis \vec{e}_{\theta} $ \newline
\\ \hline
$ \frac{\dif}{\dif t} \nocdot \vec{e_r} = \dot{\theta} \nocdot \vec{e}_{\theta} + \dot{\varphi} \nocdot \sin \left( \theta \right) \nocdot \vec{e}_{\varphi} $ \newline
$ \frac{\dif}{\dif t} \nocdot \vec{e}_{\theta} = -\dot{\theta} \nocdot \vec{e_r} + \dot{\varphi} \nocdot \cos \left( \theta \right) \nocdot \vec{e}_{\varphi} $ \newline
$ \frac{\dif}{\dif t} \nocdot \vec{e}_{\varphi} = -\dot{\varphi} \nocdot \sin \left( \theta \right) \nocdot \vec{e_r} - \dot{\varphi} \nocdot \cos \left( \theta \right) \nocdot \vec{e}_{\theta} $
\\\hline
\textbf{Équations de base} \newline
$ \sum \vec{F} = m \cdot \vec{a} $ \newline
$ \sum \vec{M}_O = \frac{\mathrm{d}}{\mathrm{d}t} \vec{L}_O $ \newline
$ \sum \vec{p} = cte $ \newline
$ E_i - E_f = 0 $ \newline
$ \sum \vec{M}_O = \frac{\dif}{\dif t} \vec{L}_O $ \newline
$ \sum \vec{p} = \cte $ \newline
$ E_i - E_f = 0 $
&
\textbf{} \newline
% \textbf{Signes} \newline
% $ r, v, a, \omega, \alpha, F $ \hfill avec \newline
% $ M, L, p $ \hfill sans \newline
% $ M, L, p $ \hfill sans
&
\textbf{Angles} \newline
$ \cos(\pi \pm \alpha) = - \cos(\alpha) $ \newline
$ \cos(\frac{\pi}{2} + \alpha) = - \sin(\alpha) $ \newline
$ \cos(\frac{\pi}{2} - \alpha) = \sin(\alpha) $ \newline
$ \sin(\pi + \alpha) = - \sin(\alpha) $ \newline
$ \sin(\pi - \alpha) = \sin(\alpha) $ \newline
$ \sin(\frac{\pi}{2} \pm \alpha) = \cos(\alpha) $ \newline
\\ \hline
% &
% \textbf{Configurabilité} \newline
% $ a \oldcdot b $ ou $ a b$ \newline
% $ \frac{a}{b} $ ou $ a/b$ \newline
% $ \vec{a} \oldbullet \vec{b} $ ou $ \vec{a} \circ \vec{b} $ \newline
% $ \oldvec{a} $ ou $ \overrightarrow{a} $ ou $ \mathbf{a} $ ou $ \oldvec{\mathbf{a}} $ \newline
% $ \dot{x} $ ou $ \frac{\mathrm{d}x}{\mathrm{d}t} $ \newline
% $ \ddot{x} $ ou $ \frac{\mathrm{d^2}x}{\mathrm{d}t^2} $ \newline
% &
% \\ \hline
\end{tabularx}
$ \cos \left( \pi \pm \alpha \right) = - \cos \left( \alpha \right) $ \hfill $ \sin \left( \pi + \alpha \right) = - \sin \left( \alpha \right) $ \newline
$ \cos \left( \frac{\pi}{2} + \alpha \right) = - \sin \left( \alpha \right) $ \hfill $ \sin \left( \pi - \alpha \right) = \sin \left( \alpha \right) $ \newline
$ \cos \left( \frac{\pi}{2} - \alpha \right) = \sin \left( \alpha \right) $ \hfill $ \sin \left( \frac{\pi}{2} \pm \alpha \right) = \cos \left( \alpha \right) $
\\\hline
\end{tabu}
\end{document}

174
BA2 - Chimie/BA2 - Chimie.tex
View File

@ -1,18 +1,19 @@
\documentclass[a4paper,10pt]{article}
%\documentclass[a4paper,10pt]{scrartcl}
\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}
\input{../Common.tex}
\input{../Base.tex}
\title{Formulaire de Chimie}
\begin{document}
\begin{tabularx}{\textwidth}{ |X|X| }
\begin{tabu}to \textwidth{ |X|X| }
\hline
\textbf{Bohr / Hydrogène} \newline
$ E_{photon} = h \cdot \nu $ \newline
$ E_{n} = \frac{-B}{n^2} $ \newline
$ \Delta E = E_f - E_i = B \cdot \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) $ \newline
$ \lambda = \frac{h}{m \cdot v} = \frac{c}{\nu} $ \newline
$ \lambda = \frac{h}{m \cdot v} = \frac{c}{\nu} $
&
\textbf{Thermodynamique} \newline
$ \Delta_r H^0 = \sum n_P \cdot \Delta_f H^0_P - \sum n_R \cdot \Delta_f H^0_R $ \newline
@ -20,69 +21,68 @@
$ \Delta_r G^0 = \sum n_P \cdot \Delta_f G^0_P - \sum n_R \cdot \Delta_f G^0_R $ \newline
$ \Delta_r G^0 = \Delta_r H^0 - T \cdot \Delta_r S^0 $ \hfill Spont. si $ \Delta_r G^0 < 0 $ \newline
$ \Delta S_{univers} = \Delta_r S^0 - \frac{\Delta_r H^0}{T} $ \newline
$ \Delta_r H^0 = \Delta_r U^0 + P \cdot \Delta V = \Delta_r U^0 + R \cdot T \cdot \Delta n $ \newline
\\ \hline
$ \Delta_r H^0 = \Delta_r U^0 + P \cdot \Delta V = \Delta_r U^0 + R \cdot T \cdot \Delta n $
\\\hline
\textbf{Équilibres} \newline
$ K_c = \frac{\prod [P]^{n_P}}{\prod [R]^{n_R}} $ \newline
$ K_c = \frac{\prod \left[ P \right]^{n_P}}{\prod \left[ R \right]^{n_R}} $ \newline
$ K_p = \frac{\prod P_P^{n_P}}{\prod P_R^{n_R}} $ \newline
$ K_c = K_p \cdot ( R \cdot T)^{-\Delta n} $ \newline
$ K_c = K_p \cdot \left( R \cdot T \right)^{-\Delta n} $
&
\textbf{Activités} \newline
$ a_i = \frac{P_i}{P_0} $ \hfill Gaz \newline
$ a_i = \frac{c_i}{c_0} $ \hfill Solutés \newline
$ a_i = 1 $ \hfill Liquides et solides \newline
$ K = \frac{\prod a_P^{n_P}}{\prod a_R^{n_R}} $ \newline
\\ \hline
$ K = \frac{\prod a_P^{n_P}}{\prod a_R^{n_R}} $
\\\hline
\textbf{Équilibres II} \newline
$ \Delta_r G = \Delta_r G^0 + R \cdot T \cdot \ln(Q) $ \newline
$ \Delta_r G^0 = -R \cdot T \cdot \ln(K) $ \newline
$ \ln\left(\frac{K_{T_2}}{K_{T_1}}\right) = \frac{\Delta_r H^0}{R} \cdot \frac{T_2 - T_1}{T_2 \cdot T_1} $ \newline
$ \Delta n = \sum n_P - \sum n_R $ \newline
$ \Delta_r G = \Delta_r G^0 + R \cdot T \cdot \ln \left( Q \right) $ \newline
$ \Delta_r G^0 = -R \cdot T \cdot \ln \left( K \right) $ \newline
$ \ln \left( \frac{K_{T_2}}{K_{T_1}} \right) = \frac{\Delta_r H^0}{R} \cdot \frac{T_2 - T_1}{T_2 \cdot T_1} $ \newline
$ \Delta n = \sum n_P - \sum n_R $
&
\textbf{Acide-Base} \newline
$ K_a = \frac{[A^-][H_3O^+]}{[HA]} $ \newline
$ K_b = \frac{[HA][OH^-]}{[A^-]} $ \newline
$ pX = -\log([X]) $ \newline
$ K_a = \frac{\left[ A^- \right] \left[ H_3O^+ \right]}{\left[ HA \right]} $ \newline
$ K_b = \frac{\left[ HA \right] \left[ OH^- \right]}{\left[ A^- \right]} $ \newline
$ pX = -\log \left( \left[ X \right] \right) $ \newline
$ pK_e = pK_a + pK_b = pH + pOH = 14 $ \hfill Eau \newline
$ \alpha = \sqrt{\frac{K_a}{M}} $ \hfill $ \alpha \leqslant 0.05 $ si faiblement dissocié \newline
$ pH = pK_a + \log\left(\frac{[A^-]}{[HA]}\right) $ \hfill Solution tampon \newline
\\ \hline
$ pH = pK_a + \log \left( \frac{\left[ A^- \right]}{\left[ HA \right]} \right) $ \hfill Solution tampon
\\\hline
\textbf{Électrochimie} \newline
$ n = \frac{I \cdot t}{z \cdot F} $ \newline
$ \eta = \frac{\Delta_r G^0}{\Delta_r H^0} $ \newline
$ \Delta E^0 = E^0_+ - E^0_- $ \hfill Spont. si $ \Delta E^0 > 0 $ \newline
$ \Delta_r G^0 = -z \cdot F \cdot \Delta E^0 $ \newline
$ \ln(K) = -\frac{\Delta_r G^0}{R \cdot T} = \frac{z \cdot F \cdot \Delta E^0}{R \cdot T} $ \newline
$ E_{Ox/Red} = E^0_{Ox/Red} + 2.3 \cdot \frac{R \cdot T}{z \cdot F} \cdot \log\left(\frac{[Ox]^{n_{Ox}}}{[Red]^{n_{Red}}}\right) $ \newline
$ \ln \left( K \right) = -\frac{\Delta_r G^0}{R \cdot T} = \frac{z \cdot F \cdot \Delta E^0}{R \cdot T} $ \newline
$ E_{Ox/Red} = E^0_{Ox/Red} + 2.3 \cdot \frac{R \cdot T}{z \cdot F} \cdot \log \left( \frac{\left[ Ox \right]^{n_{Ox}}}{\left[ Red \right]^{n_{Red}}} \right) $
&
\textbf{Cinétique} \newline
$ v = -\frac{1}{n_R} \cdot \frac{\mathrm{d}[R]}{\mathrm{d}t} = \frac{1}{n_P} \cdot \frac{\mathrm{d}[P]}{\mathrm{d}t} $ \newline
$ \tau_{1/2} = \frac{\ln(2)}{k} $ \hfill Ordre 1 \newline
$ \tau_{1/2} = \frac{1}{k \cdot [A]_0} $ \hfill Ordre 2 \newline
$ k = A_f \cdot e^{-\frac{E_a}{R \cdot T}} $ \newline
$ \ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \cdot \left( \frac{1}{T_1} - \frac{1}{T_2} \right) $ \newline
$ \Delta_r H^0 = E_a^\rightarrow - E_a^\leftarrow $ \newline
\\ \hline
$ v = -\frac{1}{n_R} \cdot \frac{\dif \left[ R \right]}{\dif t} = \frac{1}{n_P} \cdot \frac{\dif \left[ P \right]}{\dif t} $ \newline
$ \tau_{1/2} = \frac{\ln \left( 2 \right)}{k} $ \hfill Ordre 1 \newline
$ \tau_{1/2} = \frac{1}{k \cdot \left[ A \right]_0} $ \hfill Ordre 2 \newline
$ k = A_f \cdot \e^{-\frac{E_a}{R \cdot T}} $ \newline
$ \ln \left( \frac{k_2}{k_1} \right) = \frac{E_a}{R} \cdot \left( \frac{1}{T_1} - \frac{1}{T_2} \right) $ \newline
$ \Delta_r H^0 = E_a^ \rightarrow - E_a^ \leftarrow $
\\\hline
\end{tabularx}
\end{tabu}
\offinterlineskip
\nointerlineskip
\begin{tabularx}{\textwidth}{ |X|X|X| }
\begin{tabu}to \textwidth{ |X|X|X| }
\textbf{Loi de vitesse} & \textbf{Loi intégrée} & \textbf{Forme linéaire} \\
$ -\frac{\mathrm{d}[A]}{\mathrm{d}t} = k $ \hfill Ordre 0 & $ [A]_t = [A]_0 - k \cdot t $ & $ [A]_t = [A]_0 - k \cdot t $ \\
$ -\frac{\mathrm{d}[A]}{\mathrm{d}t} = k \cdot [A] $ \hfill Ordre 1 & $ [A]_t = [A]_0 \cdot e^{-k \cdot t} $ & $ \ln([A]_t) = \ln([A]_0) - k \cdot t $ \\
$ -\frac{\mathrm{d}[A]}{\mathrm{d}t} = k \cdot [A]^2 $ \hfill Ordre 2 & $ [A]_t = \frac{[A]_0}{1 + k \cdot t \cdot [A]_0} $ & $ \frac{1}{[A]_t} = \frac{1}{[A]_0} + k \cdot t $ \newline \\
$ -\frac{\dif \left[ A \right]}{\dif t} = k $ \hfill Ordre 0 & $ \left[ A \right]_t = \left[ A \right]_0 - k \cdot t $ & $ \left[ A \right]_t = \left[ A \right]_0 - k \cdot t $ \\
$ -\frac{\dif \left[ A \right]}{\dif t} = k \cdot \left[ A \right] $ \hfill Ordre 1 & $ \left[ A \right]_t = \left[ A \right]_0 \cdot \e^{-k \cdot t} $ & $ \ln \left( \left[ A \right]_t \right) = \ln \left( \left[ A \right]_0 \right) - k \cdot t $ \\
$ -\frac{\dif \left[ A \right]}{\dif t} = k \cdot \left[ A \right]^2 $ \hfill Ordre 2 & $ \left[ A \right]_t = \frac{\left[ A \right]_0}{1 + k \cdot t \cdot \left[ A \right]_0} $ & $ \frac{1}{\left[ A \right]_t} = \frac{1}{\left[ A \right]_0} + k \cdot t $ \newline \\
\hline
\end{tabularx}
\end{tabu}
\nointerlineskip
\offinterlineskip
\begin{tabularx}{\textwidth}{ |X|X| }
\begin{tabu}to \textwidth{ |X|X| }
\textbf{Constantes} \newline
$ N_A = \SI{6.02e23}{mol^{-1}} $ \newline
@ -90,18 +90,19 @@
$ B = \SI{2.179e-18}{J} $ \newline
$ F = \SI{96487}{C.mol^{-1}} $ \newline
$ R = \SI{0.0821}{L.atm.K^{-1}.mol^{-1}} $ \newline
$ R = \SI{0.0831}{L.bar.K^{-1}.mol^{-1}} $ \newline
$ R = \SI{0.0831}{L.bar.K^{-1}.mol^{-1}} $
&
\textbf{Conditions} \newline
Conditions normales : \SI{101.3}{kPa} et \SI{0}{°C} \newline
Conditions standards : \SI{1}{bar} et \SI{25}{°C} \newline \newline
Conditions normales~: \SI{101.3}{kPa} et \SI{0}{°C} \newline
Conditions standards~: \SI{1}{bar} et \SI{25}{°C} \newline
$ R = \SI{8.314}{L.kPa.K^{-1}.mol^{-1}} $ \newline
$ R = \SI{8.314}{J.K^{-1}.mol^{-1}} $ \newline
$ R = \SI{8.314}{m^3.Pa.K^{-1}.mol^{-1}} $ \newline
\\ \hline
$ R = \SI{8.314}{m^3.Pa.K^{-1}.mol^{-1}} $
\\\hline
\textbf{Construction d'une molécule} \newline
\begin{itemize}
\vspace{-\baselineskip}
\begin{itemize}[noitemsep, topsep=0pt]
\item Dénombrer les électrons de valence de tous les atomes de la molécule ou de l’ion.
\item Dessiner le squelette de la molécule en reliant les atomes les un aux autres par une pair d’électrons; l’atome le moins électronégatif occupe la place centrale.
\item Compléter les octets des atomes liés à l’atome central.
@ -110,16 +111,18 @@
\end{itemize}
&
\textbf{Équilibrage d'une réaction} \newline
\begin{itemize}
\vspace{-\baselineskip}
\begin{itemize}[noitemsep, topsep=0pt]
\item Repérer les éléments dont le degré d’oxydation (DO) change au cours de la réaction.
\item Le nombre d’électrons cédés par le réducteur doit être égal au nombre d’électrons acquis par l’oxydant. Ceci permet de trouver quatre coefficients.
\item S’il figure dans l’équation d’autres substances dont le DO n’est pas modifié, le coefficient de ces substances est déterminé par un bilan de masse.
\item Si des réactifs et/ou des produits sont des ions, il faut vérifier le calcul par un bilan de charges.
\end{itemize}
\\ \hline
\\\hline
\textbf{Formes} \newline
\begin{itemize}
\vspace{-\baselineskip}
\begin{itemize}[noitemsep, topsep=0pt]
\item Linéaire (sp).
\item Coudée (sp²).
\item Trigonale plane (sp²).
@ -128,57 +131,58 @@
\end{itemize}
&
\textbf{Nombres quantiques} \newline
\begin{itemize}
\item Principal : $ n \geqslant 1 $ \hfill Couche
\item Secondaire : $ 0 \leqslant l \leqslant n-1 $ \hfill Forme
\item Magnétique : $ -l \leqslant m_l \leqslant l $ \hfill Orientation
\item Spin : $ m_s = \pm 1/2 $ \hfill Sens de rotation sur lui-même
\vspace{-\baselineskip}
\begin{itemize}[noitemsep, topsep=0pt]
\item Principal~: $ n \geqslant 1 $ \hfill Couche
\item Secondaire~: $ 0 \leqslant l \leqslant n-1 $ \hfill Forme
\item Magnétique~: $ -l \leqslant m_l \leqslant l $ \hfill Orientation
\item Spin~: $ m_s = \pm 1/2 $ \hfill Sens de rotation sur lui-même
\end{itemize}
\\ \hline
\end{tabularx}
\\\hline
\end{tabu}
\begin{tabularx}{\textwidth}{ |X|X| }
\begin{tabu}to \textwidth{ |X|X| }
\hline
\textbf{Rayon atomique} \newline\newline
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./Rayon atomique.png} \newline
\textbf{Rayon atomique} \newline
\includegraphics[width=0.4\textwidth, keepaspectratio=true]{./Rayon atomique.png}
&
\textbf{Électronégativité} \newline\newline
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./Électronégativité.png} \newline
\\ \hline
\textbf{Électronégativité} \newline
\includegraphics[width=0.4\textwidth, keepaspectratio=true]{./Électronégativité.png}
\\\hline
\textbf{Pouvoir oxydant} \newline\newline
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./Pouvoir oxydant.png} \newline
\textbf{Pouvoir oxydant} \newline
\includegraphics[width=0.4\textwidth, keepaspectratio=true]{./Pouvoir oxydant.png}
&
\textbf{Énergie de ionisation} \newline\newline
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./Énergie de ionisation.png} \newline
\\ \hline
\textbf{Énergie de ionisation} \newline
\includegraphics[width=0.4\textwidth, keepaspectratio=true]{./Énergie de ionisation.png}
\\\hline
\textbf{Caractère métallique} \newline\newline
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./Caractère métallique.png} \newline
\textbf{Caractère métallique} \newline
\includegraphics[width=0.4\textwidth, keepaspectratio=true]{./Caractère métallique.png}
&
\textbf{Résumé} \newline\newline
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./Résumé.png} \newline
\\ \hline
\textbf{Résumé} \newline
\includegraphics[width=0.4\textwidth, keepaspectratio=true]{./Résumé.png}
\\\hline
\textbf{Géométrie} \newline\newline
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./Géométrie.png} \newline
\textbf{Géométrie} \newline
\includegraphics[width=0.4\textwidth, keepaspectratio=true]{./Géométrie.png}
&
\textbf{Titrage} \newline\newline
\textbf{Titrage} \newline
{
\begin{tabularx}{\textwidth}{cc}
\includegraphics[width=0.2\textwidth,keepaspectratio=true]{./Titrage acide fort.png} \newline &
\includegraphics[width=0.2\textwidth,keepaspectratio=true]{./Titrage acide faible.png} \newline
\end{tabularx}
\begin{tabu}to \textwidth{cc}
\includegraphics[width=0.2\textwidth, keepaspectratio=true]{./Titrage acide fort.png} \newline &
\includegraphics[width=0.2\textwidth, keepaspectratio=true]{./Titrage acide faible.png} \newline
\end{tabu}
}
\\ \hline
\\\hline
\textbf{Remplissage} \newline\newline
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./Remplissage.png} \newline
\textbf{Remplissage} \newline
\includegraphics[width=0.4\textwidth, keepaspectratio=true]{./Remplissage.png}
&
\textbf{} \newline\newline
\\ \hline
\textbf{}
\\\hline
\end{tabularx}
\end{tabu}
\end{document}
\end{document}

191
BA2 - Physique II/BA2 - Physique II.tex
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@ -1,236 +1,237 @@
\documentclass[a4paper,10pt]{article}
%\documentclass[a4paper,10pt]{scrartcl}
\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}
\input{../Common.tex}
\input{../Base.tex}
\title{Formulaire de Physique II}
\begin{document}
\begin{tabularx}{\textwidth}{ |X|X| }
\begin{tabu}to \textwidth{ |X|X| }
\hline
\textbf{Potentiels} \newline
$ F_x = -\frac{\partial U}{\partial x} $ \newline
$ \frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x} $ \newline
$ \frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x} $
&
\textbf{Lagrange} \newline
$ U = \sum m \cdot g \cdot h + \sum \frac{1}{2} \cdot k \cdot x^2 $ \newline
$ T = \sum \frac{1}{2} \cdot m \cdot v^2 + \sum \frac{1}{2} \cdot I \cdot \omega^2 $ \newline
$ L = T -U $ \newline
$ \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial}{\partial \dot{q_j}} L \right) - \frac{\partial}{\partial q_j} L = 0 $ \newline
\\ \hline
$ \frac{\dif}{\dif t} \left( \frac{\partial}{\partial \dot{q_j}} L \right) - \frac{\partial}{\partial q_j} L = 0 $
\\\hline
\textbf{Gaz} \newline
$ P \cdot V = n \cdot R \cdot T = N \cdot k_B \cdot T $ \hfill Parfait \newline
$ \left( p + \frac{n^2 \cdot a}{V^2} \right) (V -n \cdot b) = n \cdot R \cdot T $ \hfill Van der Waals \newline
$ \left( p + \frac{n^2 \cdot a}{V^2} \right) \left( V -n \cdot b \right) = n \cdot R \cdot T $ \hfill Van der Waals
&
\textbf{Maxwell-Boltzmann} \newline
$ P_i = Cst \cdot e^{-\frac{E_i}{k_B \cdot T}} $ \newline
$ \sum P_i = 1 $ \newline
\\ \hline
$ P_i = \cte \cdot \e^{-\frac{E_i}{k_B \cdot T}} $ \newline
$ \sum P_i = 1 $
\\\hline
\textbf{Lois thermodynamiques} \newline
$ \mathrm{d} U = \delta W + \delta Q $ \hfill 1\textsuperscript{ère} \newline
$ \mathrm{d} S = \delta S_{ext} + \delta S_{int} = \frac{\delta Q}{T} + \delta S_{int} $ \hfill 2\textsuperscript{ème} \newline
$ \dif U = \delta W + \delta Q $ \hfill 1\textsuperscript{ère} \newline
$ \dif S = \delta S_{ext} + \delta S_{int} = \frac{\delta Q}{T} + \delta S_{int} $ \hfill 2\textsuperscript{ème}
&
\textbf{Énergies} \newline
$ U = \frac{f}{2} \cdot n \cdot R \cdot T $ \newline
$ H = U + P \cdot V = \frac{f}{2} \cdot n \cdot R \cdot T + n \cdot R \cdot T $ \newline
\\ \hline
$ H = U + P \cdot V = \frac{f}{2} \cdot n \cdot R \cdot T + n \cdot R \cdot T $
\\\hline
\textbf{Isentropie} \newline
$ P \cdot V^\gamma = cte $ \newline
$ T \cdot V^{\gamma - 1} = cte $ \newline
$ P \cdot V^\gamma = \cte $ \newline
$ T \cdot V^{\gamma - 1} = \cte $
&
\textbf{Énergies II} \newline
$ U = C_v \cdot \Delta T $ \newline
$ Q = C_v \cdot \Delta T $ \hfill Isochore \newline
$ Q = C_p \cdot \Delta T $ \hfill Isobare \newline
$ W = - \int p_{ext} \cdot \mathrm{d}V = -W_{ext}$ \newline
\\ \hline
$ W = - \int p_{ext} \cdot \dif V = -W_{ext} $
\\\hline
\textbf{Chaleurs} \newline
$ C_p = C_v \cdot \gamma $ \newline
$ C_p = C_v + n \cdot R $ \newline
$ C_v = \frac{\partial U}{\partial T} = \frac{n \cdot R}{\gamma -1} $ \newline
$ C_p = \frac{\partial H}{\partial T} = \frac{\gamma \cdot n \cdot R}{\gamma -1} $ \newline
$ C_p = \frac{\partial H}{\partial T} = \frac{\gamma \cdot n \cdot R}{\gamma -1} $
&
\textbf{Rendements} \newline
$ \eta_{Carnot} = \frac{T_c - T_f}{T_c} $ \newline
$ \eta = -\frac{W}{Q_c} $ \hfill Moteur \newline
$ \eta = -\frac{Q_c}{W} $ \hfill Récepteur chauffant \newline
$ \eta = \frac{Q_f}{W} $ \hfill Récepteur refroidissant \newline
\\ \hline
$ \eta = \frac{Q_f}{W} $ \hfill Récepteur refroidissant
\\\hline
\textbf{Cycle} \newline
$ \circlearrowright $ Cycle moteur \newline
$ \circlearrowleft $ Cycle récepteur \newline
$ \circlearrowleft $ Cycle récepteur
&
\textbf{Cycle II} \newline
$ \Delta U = 0 = W + Q_c + Q_f $ \newline
$ \Delta S = 0 = \int \frac{\delta Q_c}{T} + \int \frac{\delta Q_f}{T} + S_{int}$ \newline
$ W = - (Q_c + Q_f) $ \newline
\\ \hline
$ \Delta S = 0 = \int \frac{\delta Q_c}{T} + \int \frac{\delta Q_f}{T} + S_{int} $ \newline
$ W = - \left( Q_c + Q_f \right) $
\\\hline
\textbf{Conductibilité} \newline
$ \lambda = \frac{1}{\rho \cdot 4 \cdot \sqrt{2} \cdot \pi \cdot R^2} $ \newline
$ \rho = \frac{p}{k_B \cdot T} $ \newline
$ J_Q = -k \cdot \frac{\partial T}{\partial x} $ \newline
$ \frac{\partial Q}{\partial T} = A \cdot \alpha \cdot \frac{\partial T}{\partial x} $ \hfill $ \lambda \ll d $ \newline
$ \frac{\partial Q}{\partial T} = \mathrm{d}A \cdot \kappa \cdot \Delta T $ \hfill $ \lambda \gg d $ \newline
$ \frac{\partial Q}{\partial T} = \dif A \cdot \kappa \cdot \Delta T $ \hfill $ \lambda \gg d $
&
\textbf{Diffusion} \newline
$ \frac{\partial \rho \cdot u}{\partial t} + \frac{\partial J_U}{\partial x} = \sigma_U $ \newline
$ J_U = -\lambda \cdot \frac{\partial T}{\partial x} $ \newline
$ \frac{\partial \rho \cdot u}{\partial t} - \lambda \cdot \frac{\partial^2 T}{\partial x^2} = \sigma_U $ \newline
\\ \hline
$ \frac{\partial \rho \cdot u}{\partial t} - \lambda \cdot \frac{\partial^2 T}{\partial x^2} = \sigma_U $
\\\hline
\textbf{Lennard-Jones} \newline
$ E = 4 \cdot \varepsilon_0 \cdot \left( \left( \frac{r_1}{r} \right)^{12} - \left( \frac{r_1}{r} \right)^6 \right) $ \newline
$ E = \varepsilon_0 \cdot \left( \left( \frac{r_0}{r} \right)^{12} - 2 \cdot \left( \frac{r_0}{r} \right)^6 \right) $ \newline
$ E = \varepsilon_0 \cdot \left( \left( \frac{r_0}{r} \right)^{12} - 2 \cdot \left( \frac{r_0}{r} \right)^6 \right) $
&
\textbf{Lennard-Jones II} \newline
\includegraphics[width=0.2\textwidth,keepaspectratio=true]{./Potentiel de Lennard-Jones.png} \newline
\\ \hline
\end{tabularx}
\offinterlineskip
\begin{tabularx}{\textwidth}{ |X|X|X| }
\textbf{Diagramme de phase} \newline\newline
\includegraphics[width=0.3\textwidth,keepaspectratio=true]{./Diagramme de phase.png} \newline
&
\textbf{Diagramme P-V} \newline\newline
\includegraphics[width=0.3\textwidth,keepaspectratio=true]{./Diagramme P-V.png} \newline
&
\textbf{Diagramme P-T} \newline\newline
\includegraphics[width=0.3\textwidth,keepaspectratio=true]{./Diagramme P-T.png} \newline
\includegraphics[width=0.2\textwidth, keepaspectratio=true]{./Potentiel de Lennard-Jones.png}
\\\hline
\end{tabularx}
\begin{tabularx}{\textwidth}{ |X|X|X|X|X| }
\end{tabu}
\nointerlineskip
\begin{tabu}to \textwidth{ |X|X|X| }
\textbf{Diagramme de phase} \newline
\includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Diagramme de phase.png}
&
\textbf{Diagramme P-V} \newline
\includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Diagramme P-V.png}
&
\textbf{Diagramme P-T} \newline
\includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Diagramme P-T.png}
\\\hline
\end{tabu}
\begin{tabu}to \textwidth{ |X|X|X|X|X| }
\hline
\textit{Résultats uniquement pour le cas réversible} & Isotherme & Isobare & Isochore & Adiabatique
\\\hline
Constantes &
$\begin{aligned} P \cdot V = cte \end{aligned}$ &
$\begin{aligned} \frac{V}{T} = cte \end{aligned}$ &
$\begin{aligned} \frac{P}{T} = cte \end{aligned}$ &
$\begin{aligned} P \cdot V^\gamma = cte \\ T \cdot V^{\gamma - 1} = cte \end{aligned}$
$ \begin{aligned} P \cdot V = \cte \end{aligned} $ &
$ \begin{aligned} \frac{V}{T} = \cte \end{aligned} $ &
$ \begin{aligned} \frac{P}{T} = \cte \end{aligned} $ &
$ \begin{aligned} P \cdot V^\gamma = \cte \\ T \cdot V^{\gamma - 1} = \cte \end{aligned} $
\\\hline
Énergie interne &
$
\begin{aligned}
\Delta U &= 0
\Delta U & = 0
\end{aligned}
$ &
$
\begin{aligned}
\Delta U &= C_v \cdot \Delta T \\
&= \frac{n \cdot R}{\gamma - 1} \Delta T \\
&= \frac{p_0}{\gamma - 1} \Delta V \\
&= C_v \cdot \frac{T_0}{V_0} \cdot \Delta V
\Delta U & = C_v \cdot \Delta T \\
& = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
& = \frac{p_0}{\gamma - 1} \cdot \Delta V \\
& = C_v \cdot \frac{T_0}{V_0} \cdot \Delta V
\end{aligned}
$ &
$
\begin{aligned}
\Delta U &= C_v \cdot \Delta T \\
&= \frac{n \cdot R}{\gamma - 1} \Delta T \\
&= \frac{V_0}{\gamma - 1} \Delta p \\
&= C_v \cdot \frac{T_0}{p_0} \cdot \Delta p
\Delta U & = C_v \cdot \Delta T \\
& = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
& = \frac{V_0}{\gamma - 1} \cdot \Delta p \\
& = C_v \cdot \frac{T_0}{p_0} \cdot \Delta p
\end{aligned}
$ &
$
\begin{aligned}
\Delta U &= C_v \cdot \Delta T \\
&= \frac{n \cdot R}{\gamma - 1} \Delta T \\
&= \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \Delta (V^{1-\gamma})
\Delta U & = C_v \cdot \Delta T \\
& = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
& = \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \cdot \Delta \left( V^{1-\gamma} \right)
\end{aligned}
$
\\\hline
Chaleur &
$
\begin{aligned}
Q &= n \cdot R \cdot T_0 \cdot \ln \frac{V_1}{V_0} \\
&= n \cdot R \cdot T_0 \cdot \ln \frac{p_1}{p_0} \\
Q & = n \cdot R \cdot T_0 \cdot \ln \left( \frac{V_1}{V_0} \right) \\
& = n \cdot R \cdot T_0 \cdot \ln \left( \frac{p_1}{p_0} \right) \\
\end{aligned}
$ &
$
\begin{aligned}
Q &= C_p \cdot \Delta T \\
&= \frac{\gamma \cdot n \cdot R}{\gamma - 1} \Delta T \\
&= \frac{\gamma \cdot p_0}{\gamma - 1} \Delta V \\
Q & = C_p \cdot \Delta T \\
& = \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \Delta T \\
& = \frac{\gamma \cdot p_0}{\gamma - 1} \cdot \Delta V \\
\end{aligned}
$ &
$
\begin{aligned}
Q &= C_v \cdot \Delta T \\
&= \frac{n \cdot R}{\gamma - 1} \Delta T \\
&= \frac{V_0}{\gamma - 1} \Delta p \\
Q & = C_v \cdot \Delta T \\
& = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
& = \frac{V_0}{\gamma - 1} \cdot \Delta p \\
\end{aligned}
$ &
$
\begin{aligned}
Q &= 0
Q & = 0
\end{aligned}
$
\\\hline
Travail &
$
\begin{aligned}
W &= -n \cdot R \cdot T_0 \cdot \ln \frac{V_1}{V_0} \\
&= -n \cdot R \cdot T_0 \cdot \ln \frac{p_1}{p_0} \\
W & = -n \cdot R \cdot T_0 \cdot \ln \left( \frac{V_1}{V_0} \right) \\
& = -n \cdot R \cdot T_0 \cdot \ln \left( \frac{p_1}{p_0} \right) \\
\end{aligned}
$ &
$
\begin{aligned}
W &= -p_0 \cdot \Delta V \\
&= -n \cdot R \cdot \Delta T \\
&= -p_0 \cdot \frac{V_0}{T_0} \cdot \Delta V \\
W & = -p_0 \cdot \Delta V \\
& = -n \cdot R \cdot \Delta T \\
& = -p_0 \cdot \frac{V_0}{T_0} \cdot \Delta V \\
\end{aligned}
$ &
$
\begin{aligned}
W &= 0 \\
W & = 0 \\
\end{aligned}
$ &
$
\begin{aligned}
W &= C_v \cdot \Delta T \\
&= \frac{n \cdot R}{\gamma - 1} \Delta T \\
&= \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \Delta (V^{1-\gamma})
W & = C_v \cdot \Delta T \\
& = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
& = \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \cdot \Delta \left( V^{1-\gamma} \right)
\end{aligned}
$
\\\hline
Entropie &
$
\begin{aligned}
\Delta S &= n \cdot R \cdot \ln \frac{V_1}{V_0} \\
&= n \cdot R \cdot \ln \frac{p_1}{p_0} \\
\Delta S & = n \cdot R \cdot \ln \left( \frac{V_1}{V_0} \right) \\
& = n \cdot R \cdot \ln \left( \frac{p_1}{p_0} \right) \\
\end{aligned}
$ &
$
\begin{aligned}
\Delta S &= C_p \cdot \ln \frac{V_1}{V_0} \\
&= \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \frac{V_1}{V_0} \\
&= C_p \cdot \ln \frac{T_1}{T_0} \\
&= \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \frac{T_1}{T_0}
\Delta S & = C_p \cdot \ln \left( \frac{V_1}{V_0} \right) \\
& = \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{V_1}{V_0} \right) \\
& = C_p \cdot \ln \left( \frac{T_1}{T_0} \right) \\
& = \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{T_1}{T_0} \right)
\end{aligned}
$ &
$
\begin{aligned}
\Delta S &= C_v \cdot \ln \frac{p_1}{p_0} \\
&= \frac{n \cdot R}{\gamma - 1} \cdot \ln \frac{p_1}{p_0} \\
&= C_v \cdot \ln \frac{T_1}{T_0} \\
&= \frac{n \cdot R}{\gamma - 1} \cdot \ln \frac{T_1}{T_0}
\Delta S & = C_v \cdot \ln \left( \frac{p_1}{p_0} \right) \\
& = \frac{n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{p_1}{p_0} \right) \\
& = C_v \cdot \ln \left( \frac{T_1}{T_0} \right) \\
& = \frac{n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{T_1}{T_0} \right)
\end{aligned}
$ &
$
\begin{aligned}
\Delta S &= 0
\Delta S & = 0
\end{aligned}
$
\\\hline
\end{tabularx}
\end{tabu}
\end{document}
\end{document}

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\documentclass[a4paper,10pt]{article}
%\documentclass[a4paper,10pt]{scrartcl}
\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}
\input{../Common.tex}
\input{../Base.tex}
\geometry{top=6.5pt, bottom=6pt, left=6.5pt, right=6pt}
\geometry{top=18pt, bottom=18pt, left=6pt, right=6pt, headsep=-5pt, headheight=12pt, footskip=7pt}
\title{Formulaire d'Analyse numérique}
\begin{document}
% \pagestyle{plain}
\includepdf[width=0.5\textwidth,pages={-},nup=2x2]{BA3 - Analyse numérique.pdf}
\includepdf[width=0.5\textwidth, pages={-}, nup=2x2, pagecommand={\thispagestyle{scrheadings}}]{BA3 - Analyse numérique - Pour inclusion.pdf}
\end{document}
\end{document}

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\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}
\input{../Base.tex}
\title{Formulaire d'Analyse numérique}
\chead{}
\cfoot{}
\begin{document}
\input{"BA3 - Analyse numérique - Contenu.tex"}
\end{document}

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\documentclass[a4paper,10pt]{article}
%\documentclass[a4paper,10pt]{scrartcl}
\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}
\input{../Common.tex}
\input{../Base.tex}
\title{Formulaire de Physique III}
\begin{document}
\begin{tabularx}{\textwidth}{ |X|X| }
\begin{tabu}to \textwidth{ |X|X| }
\hline
\textbf{Fluides} \newline
$ \dif\vec{F} = - P \cdot \dif\vec{\sigma} $ \newline
$ \frac{\partial\rho}{\partial t} + \nabla \bullet \left( \rho \cdot \vec{v} \right) = 0 $ \hfill Éq. de continuité \newline
$ \frac{1}{2} \cdot \rho \cdot v^2 + \rho \cdot g \cdot z + P = const $ \hfill Éq. de Bernoulli \newline
$ - \nabla P + \rho \cdot \vec{g} + \eta \cdot \nabla^2 \vec{v} = \rho \cdot \left( \frac{\partial \vec{v}}{\partial t} + \left( \vec{v} \bullet \nabla \right) \vec{v} \right) $ \hfill Éq. d'Euler \newline
$ \dif \vec{x} \parallel \vec{v} \quad \Leftrightarrow \quad \frac{\dif x}{\dif y} = \frac{v_x}{v_y} $ \hfill Lignes de courant
&
\textbf{Fluides II} \newline
$ \Delta P = \frac{8 \cdot \eta \cdot L \cdot D}{\pi \cdot R^4} $ \hfill Loi de Poiseuille \newline
$ v \left( r \right) = \frac{\Delta P}{4 \cdot \eta \cdot L} \cdot \left( R^2 - r^2 \right) $ \hfill Profil de vitesse de Poiseuille \newline
$ \vec{F}_{visc} = \eta \cdot \frac{S \cdot \left( \vec{v}_{sup} - \vec{v}_{inf} \right)}{d} $ \newline
$ \dif \vec{F}_{visc} = \eta \cdot \nabla^2 \vec{v} \cdot \dif V $ \newline
$ \frac{\dif E}{\dif t} = -\Phi_{en} + \frac{\dif W}{\dif t} $
\\\hline
\textbf{Équations de Maxwell} \newline
$ \nabla \bullet \vec{E} = \frac{\rho}{\varepsilon_0} \hspace{15mm} \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} $ \newline
$ \nabla \bullet \vec{B} = 0 \hspace{17mm} \nabla \times \vec{B} = \mu_0 \cdot \vec{j} + \frac{1}{c^2} \cdot \frac{\partial \vec{E}}{\partial t} $ \newline
$ \begin{array}{@{}l@{\qquad\qquad}l} \nabla \bullet \vec{E} = \frac{\rho}{\varepsilon_0} & \nabla \times \vec{E} = - \frac{\partial}{\partial t} \vec{B} \\ \nabla \bullet \vec{B} = 0 & \nabla \times \vec{B} = \mu_0 \cdot \vec{j} + \frac{1}{c^2} \cdot \frac{\partial}{\partial t} \vec{E} \end{array} $
&
\textbf{Formes intégrales} \newline
$ \oiint_\Sigma \vec{E} \bullet \dif\vec{\sigma} = \frac{Q_{int}}{\varepsilon_0} \hspace{21mm} = \Phi_E $ \hfill Th. de Gauss \newline
$ \oiint_\Sigma \vec{E} \bullet \dif\vec{\sigma} = \frac{Q_{int}}{\varepsilon_0} = \Phi_E $ \hfill Th. de Gauss \newline
$ \oint_\Gamma \vec{B} \bullet \dif\vec{l} = \mu_0 \cdot I + \frac{1}{c^2} \cdot \frac{\dif \Phi_E}{\dif t} \hspace{8mm} I_d = \varepsilon_0 \cdot \frac{\dif \Phi_E}{\dif t} $ \hfill Th. d'Ampère \newline
$ V = \oint_\Gamma \vec{E} \bullet \dif\vec{l} = - \frac{\dif \Phi_M}{\dif t} $ \hfill Induction \newline
\\ \hline
$ V = \oint_\Gamma \vec{E} \bullet \dif\vec{l} = - \frac{\dif \Phi_M}{\dif t} $ \hfill Induction
\\\hline
\end{tabularx}
\end{tabu}
\offinterlineskip
\nointerlineskip
\begin{tabularx}{\textwidth}{ |X| }
\begin{tabu}to \textwidth{ |X| }
\textbf{Électrostatique} \newline
$ \vec{E} = \frac{1}{4 \cdot \pi \cdot \varepsilon_0} \left( \sum q_i \cdot \frac{\vec{r}-\vec{r_i}}{|\vec{r}-\vec{r_i}|^3} + \int_\Gamma \frac{\lambda(\vec{r}') \cdot (\vec{r}-\vec{r}')}{|\vec{r}-\vec{r}'|^3} \cdot \dif l + \iint_\Sigma \frac{\sigma(\vec{r}') \cdot (\vec{r}-\vec{r}')}{|\vec{r}-\vec{r}'|^3} \cdot \dif\sigma + \iiint_V \frac{\rho(\vec{r}') \cdot (\vec{r}-\vec{r}')}{|\vec{r}-\vec{r}'|^3} \cdot \dif V \right) $ \newline
$ V = \frac{1}{4 \cdot \pi \cdot \varepsilon_0} \left( \sum q_i \cdot \frac{1}{|\vec{r}-\vec{r_i}|} + \int_\Gamma \frac{\lambda(\vec{r}')}{|\vec{r}-\vec{r}'|} \cdot \dif l + \iint_\Sigma \frac{\sigma(\vec{r}')}{|\vec{r}-\vec{r}'|} \cdot \dif\sigma + \iiint_V \frac{\rho(\vec{r}')}{|\vec{r}-\vec{r}'|} \cdot \dif V \right) $ \newline
\\ \hline
$ \vec{E} = \frac{1}{4 \cdot \pi \cdot \varepsilon_0} \cdot \left( \sum q_i \cdot \frac{\vec{r}-\vec{r_i}}{\abs{\vec{r}-\vec{r_i}}^3} + \int_\Gamma \frac{\lambda \left( \vec{r}' \right) \cdot \left( \vec{r}-\vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}^3} \cdot \dif l + \iint_\Sigma \frac{\sigma \left( \vec{r}' \right) \cdot \left( \vec{r}-\vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}^3} \cdot \dif\sigma + \iiint_V \frac{\rho \left( \vec{r}' \right) \cdot \left( \vec{r}-\vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}^3} \cdot \dif V \right) $ \newline
$ V = \frac{1}{4 \cdot \pi \cdot \varepsilon_0} \cdot \left( \sum q_i \cdot \frac{1}{\abs{\vec{r}-\vec{r_i}}} + \int_\Gamma \frac{\lambda \left( \vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}} \cdot \dif l + \iint_\Sigma \frac{\sigma \left( \vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}} \cdot \dif\sigma + \iiint_V \frac{\rho \left( \vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}} \cdot \dif V \right) $
\\\hline
\textbf{Magnétostatique} \newline
$ \vec{B} = \frac{\mu_0 \cdot I}{4 \cdot \pi} \oint_\Gamma \frac{\vec{u}_t \times \vec{u}_r}{r^2} \cdot \dif l = \frac{\mu_0}{4 \cdot \pi} \iiint_V \frac{\vec{j}(\vec{x}') \times \vec{u}_r}{r^2} \cdot \dif^3 x' $ \hfill Loi de Biot-Savart \newline
\\ \hline
$ \vec{B} = \frac{\mu_0 \cdot I}{4 \cdot \pi} \cdot \oint_\Gamma \frac{\vec{u}_t \times \vec{u}_r}{r^2} \cdot \dif l = \frac{\mu_0}{4 \cdot \pi} \cdot \iiint_V \frac{\vec{j} \left( \vec{x}' \right) \times \vec{u}_r}{r^2} \cdot \dif^3 x' $ \hfill Loi de Biot-Savart
\\\hline
\end{tabularx}
\end{tabu}
\offinterlineskip
\nointerlineskip
\begin{tabularx}{\textwidth}{ |X|X| }
\begin{tabu}to \textwidth{ |X|X| }
\textbf{Dipôle électrique} \newline
$ \vec{p} = q \cdot \vec{r}_+ - q \cdot \vec{r}_- = q \cdot \vec{a} $ \newline
$ \vec{\tau} = \vec{p} \times \vec{E}_{ext} $ \newline
$ U_{\acute el} = - \vec{p} \bullet \vec{E}_{ext} $ \newline
$ E_r = - \frac{\partial V}{\partial r} = \frac{2 \cdot p \cdot \cos \theta}{4 \cdot \pi \cdot \varepsilon_0 \cdot r^3} $ \newline
$ E_\theta = - \frac{1}{r} \frac{\partial V}{\partial \theta} = \frac{p \cdot \sin \theta}{4 \cdot \pi \cdot \varepsilon_0 \cdot r^3} $ \newline
$ E_r = - \frac{\partial V}{\partial r} = \frac{2 \cdot p \cdot \cos \left( \theta \right)}{4 \cdot \pi \cdot \varepsilon_0 \cdot r^3} $ \newline
$ E_\theta = - \frac{1}{r} \cdot \frac{\partial V}{\partial \theta} = \frac{p \cdot \sin \left( \theta \right)}{4 \cdot \pi \cdot \varepsilon_0 \cdot r^3} $
&
\textbf{Dipôle magnétique} \newline
$ \vec{M} = I \cdot \vec{S} $ \newline
$ \vec{\tau} = \vec{M} \times \vec{B}_{ext} $ \newline
$ U_{mag} = - \vec{M} \bullet \vec{B}_{ext} $ \newline
\\ \hline
$ U_{mag} = - \vec{M} \bullet \vec{B}_{ext} $
\\\hline
\textbf{Polarisation} \newline
$ \sigma_P = \vec{P} \bullet \vec{e}_n $ \newline
$ <\vec{E}> = \frac{E_{ext}}{\varepsilon_r} $ \newline
$ \vec{P} = n \cdot <\vec{p}> $ \newline
$ \vec{\left\langle E \right\rangle} = \frac{E_{ext}}{\varepsilon_r} $ \newline
$ \vec{P} = n \cdot \vec{\left\langle p \right\rangle} $
&
\textbf{Aimantation} \newline
$ j_{lie} = \vec{M} \bullet \vec{e}_n $ \newline
$ <\vec{B}> = \mu_r \cdot B_{ext} $ \newline
\\ \hline
$ j_{li\acute e} = \vec{M} \bullet \vec{e}_n $ \newline
$ \vec{\left\langle B \right\rangle} = \mu_r \cdot B_{ext} $
\\\hline
\textbf{Champ électrique D} \newline
$ \vec{D} = \varepsilon_0 \cdot \vec{E} + \vec{P} $ \newline
$ \nabla \bullet \vec{D} = \rho_{libre} $ \newline
$ \vec{D} = \varepsilon_0 \cdot \vec{E} + \vec{P} = \varepsilon_0 \cdot (1 + \chi) \cdot \vec{E} = \varepsilon \cdot \vec{E}$ \newline
$ \vec{D} = \varepsilon_0 \cdot \vec{E} + \vec{P} = \varepsilon_0 \cdot \left( 1 + \chi \right) \cdot \vec{E} = \varepsilon \cdot \vec{E} $
&
\textbf{Champ magnétisant H} \newline
$ \vec{H} = \frac{1}{\mu_0} \cdot \vec{B} - \vec{M} $ \newline
$ \nabla \times \vec{H} = \vec{j}_{libre} $ \newline
$ \vec{B} = \mu_0 \cdot (\vec{H} + \vec{M}) = \mu_0 \cdot (1 + \chi) \cdot \vec{H} = \mu \cdot \vec{H}$ \newline
\\ \hline
$ \vec{B} = \mu_0 \cdot \left( \vec{H} + \vec{M} \right) = \mu_0 \cdot \left( 1 + \chi \right) \cdot \vec{H} = \mu \cdot \vec{H} $
\\\hline
\textbf{Conditions au bord} \newline
$ E_{1t} = E_{2t} $ \newline
$ D_{1n} = D_{2n} \Rightarrow \varepsilon_{r1} \cdot E_{1n} = \varepsilon_{r2} \cdot E_{2n} $ \hfill Isolant-Isolant \newline
$ D_{1n} = \sigma_{libre} \Rightarrow E_{1n} = \frac{\sigma_{libre}}{\varepsilon_0 \cdot \varepsilon_{r1}} $ \hfill Isolant-Métal \newline
$ D_{1n} = D_{2n} \quad \Rightarrow \quad \varepsilon_{r1} \cdot E_{1n} = \varepsilon_{r2} \cdot E_{2n} $ \hfill Isolant-Isolant \newline
$ D_{1n} = \sigma_{libre} \quad \Rightarrow \quad E_{1n} = \frac{\sigma_{libre}}{\varepsilon_0 \cdot \varepsilon_{r1}} $ \hfill Isolant-Métal
&
\textbf{Conditions au bord} \newline
$ H_{1t} = H_{2t} \Rightarrow \frac{B_{1t}}{\mu_{r1}} = \frac{B_{2t}}{\mu_{r2}} $ \newline
$ B_{1n} = B_{2n} $ \newline
\\ \hline
$ H_{1t} = H_{2t} \quad \Rightarrow \quad \frac{B_{1t}}{\mu_{r1}} = \frac{B_{2t}}{\mu_{r2}} $ \newline
$ B_{1n} = B_{2n} $
\\\hline
\textbf{Électrostatique} \newline
$ \vec{F} = q \cdot (\vec{E} + \vec{v} \times \vec{B}) $ \hfill Force de Lorentz \newline
$ \vec{F} = q \cdot \left( \vec{E} + \vec{v} \times \vec{B} \right) $ \hfill Force de Lorentz \newline
$ \vec{E} = - \nabla V $ \newline
$ V(\vec{r}) = V(\vec{r_0}) - \int_{\vec{r_0}}^{\vec{r}} \vec{E} \bullet \dif\vec{l} $ \newline
$ \nabla^2 V(\vec{r})= - \frac{\rho}{\varepsilon_0} $ \hfill Équation de Poisson \newline
$ W_{AB} = \int_{\vec{r}_A}^{\vec{r}_B} q \cdot \vec{E} \cdot \dif \vec{l} = q \cdot V(\vec{r}_A) - q \cdot V(\vec{r}_B) $ \newline
$ U_E = \frac{1}{4 \cdot \pi \cdot \varepsilon_0} \cdot \frac{1}{2} \cdot \sum_{i=1}^N \sum_{j=1,j \neq i}^N \frac{q_i \cdot q_j}{|\vec{r}_i - \vec{r}_j|} $ \hfill Distribution discrète \newline
$ U_E = \frac{1}{2} \cdot \iiint_V \rho(\vec{r}) \cdot V(\vec{r}) \cdot \dif V $ \hfill Distribution continue \newline
$ \vec{j} = n \cdot q \cdot \vec{v} = \rho \cdot \vec{v} $ \hfill Densité de courant \newline
$ \vec{j} = \sigma \cdot \vec{E} $ \hfill $ \sigma $ conductivité \newline
$ \vec{E} = 0 \text{, } V = cte $ \hfill Dans un conducteur \newline
$ V \left( \vec{r} \right) = V \left( \vec{r_0} \right) - \int_{\vec{r_0}}^{\vec{r}} \vec{E} \bullet \dif\vec{l} $ \newline
$ \nabla^2 V \left( \vec{r} \right) = - \frac{\rho}{\varepsilon_0} $ \hfill Équation de Poisson \newline
$ W_{AB} = \int_{\vec{r}_A}^{\vec{r}_B} q \cdot \vec{E} \cdot \dif \vec{l} = q \cdot V \left( \vec{r}_A \right) - q \cdot V \left( \vec{r}_B \right) $ \newline
$ U_E = \frac{1}{4 \cdot \pi \cdot \varepsilon_0} \cdot \frac{1}{2} \cdot \sum_{i = 1}^N \sum_{j = 1, j \neq i}^N \frac{q_i \cdot q_j}{\abs{\vec{r}_i - \vec{r}_j}} $ \hfill Distribution discrète \newline
$ U_E = \frac{1}{2} \cdot \iiint_V \rho \left( \vec{r} \right) \cdot V \left( \vec{r} \right) \cdot \dif V $ \hfill Distribution continue \newline
$ \vec{j} = n \cdot q \cdot \vec{v} = \rho \cdot \vec{v} = \sigma \cdot \vec{E} $ \hfill Densité de courant, $ \sigma $ conductivité \newline
$ \vec{E} = 0 \comma V = \cte $ \hfill Dans un conducteur
&
\textbf{Magnétostatique} \newline
$ r = \frac{m \cdot v}{q \cdot B_0} $ \hfill Rayon de Larmor \newline
@ -102,226 +116,208 @@
$ \vec{F} = I \cdot \int_\Gamma \dif\vec{l} \times \vec{B} $ \hfill Force de Laplace \newline
$ \frac{F}{L} = \frac{\mu_0 \cdot I_1 \cdot I_2}{2 \cdot \pi \cdot d} $ \hfill Force entre deux conducteurs \newline
$ B = \mu_0 \cdot I \cdot n $ \hfill Champ dans une bobine \newline
$ \vec{B}(\vec{x}) = \frac{1}{c^2} \cdot \vec{v} \times \vec{E}(\vec{x}) $ \hfill Charge en mouvement \newline
$ \vec{B} \left( \vec{x} \right) = \frac{1}{c^2} \cdot \vec{v} \times \vec{E} \left( \vec{x} \right) $ \hfill Charge en mouvement \newline
$ F_{\acute el} = \gamma \cdot F_{Lorentz} $ \hfill Effet relatif \newline
$ \nabla^2 \vec{A} = - \mu_0 \cdot \vec{j} $ \hfill Potentiel Vecteur \newline
\\ \hline
$ \nabla^2 \vec{A} = - \mu_0 \cdot \vec{j} $ \hfill Potentiel Vecteur
\\\hline
\textbf{Condensateur} \newline
$ Q = C \cdot \Delta V $ \newline
$ U = \frac{1}{2} \cdot C \cdot V^2 = \frac{Q^2}{2 \cdot C} $ \newline
$ V = \frac{1}{C} \cdot \int I \cdot \dif t $ \newline
$ C = \frac{\varepsilon_0 \cdot A}{d} $ \hfill Pour un condensateur plan \newline
$ C = 4 \cdot \pi \cdot \varepsilon_0 \cdot \frac{R_b \cdot R_a}{R_b - R_a} $ \hfill Pour un condensateur sphère \newline
$ C = 4 \cdot \pi \cdot \varepsilon_0 \cdot \frac{R_b \cdot R_a}{R_b - R_a} $ \hfill Pour un condensateur sphère
&
\textbf{Inductance} \newline
$ \Phi_M = L \cdot I $ \newline
$ U = \frac{1}{2} \cdot L \cdot I^2 $ \newline
$ V = L \cdot \frac{\dif I}{\dif t} $ \newline
\\ \hline
$ V = L \cdot \frac{\dif I}{\dif t} $
\\\hline
\end{tabu}
\nointerlineskip
\begin{tabu}to \textwidth{ |X|X| }
\hline
\textbf{Ondes} \newline
$ \xi(x,t) = f(x - v \cdot t) + g(x + v \cdot t) $ \newline
$ \xi(x,t) = \xi_0 \cdot \sin(k \cdot x - \omega \cdot t) $ \newline
$ \frac{\partial^2 \xi}{\partial t^2} = v^2 \cdot \nabla^2 \xi $ \hfill Équation d'Alembert \newline
$ \xi \left( x, t \right) = f \left( x - v \cdot t \right) + g \left( x + v \cdot t \right) $ \newline
$ \xi \left( x, t \right) = \xi_0 \cdot \sin \left( k \cdot x - \omega \cdot t \right) $ \newline
$ v = \frac{\omega}{k} = \lambda \cdot \nu $ \newline
$ v_g = v + k \cdot \frac{\dif v}{\dif t} $ \newline
$ v_{tr} = - \omega \cdot \xi_0 \cdot \cos(k \cdot x - \omega \cdot t) $ \newline
$ v_{tr} = - \omega \cdot \xi_0 \cdot \cos \left( k \cdot x - \omega \cdot t \right) $ \newline
$ k \cdot \lambda = 2 \cdot \pi $ \newline
$ \frac{\partial^2 \xi}{\partial t^2} = v^2 \cdot \nabla^2 \xi $ \hfill Équation d'Alembert \newline
$ \nu' = \left( \frac{v - v_O}{v - v_S} \right) \cdot \nu $ \hfill Effet Doppler \newline
$ \nu' = \left( \frac{\sqrt{1 - v_R/c}}{\sqrt{1 + v_R/c}} \right) \cdot \nu $ \hfill Effet Doppler (lumière) \newline
$ I = \frac{P}{A} = \frac{1}{A} \cdot \frac{\dif W}{\dif t} \propto \xi^2 $ \newline
$ n = 10 \cdot \log_{10} \frac{I}{I_0} $ \newline
$ n = 10 \cdot \log_{10} \left( \frac{I}{I_0} \right) $
&
\textbf{Électromagnétisme} \newline
$ E = c \cdot B $ \newline
$ c^2 = \frac{1}{\mu_0 \cdot \varepsilon_0} $ \newline
$ I = S = c \cdot u_{EM} $ \newline
$ u_E = \frac{1}{2} \cdot \vec{E} \bullet \vec{D} = \frac{1}{2} \cdot \varepsilon_0 \cdot |\vec{E}|^2 $ \newline
$ u_M = \frac{1}{2} \cdot \vec{B} \bullet \vec{H} = \frac{1}{2 \cdot \mu_0} \cdot |\vec{B}|^2 $ \newline
$ u_E = \frac{1}{2} \cdot \vec{E} \bullet \vec{D} = \frac{1}{2} \cdot \varepsilon_0 \cdot \abs{\vec{E}}^2 $ \newline
$ u_M = \frac{1}{2} \cdot \vec{B} \bullet \vec{H} = \frac{1}{2 \cdot \mu_0} \cdot \abs{\vec{B}}^2 $ \newline
$ u_E = u_M = \frac{1}{2} \cdot u_{EM} $ \newline
$ \vec{S} = \frac{1}{\mu_0} \cdot \vec{E} \times \vec{B} $ \newline
$ \frac{\partial u_{EM}}{\partial t} + \nabla \bullet \vec{S} = 0 $ \hfill Théorème de Poynting \newline
$ P = \frac{I}{c} $ \hfill Pression de radiation (absorbtion) \newline
$ P = \frac{2 \cdot I}{c} $ \hfill Pression de radiation (réflexion) \newline
$ \vec{p} = \varepsilon_0 \cdot \vec{E} \times \vec{B} = \frac{\vec{S}}{c} $ \newline
\\ \hline
\end{tabularx}
\offinterlineskip
\begin{tabularx}{\textwidth}{ |X|X| }
\hline
$ \vec{p} = \varepsilon_0 \cdot \vec{E} \times \vec{B} = \frac{\vec{S}}{c} $
\\\hline
\textbf{Onde stationnaire} \newline
$ \xi = 2 \cdot \xi_0 \cdot \sin(k \cdot x) \cdot \cos(\omega \cdot t) $ \newline
$ \xi = 2 \cdot \xi_0 \cdot \sin \left( k \cdot x \right) \cdot \cos \left( \omega \cdot t \right) $ \newline
$ L = m \cdot \frac{\lambda}{2} $ \hfill Corde fixée aux 2 ext. / Tuyeau ouvert \newline
$ L = (2 \cdot m + 1) \cdot \frac{\lambda}{4} $ \hfill Corde fixée à 1 ext. / Tuyeau fermé \newline
$ L = \left( 2 \cdot m + 1 \right) \cdot \frac{\lambda}{4} $ \hfill Corde fixée à 1 ext. / Tuyeau fermé \newline
$ k \cdot x = m \cdot \pi $ \hfill Noeud ou Ventre \newline
$ k \cdot x = (m + \frac{1}{2}) \cdot \pi $ \hfill Ventre ou Noeud \newline
$ k \cdot x = \left( m + \frac{1}{2} \right) \cdot \pi $ \hfill Ventre ou Noeud
&
\textbf{Interférences} \newline
$ \xi_0^2 = \xi_{01}^2 + \xi_{02}^2 + 2 \cdot \xi_{01} \cdot \xi_{02} \cdot \cos \delta $ \newline
$ \xi_0^2 = 4 \cdot \xi_{01}^2 \cdot \cos^2 \frac{\delta}{2} $ \hfill Même amplitude \newline
$ \xi(t) = \xi_0 \cdot \cos(\omega \cdot t - k\cdot r_1 + \delta/2) $ \hfill Même amplitude \newline
$ I = I_0 \cdot \cos^2 \frac{\delta}{2} $ \hfill Même amplitude \newline
$ \delta = k \cdot \Delta r = k \cdot a \cdot \sin \theta $ \newline
$ \xi_0^2 = \xi_{01}^2 + \xi_{02}^2 + 2 \cdot \xi_{01} \cdot \xi_{02} \cdot \cos \left( \delta \right) $ \newline
$ \xi_0^2 = 4 \cdot \xi_{01}^2 \cdot \cos^2 \left( \frac{\delta}{2} \right) $ \hfill Même amplitude \newline
$ \xi \left( t \right) = \xi_0 \cdot \cos \left( \omega \cdot t - k \cdot r_1 + \delta/2 \right) $ \hfill Même amplitude \newline
$ I = I_0 \cdot \cos^2 \left( \frac{\delta}{2} \right) $ \hfill Même amplitude \newline
$ \delta = k \cdot \Delta r = k \cdot a \cdot \sin \left( \theta \right) $ \newline
$ \delta = 2 \cdot m \cdot \pi $ \hfill Max \newline
$ \delta = (2 \cdot m + 1) \cdot \pi $ \hfill Min \newline
\\ \hline
$ \delta = \left( 2 \cdot m + 1 \right) \cdot \pi $ \hfill Min
\\\hline
\textbf{Diffraction} \newline
$ I = I_0 \cdot \left( \frac{\sin(\pi \cdot b \cdot \sin \theta / \lambda)}{\pi \cdot b \cdot \sin \theta / \lambda} \right)^2 $ \newline
$ b \cdot \sin \theta = \pm m \cdot \lambda \hspace{15mm} (m \neq 0) $ \hfill Zéro \newline
$ b \cdot \sin \theta = \pm (m + \frac{1}{2}) \cdot \lambda \hspace{5mm} (m \neq 0) $ \hfill Max \newline
$ I = I_0 \cdot \left( \frac{\sin \left( \pi \cdot b \cdot \sin \left( \theta / \lambda \right) \right)}{\pi \cdot b \cdot \sin \left( \theta / \lambda \right)} \right)^2 $ \newline
$ b \cdot \sin \left( \theta \right) = \pm m \cdot \lambda \hspace{15mm} \left( m \neq 0 \right) $ \hfill Zéro \newline
$ b \cdot \sin \left( \theta \right) = \pm \left( m + \frac{1}{2} \right) \cdot \lambda \hspace{5mm} \left( m \neq 0 \right) $ \hfill Max \newline
$ \theta \geqslant \frac{\lambda}{b} $ \hfill Critère de Rayleigh (fente) \newline
$ \theta \geqslant 1.22 \cdot \frac{\lambda}{D} $ \hfill Critère de Rayleigh (ouv. circ.) \newline
$ 2 \cdot d \cdot \sin \theta = m \cdot \lambda $ \hfill Condition de Bragg \newline
$ 2 \cdot d \cdot \sin \left( \theta \right) = m \cdot \lambda $ \hfill Condition de Bragg
&
\textbf{Optique} \newline
$ n_i \cdot \sin \theta_i = n_r \cdot \sin \theta_r $ \hfill Loi de Snell-Descartes \newline
$ \sin \theta_i > \frac{n_r}{n_i} $ \hfill Réflexion totale \newline
$ n_i \cdot \sin \left( \theta_i \right) = n_r \cdot \sin \left( \theta_r \right) $ \hfill Loi de Snell-Descartes \newline
$ \sin \left( \theta_i \right) > \frac{n_r}{n_i} $ \hfill Réflexion totale \newline
$ v = \frac{c}{n} $ \newline
$ \lambda_n = \frac{\lambda}{n} $ \newline
$ k_n = n \cdot k $ \newline
$ n = \sqrt{\varepsilon_r \cdot \mu_r} \sim \sqrt{\varepsilon_r} $ \newline
\\ \hline
$ n = \sqrt{\varepsilon_r \cdot \mu_r} \sim \sqrt{\varepsilon_r} $
\\\hline
\textbf{Polarisation} \newline
$ \tan(\theta) = \frac{n_r}{n_i} $ \hfill Angle de Brewster \newline
Angle de Brewster \hfill $ \Rightarrow $ \hfill $ \pi $ 100\% transmis et 0\% réfléchi \newline
$ I = I_m \cdot \cos^2 \theta $ \hfill Loi de Malus \newline
\includegraphics[width=0.48\textwidth,keepaspectratio=true]{./Polarisation.png} \newline
Polarisation $ \sigma $ \hfill Polarisation $ \pi $ \newline
$ \tan \left( \theta \right) = \frac{n_r}{n_i} $ \hfill Angle de Brewster \newline
Angle de Brewster \hfill $ \quad \Rightarrow \quad $ \hfill Polarisation $ \pi $ 100\% transmise et 0\% réfléchie \newline
$ I = I_m \cdot \cos^2 \left( \theta \right) $ \hfill Loi de Malus
&
\textbf{Interférences à N sources} \newline
$ I = I_0 \cdot \left( \frac{\sin(N \cdot \pi \cdot a \cdot \sin \theta / \lambda)}{\sin(\pi \cdot a \cdot \sin \theta / \lambda)} \right) $ \newline
$ a \cdot \sin \theta = m \cdot \lambda, \hspace{1em} I = N^2 \cdot I_0 $ \hfill Max \newline
$ a \cdot \sin \theta = \frac{m'}{N} \cdot \lambda, \hspace{1em} \frac{m'}{N} \neq m $ \hfill Min \newline
\\ \hline
$ I = I_0 \cdot \left( \frac{\sin \left( N \cdot \pi \cdot a \cdot \sin \left( \theta / \lambda \right) \right)}{\sin \left( \pi \cdot a \cdot \sin \left( \theta / \lambda \right) \right)} \right) $ \newline
$ a \cdot \sin \left( \theta \right) = m \cdot \lambda, \qquad I = N^2 \cdot I_0 $ \hfill Max \newline
$ a \cdot \sin \left( \theta \right) = \frac{m'}{N} \cdot \lambda, \qquad \frac{m'}{N} \neq m $ \hfill Min
\\\hline
\textbf{Fluides} \newline
$ \dif\vec{F} = - P \cdot \dif\vec{\sigma} $ \newline
$ \frac{\partial\rho}{\partial t} + \nabla \bullet (\rho \cdot \vec{v}) = 0 $ \hfill Éq. de continuité \newline
$ \frac{1}{2} \cdot \rho \cdot v^2 + \rho \cdot g \cdot z + P = const $ \hfill Éq. de Bernoulli \newline
$ - \nabla P + \rho \cdot \vec{g} + \eta \cdot \nabla^2 \vec{v} = \rho \cdot (\frac{\partial \vec{v}}{\partial t} + (\vec{v} \bullet \nabla)\vec{v}) $ \hfill Éq. d'Euler \newline
$ \dif \vec{x} \parallel \vec{v} \Leftrightarrow \frac{\dif x}{\dif y} = \frac{v_x}{v_y} $ \hfill Lignes de courant \newline
\textbf{Polarisation $ \boldsymbol{\sigma} $ et polarisation $ \boldsymbol{\pi} $} \newline
\includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Polarisation.png}
&
\textbf{Fluides II} \newline
$ \Delta P = \frac{8 \cdot \eta \cdot L \cdot D}{\pi \cdot R^4} $ \hfill Loi de Poiseuille \newline
$ v(r) = \frac{\Delta P}{4 \cdot \eta \cdot L} \cdot (R^2 - r^2) $ \hfill Profil de vitesse de Poiseuille \newline
$ \vec{F}_{visc} = \eta \cdot \frac{S \cdot (\vec{v}_{sup} - \vec{v}_{inf})}{d} $ \newline
$ \dif \vec{F}_{visc} = \eta \cdot \nabla^2 \vec{v} \cdot \dif V $ \newline
$ \frac{\dif E}{\dif t} = -\Phi_{en} + \frac{\dif W}{\dif t} $ \newline
\\ \hline
\textbf{Opérateurs en coordonées cylindriques} \newline
$ \nabla U =
\begin{pmatrix}
\frac{\partial U}{\partial \rho} \\
\frac{1}{\rho} \frac{\partial U}{\partial \phi} \\
\frac{\partial U}{\partial z} \\
\end{pmatrix}
$ \newline
$ \nabla \bullet \vec{A}
= \frac{1}{\rho} \frac{\partial (\rho A_\rho)}{\partial \rho}
+ \frac{1}{\rho} \frac{\partial A_\phi}{\partial \phi}
+ \frac{\partial A_z}{\partial z}
$ \newline
$ \nabla \times \vec{A} =
\begin{pmatrix}
\frac{1}{\rho} \frac{\partial A_z}{\partial \phi} - \frac{\partial A_\phi}{\partial z} \\
\frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \\
\frac{1}{\rho} \frac{\partial (\rho A_\phi)}{\partial \rho} - \frac{1}{\rho} \frac{\partial A_\rho}{\partial \phi} \\
\end{pmatrix}
$ \newline
$ \nabla^2 U
= \frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho \frac{\partial U}{\partial \rho} \right)
+ \frac{1}{\rho^2} \frac{\partial^2 U}{\partial \phi^2}
+ \frac{\partial^2 U}{\partial z^2}
= \frac{\partial^2 U}{\partial \rho^2}
+ \frac{1}{\rho} \frac{\partial U}{\partial \rho}
+ \frac{1}{\rho^2} \frac{\partial^2 U}{\partial \phi^2}
+ \frac{\partial^2 U}{\partial z^2}
$ \newline
$ \vec{\nabla}^2 \vec{A} =
\begin{pmatrix}
\nabla^2 A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \frac{\partial A_\phi}{\partial \phi} \\
\nabla^2 A_\phi - \frac{A_\phi}{\rho^2} + \frac{2}{\rho^2} \frac{\partial A_\rho}{\partial \phi} \\
\nabla^2 A_z \\
\end{pmatrix}
$ \newline
&
\textbf{Opérateurs en coordonées sphériques} \newline
$ \nabla U =
\begin{pmatrix}
\frac{\partial U}{\partial r} \\
\frac{1}{r} \frac{\partial U}{\partial \theta} \\
\frac{1}{r \sin \theta} \frac{\partial U}{\partial \phi} \\
\end{pmatrix}
$ \newline
$ \nabla \bullet \vec{A}
= \frac{1}{r^2} \frac{\partial (r^2 A_r)}{\partial r}
+ \frac{1}{r \sin \theta} \frac{\partial (\sin \theta A_\theta)}{\partial \theta}
+ \frac{1}{r \sin \theta} \frac{\partial A_\phi}{\partial \phi}
$ \newline
$ \nabla \times \vec{A} =
\begin{pmatrix}
\frac{1}{r \sin \theta} \left[ \frac{\partial (\sin \theta A_\phi)}{\partial \theta} - \frac{\partial A_\theta}{\partial \phi} \right] \\
\frac{1}{r \sin \theta} \frac{\partial A_r}{\partial \phi} - \frac{1}{r} \frac{\partial (r A_\phi)}{\partial r} \\
\frac{1}{r} \left[ \frac{\partial (r A_\theta)}{\partial r} - \frac{\partial A_r}{\partial \theta} \right] \\
\end{pmatrix}
$ \newline
$ \nabla^2 U
= \frac{1}{r^2 \sin \theta} \left[ \frac{\partial}{\partial r} \left( r^2 \sin \theta \frac{\partial U}{\partial r} \right)
+ \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial U}{\partial \theta} \right)
+ \frac{\partial}{\partial \phi} \left( \frac{1}{\sin \theta} \frac{\partial U}{\partial \phi} \right) \right]
$ \newline
$ \nabla^2 U
= \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial U}{\partial r} \right)
+ \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial U}{\partial \theta} \right)
+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 U}{\partial \phi^2}
$ \newline
$ \nabla^2 U
= \frac{\partial^2 U}{\partial r^2}
+ \frac{2}{r} \frac{\partial U}{\partial r}
+ \frac{1}{r^2} \frac{\partial^2 U}{\partial \theta^2}
+ \frac{1}{r^2} \cot \theta \frac{\partial U}{\partial \theta}
+ \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 U}{\partial \phi^2}
$ \newline
$ \vec{\nabla}^2 \vec{A} =
\begin{pmatrix}
\nabla^2 A_r - \frac{2}{r^2} \left( A_r + \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} (\sin \theta A_\theta) + \frac{1}{\sin \theta} \frac{\partial A_\phi}{\partial \phi} \right) \\
\nabla^2 A_\theta + \frac{2}{r^2} \left( \frac{\partial A_r}{\partial \theta} - \frac{A_\theta}{2 \sin^2 \theta} - \frac{\cot \theta}{\sin \theta} \frac{\partial A_\phi}{\partial \phi} \right) \\
\nabla^2 A_\phi + \frac{2}{r^2 \sin \theta} \left( \frac{\partial A_r}{\partial \phi} + \cot \theta \frac{\partial A_\theta}{\partial \phi} - \frac{A_\phi}{2 \sin \theta} \right) \\
\end{pmatrix}
$ \newline
\\ \hline
\textbf{Théorèmes} \newline
$ \iiint_V \nabla f \cdot \dif V = \oiint_\Sigma f \cdot \dif\vec{\sigma} $ \hfill Th. du Gradient \newline
$ \iiint_V \nabla \bullet \vec{F} \cdot \dif V = \oiint_\Sigma \vec{F} \bullet \dif\vec{\sigma} $ \hfill Th. de la Divergence \newline
$ \iint_\Sigma (\nabla \times \vec{F}) \bullet \dif\vec{\sigma} = \oint_\Gamma \vec{F} \bullet \dif\vec{l} $ \hfill Th. de Stokes \newline
$ \iint_\Sigma \left( \nabla \times \vec{F} \right) \bullet \dif\vec{\sigma} = \oint_\Gamma \vec{F} \bullet \dif\vec{l} $ \hfill Th. de Stokes \newline
$ \frac{\dif F}{\dif t}
= \frac{\partial F}{\partial t}
+ \frac{\partial F}{\partial x} \cdot \frac{\dif x}{\dif t}
+ \frac{\partial F}{\partial y} \cdot \frac{\dif y}{\dif t}
+ \frac{\partial F}{\partial z} \cdot \frac{\dif z}{\dif t}
= \frac{\partial F}{\partial t} + (\vec{v} \bullet \nabla) F $ \newline
= \frac{\partial F}{\partial t} + \left( \vec{v} \bullet \nabla \right) F $
\\\hline
\textbf{Opérateurs en coordonées cylindriques} \newline
\footnotesize{
$ \nabla U =
\begin{pmatrix}
\frac{\partial U}{\partial \rho} \\
\frac{1}{\rho} \nocdot \frac{\partial U}{\partial \phi} \\
\frac{\partial U}{\partial z} \\
\end{pmatrix}
$ \newline
$ \nabla \bullet \vec{A}
= \frac{1}{\rho} \nocdot \frac{\partial \left( \rho \nocdot A_\rho \right)}{\partial \rho}
+ \frac{1}{\rho} \nocdot \frac{\partial A_\phi}{\partial \phi}
+ \frac{\partial A_z}{\partial z}
$ \newline
$ \nabla \times \vec{A} =
\begin{pmatrix}
\frac{1}{\rho} \nocdot \frac{\partial A_z}{\partial \phi} - \frac{\partial A_\phi}{\partial z} \\
\frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \\
\frac{1}{\rho} \nocdot \frac{\partial \left( \rho \nocdot A_\phi \right)}{\partial \rho} - \frac{1}{\rho} \nocdot \frac{\partial A_\rho}{\partial \phi} \\
\end{pmatrix}
$ \newline
$ \nabla^2 U
= \frac{1}{\rho} \nocdot \frac{\partial}{\partial \rho} \left( \rho \nocdot \frac{\partial U}{\partial \rho} \right)
+ \frac{1}{\rho^2} \nocdot \frac{\partial^2 U}{\partial \phi^2}
+ \frac{\partial^2 U}{\partial z^2}
$ \newline
$ \hphantom{\nabla^2 U}
= \frac{\partial^2 U}{\partial \rho^2}
+ \frac{1}{\rho} \nocdot \frac{\partial U}{\partial \rho}
+ \frac{1}{\rho^2} \nocdot \frac{\partial^2 U}{\partial \phi^2}
+ \frac{\partial^2 U}{\partial z^2}
$ \newline
$ \vec{\nabla}^2 \vec{A} =
\begin{pmatrix}
\nabla^2 A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \nocdot \frac{\partial A_\phi}{\partial \phi} \\
\nabla^2 A_\phi - \frac{A_\phi}{\rho^2} + \frac{2}{\rho^2} \nocdot \frac{\partial A_\rho}{\partial \phi} \\
\nabla^2 A_z \\
\end{pmatrix}
$
}
&
\textbf{} \newline
\\ \hline
\textbf{Opérateurs en coordonées sphériques} \newline
\footnotesize{
$ \nabla U =
\begin{pmatrix}
\frac{\partial U}{\partial r} \\
\frac{1}{r} \nocdot \frac{\partial U}{\partial \theta} \\
\frac{1}{r \nocdot \sin \left( \theta \right)} \nocdot \frac{\partial U}{\partial \phi} \\
\end{pmatrix}
$ \newline
$ \nabla \bullet \vec{A}
= \frac{1}{r^2} \nocdot \frac{\partial \left( r^2 \nocdot A_r \right)}{\partial r}
+ \frac{1}{r \nocdot \sin \left( \theta \right)} \nocdot \frac{\partial \left( \sin \left( \theta \right) \nocdot A_\theta \right)}{\partial \theta}
+ \frac{1}{r \nocdot \sin \left( \theta \right)} \nocdot \frac{\partial A_\phi}{\partial \phi}
$ \newline
$ \nabla \times \vec{A} =
\begin{pmatrix}
\frac{1}{r \nocdot \sin \left( \theta \right)} \nocdot \left[ \frac{\partial \left( \sin \left( \theta \right) \nocdot A_\phi \right)}{\partial \theta} - \frac{\partial A_\theta}{\partial \phi} \right] \\
\frac{1}{r \nocdot \sin \left( \theta \right)} \nocdot \frac{\partial A_r}{\partial \phi} - \frac{1}{r} \nocdot \frac{\partial \left( r \nocdot A_\phi \right)}{\partial r} \\
\frac{1}{r} \nocdot \left[ \frac{\partial \left( r \nocdot A_\theta \right)}{\partial r} - \frac{\partial A_r}{\partial \theta} \right] \\
\end{pmatrix}
$ \newline
$ \nabla^2 U
= \frac{1}{r^2 \nocdot \sin \left( \theta \right)} \nocdot \left[ \frac{\partial}{\partial r} \left( r^2 \nocdot \sin \left( \theta \right) \nocdot \frac{\partial U}{\partial r} \right)
+ \frac{\partial}{\partial \theta} \left( \sin \left( \theta \right) \nocdot \frac{\partial U}{\partial \theta} \right)
+ \frac{\partial}{\partial \phi} \left( \frac{1}{\sin \left( \theta \right)} \nocdot \frac{\partial U}{\partial \phi} \right) \right]
$ \newline
$ \hphantom{\nabla^2 U}
= \frac{1}{r^2} \nocdot \frac{\partial}{\partial r} \left( r^2 \nocdot \frac{\partial U}{\partial r} \right)
+ \frac{1}{r^2 \nocdot \sin \left( \theta \right)} \nocdot \frac{\partial}{\partial \theta} \left( \sin \left( \theta \right) \nocdot \frac{\partial U}{\partial \theta} \right)
+ \frac{1}{r^2 \nocdot \sin^2 \left( \theta \right)} \nocdot \frac{\partial^2 U}{\partial \phi^2}
$ \newline
$ \hphantom{\nabla^2 U}
= \frac{\partial^2 U}{\partial r^2}
+ \frac{2}{r} \nocdot \frac{\partial U}{\partial r}
+ \frac{1}{r^2} \nocdot \frac{\partial^2 U}{\partial \theta^2}
+ \frac{1}{r^2} \nocdot \cot \theta \nocdot \frac{\partial U}{\partial \theta}
+ \frac{1}{r^2 \nocdot \sin^2 \left( \theta \right)} \nocdot \frac{\partial^2 U}{\partial \phi^2}
$ \newline
$ \vec{\nabla}^2 \vec{A} =
\begin{pmatrix}
\nabla^2 A_r - \frac{2}{r^2} \nocdot \left( A_r + \frac{1}{\sin \left( \theta \right)} \nocdot \frac{\partial}{\partial \theta} \left( \sin \left( \theta \right) \nocdot A_\theta \right) + \frac{1}{\sin \left( \theta \right)} \nocdot \frac{\partial A_\phi}{\partial \phi} \right) \\
\nabla^2 A_\theta + \frac{2}{r^2} \nocdot \left( \frac{\partial A_r}{\partial \theta} - \frac{A_\theta}{2 \nocdot \sin^2 \left( \theta \right)} - \frac{\cot \theta}{\sin \left( \theta \right)} \nocdot \frac{\partial A_\phi}{\partial \phi} \right) \\
\nabla^2 A_\phi + \frac{2}{r^2 \nocdot \sin \left( \theta \right)} \nocdot \left( \frac{\partial A_r}{\partial \phi} + \cot \theta \nocdot \frac{\partial A_\theta}{\partial \phi} - \frac{A_\phi}{2 \nocdot \sin \left( \theta \right)} \right) \\
\end{pmatrix}
$
}
\\\hline
\end{tabularx}
\end{tabu}
\end{document}
\end{document}

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\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}
\input{../Base.tex}
\title{Formulaire d'Analyse IV}
\begin{document}
\begin{tabu}to \textwidth{ |X| }
\hline
\textbf{Quelques propriétés} \newline
$ \int_0^T \cos \left( \frac{2 \cdot \pi \cdot n}{T} \cdot x \right) \cdot \cos \left( \frac{2 \cdot \pi \cdot m}{T} \cdot x \right) \cdot \dif x = \int_0^T \sin \left( \frac{2 \cdot \pi \cdot n}{T} \cdot x \right) \cdot \sin \left( \frac{2 \cdot \pi \cdot m}{T} \cdot x \right) \cdot \dif x = \left\{ \begin{array}{ll} 0 &\text{ si } n \neq m \\ T/2 &\text{ si } n = m \\ \end{array} \right. $ \newline
$ \int_0^T \cos \left( \frac{2 \cdot \pi \cdot n}{T} \cdot x \right) \cdot \sin \left( \frac{2 \cdot \pi \cdot m}{T} \cdot x \right) \cdot \dif x = 0 $ \newline
$ \int_a^{a+T} f \left( x \right) \cdot \dif x = \int_0^T f \left( x \right) \cdot \dif x $ \hspace{5em} $ f \left( x \right) \quad T \text{-périodique} $
\\\hline
\textbf{Série de Fourier d'une fonction $ \symbf{T} \text{-périodique} $} \newline
$ \symsf{F} f \left( x \right) = \frac{a_0}{2} + \sum_{n = 1}^\infty \left[ a_n \cdot \cos \left( \frac{2 \cdot \pi \cdot n}{T} \cdot x \right) + b_n \cdot \sin \left( \frac{2 \cdot \pi \cdot n}{T} \cdot x \right) \right] $ \newline
$ a_n = \frac{2}{T} \cdot \int_0^T f \left( x \right) \cdot \cos \left( \frac{2 \cdot \pi \cdot n}{T} \cdot x \right) \cdot \dif x \hspace{5em} b_n = \frac{2}{T} \cdot \int_0^T f \left( x \right) \cdot \sin \left( \frac{2 \cdot \pi \cdot n}{T} \cdot x \right) \cdot \dif x $ \newline
{\setlength{\tabcolsep}{2pt}
\begin{tabu}to \linewidth{@{}rX@{}}
Dirichlet~: & si $ f $ et $ f' $ continues par morceaux, $ \symsf{F}f \left( x \right) = \frac{1}{2} \left( f \left( x_- \right) + f \left( x_+ \right) \right) $ \\
Not. complexe~: & $ \symsf{F} f \left( x \right) = \sum_{n = -\infty}^{+\infty} c_n \cdot \e^{\im \cdot \frac{2 \cdot \pi \cdot n}{T} \cdot x} \hspace{5em} c_n = \frac{1}{T} \cdot \int_0^T f \left( x \right) \cdot \e^{-\im \cdot \frac{2 \cdot \pi \cdot n}{T} \cdot x} \cdot \dif x \in \symbb{C} $ \\
Id. de Parseval~: & $ \frac{2}{T} \cdot \int_0^T \left( f \left( x \right) \right)^2 \cdot \dif x = \frac{a_0^2}{2} + \sum_{n = 1}^\infty \left[ a_n^2 + b_n^2 \right] $ \hfill $ f $ et $ f' $ c.p.m. \\
Dérivée~: & $ \symsf{F} f' \left( x \right) = \sum_{n = 1}^\infty \left[ -a_n \cdot \frac{2 \cdot \pi \cdot n}{T} \cdot \sin \left( \frac{2 \cdot \pi \cdot n}{T} \cdot x \right) + b_n \cdot \frac{2 \cdot \pi \cdot n}{T} \cdot \cos \left( \frac{2 \cdot \pi \cdot n}{T} \cdot x \right) \right] = \frac{1}{2} \left( f' \left( x_- \right) + f' \left( x_+ \right) \right) $ \hfill $ f $ c., $ f' $ et $ f'' $ c.p.m. \\
Intégrale~: & $ \int_{x_0}^x f \left( t \right) \cdot \dif t = \int_{x_0}^x \frac{a_0}{2} \cdot \dif t + \sum_{n = 1}^\infty \left[ a_n \cdot \int_{x_0}^x \cos \left( \frac{2 \cdot \pi \cdot n}{T} \cdot t \right) \cdot \dif t + b_n \cdot \int_{x_0}^x \sin \left( \frac{2 \cdot \pi \cdot n}{T} \cdot t \right) \cdot \dif t \right] $ \hfill $ f $ et $ f' $ c.p.m.
\end{tabu}}
\\\hline
\textbf{Série de Fourier sur un intervalle $ \symbf{\left[ 0;L \right]} $} \newline
{\setlength{\tabcolsep}{10pt}
\begin{tabu}to \linewidth{@{}lXl@{}}
$ \symsf{F_c} f \left( x \right) = \frac{a_0}{2} + \sum_{n = 1}^\infty a_n \cdot \cos \left( \frac{\pi \cdot n}{L} \cdot x \right) = f \left( x \right) $ & $ a_n = \frac{2}{L} \cdot \int_0^L f \left( x \right) \cdot \cos \left( \frac{\pi \cdot n}{L} \cdot x \right) \cdot \dif x $ & $ f $ c., $ f' $ c.p.m. \\
$ \symsf{F_s} f \left( x \right) = \sum_{n = 1}^\infty b_n \cdot \sin \left( \frac{\pi \cdot n}{L} \cdot x \right) = f \left( x \right) $ & $ b_n = \frac{2}{L} \cdot \int_0^L f \left( x \right) \cdot \sin \left( \frac{\pi \cdot n}{L} \cdot x \right) \cdot \dif x $ & $ f $ c., $ f' $ c.p.m., $ f \left( 0 \right) = f \left( L \right) = 0 $
\end{tabu}}
\\\hline
\textbf{Transformée de Fourier} \newline
$ f: \symbb{R} \rightarrow \symbb{C} $ continue par morceaux et telle que $ \int_{-\infty}^{+\infty} \abs{f \left( x \right)} \cdot \dif x < +\infty $ \newline
$ \symcal{F}f \left( \alpha \right) = \hat{f} \left( \alpha \right) = \frac{1}{\sqrt{2 \cdot \pi}} \cdot \int_{-\infty}^{+\infty} f \left( x \right) \cdot \e^{-\im \cdot \alpha \cdot x} \cdot \dif x \hspace{5em} \symcal{F}^{-1}f \left( \alpha \right) = \frac{1}{\sqrt{2 \cdot \pi}} \cdot \int_{-\infty}^{+\infty} f \left( x \right) \cdot \e^{\im \cdot \alpha \cdot x} \cdot \dif \alpha $ \newline
{\setlength{\tabcolsep}{2pt}
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Réciprocité~: & $ \symcal{F}^{-1} \left( \symcal{F}f \right) \left( x \right) = \symcal{F}^{-1} \left( \hat{f} \right) \left( x \right) = \frac{1}{2} \left( f \left( x_- \right) + f \left( x_+ \right) \right) $ \hfill $ f $ et $ f' $ c.p.m., $ f $ et $ \hat{f} $ intégrables sur $ \left[ -\infty;+\infty \right] $ \\
Continuité~: & $ \hat{f}: \symbb{R} \rightarrow \symbb{C} $ est continue et $ \lim_{\alpha \rightarrow \pm\infty} \hat{f} \left( \alpha \right) = 0 $ \\
Linéarité~: & $ \symcal{F} \left( a \cdot f + b \cdot g \right) = a \cdot \symcal{F}f + b \cdot \symcal{F}g $ \\
Dérivée~: & $ \symcal{F} \left( f^{\left( k \right)} \right) \left( \alpha \right) = \left( \im \cdot \alpha \right)^k \cdot \symcal{F} \left( f \right) \left( \alpha \right) $ \\
Décalage et \newline ch. d'échelle~: & $ g \left( x \right) = f \left( a \cdot \left( x + b \right) \right) \quad \Rightarrow \quad \symcal{F} \left( g \right) \left( \alpha \right) = \e^{\im \cdot \alpha \cdot b} \cdot \frac{1}{\abs{a}} \cdot \symcal{F} \left( f \right) \left( \frac{\alpha}{a} \right) \hspace{1em} a \in \symbb{R}^*, b \in \symbb{R} $ \\
Identité de Plancherel~: & $ \int_{-\infty}^{+\infty} \left( f \left( x \right) \right)^2 \cdot \dif x = \int_{-\infty}^{+\infty} \abs{\symcal{F}f \left( \alpha \right)}^2 \cdot \dif \alpha $ \\
T. de F. en sinus/cosinus~: & $ \symcal{F}f \left( \alpha \right) = \sqrt{\frac{2}{\pi}} \cdot \int_0^{+\infty} f \left( x \right) \cdot \cos \left( \alpha \cdot x \right) \cdot \dif x $ \hfill $ f $ paire \\
& $ \hphantom{\symcal{F}}f \left( x \right) = \sqrt{\frac{2}{\pi}} \cdot \int_0^{+\infty} \hat{f} \left( x \right) \cdot \cos \left( \alpha \cdot x \right) \cdot \dif \alpha $ \hfill $ f $ paire \\
& $ \symcal{F}f \left( \alpha \right) = -\im \cdot \sqrt{\frac{2}{\pi}} \cdot \int_0^{+\infty} f \left( x \right) \cdot \sin \left( \alpha \cdot x \right) \cdot \dif x $ \hfill $ f $ impaire \\
& $ \hphantom{\symcal{F}}f \left( x \right) = \hphantom{-}\im \cdot \sqrt{\frac{2}{\pi}} \cdot \int_0^{+\infty} \hat{f} \left( x \right) \cdot \sin \left( \alpha \cdot x \right) \cdot \dif \alpha $ \hfill $ f $ impaire
\end{tabu}}
\\\hline
\textbf{Produit de convolution} \newline
$ \left( f \ast g \right) \left( x \right) = \int_{-\infty}^{+\infty} f \left( x-t \right) \cdot g \left( t \right) \cdot \dif t $ \newline
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Commutativité~: & $ \left( g \ast f \right) \left( x \right) = \left( f \ast g \right) \left( x \right) = \int_{-\infty}^{+\infty} g \left( x-t \right) \cdot f \left( t \right) \cdot \dif t $ \\
Associativité~: & $ \left( f \ast g \right) \ast h = f \ast \left( g \ast h \right) $ \\
Distributivité~: & $ f \ast \left( g + h \right) = f \ast g + f \ast h $ \\
T. de F.~: & $ \symcal{F} \left( f \ast g \right) \left( \alpha \right) = \sqrt{2 \cdot \pi} \cdot \symcal{F}f \left( \alpha \right) \cdot \symcal{F}g \left( \alpha \right) $ \\
Dérivée~: & $ \left( f \ast g \right) ' \left( x \right) = \left( f' \ast g \right) \left( x \right) = \left( f \ast g' \right) \left( x \right) $
\end{tabu}}
\\\hline
\end{tabu}
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\hline
\textbf{Transformée de Laplace} \newline
$ f: \symbb{R}_+ \rightarrow \symbb{R} $ continue par morceaux et $ \gamma_0 $ tel que $ \int_{0}^{+\infty} \abs{f \left( t \right)} \cdot \e^{-\gamma_0 \cdot t} \cdot \dif t < +\infty $ \newline
$ \symcal{L}f \left( z \right) = F \left( z \right) = \int_{0}^{+\infty} f \left( t \right) \cdot \e^{-z \cdot t} \cdot \dif t \hspace{5em} \forall z \in \symbb{C} \tq \Re \left( z \right) \geq \gamma_0 $ \hspace{5em} ($ \gamma_0 $ abscisse de convergence) \newline
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Linéarité~: & $ \symcal{L} \left( a \cdot f + b \cdot g \right) = a \cdot \symcal{L}f + b \cdot \symcal{L}g $ \\
Décalage~: & $ a > 0 \comma g \left( t \right) = \left\{ \begin{array}{ll} f \left( t-a \right) &\text{ si } t \geq a \\ 0 &\text{ sinon }\\ \end{array} \right. \quad \Rightarrow \quad \symcal{L}g \left( z \right) = \e^{-z \cdot a} \cdot \symcal{L}f \left( z \right) $ \\
Ch. d'échelle~: & $ a > 0 \comma g \left( t \right) = f \left( a \cdot t \right) \quad \Rightarrow \quad \symcal{L}g \left( z \right) = \frac{1}{a} \cdot \symcal{L}f \left( \frac{z}{a} \right) $ \\
Holomorphie \hphantom{~:} & $ F = \symcal{L}f $ est holomorphe dans $ D = \left\{ z \in \symbb{C} : \Re \left( z \right) > \gamma_0 \right\} $ \\
et dérivée~: & $ F' \left( z \right) = -\int_{0}^{+\infty} t \cdot f \left( t \right) \cdot \e^{-z \cdot t} \cdot \dif t = \symcal{L}h \left( z \right) $ où $ h \left( t \right) = -t \cdot f \left( t \right) $ \\
Dérivée~: & $ \symcal{L} \left( f^{\left( k \right)} \right) \left( z \right) = z^k \cdot \symcal{L} \left( f \right) \left( z \right) - \sum_{j = 0}^{k-1} z^j \cdot f^{\left( k-1-j \right)} \left( 0 \right) $ \\
& $ \hphantom{\symcal{L} \left( f^{\left( k \right)} \right) \left( z \right)} = z^k \cdot \symcal{L} \left( f \right) \left( z \right) - f^{\left( k-1 \right)} \left( 0 \right) - z \cdot f^{\left( k-2 \right)} \left( 0 \right) - \dots - z^{k-1} \cdot f \left( 0 \right) $ \\
Intégrale~: & $ \varphi \left( t \right) = \int_0^t f \left( s \right) \cdot \dif s \quad \Rightarrow \quad \symcal{L}\varphi \left( z \right) = \frac{1}{z} \cdot \symcal{L}f \left( z \right) $ \\
Convolution~: & $ \left( f \ast g \right) \left( t \right) = \int_{-\infty}^{+\infty} f \left( t-s \right) \cdot g \left( s \right) \cdot \dif s = \int_{0}^{t} f \left( t-s \right) \cdot g \left( s \right) \cdot \dif s \quad \Rightarrow \quad \symcal{L} \left( f \ast g \right) \left( z \right) = \symcal{L}f \left( z \right) \cdot \symcal{L}g \left( z \right) $ \\
Inversion~: & Si $ f $ et $ f' $ c.p.m et si $ \int_{-\infty}^{+\infty} \abs{F \left( \gamma + \im \cdot s \right)} \cdot \dif s < +\infty $ \\
& $ \symcal{L}^{-1}f \left( t \right) = \frac{1}{2 \cdot \pi} \cdot \int_{-\infty}^{+\infty} F \left( \gamma + \im \cdot s \right) \cdot \e^{\left( \gamma + \im \cdot s \right) \cdot t} \cdot \dif t = \frac{1}{2} \left( f \left( t_- \right) + f \left( t_+ \right) \right) $ \\
& Si $ F \left( z \right) = \frac{p \left( z \right)}{q \left( z \right)} $ et $ \deg \left( q \right) \geq \deg \left( p \right) + 2 \comma \symcal{L}^{-1}f \left( t \right) = \sum_{R \acute e s_{z_k}} \left( F \left( z \right) \cdot \e^{z \cdot t} \right) $
\end{tabu}}
\\\hline
\textbf{Distribution tempérées} \newline
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Espace de Schwartz~: & Espace vectoriel des fonctions $ \varphi \in C^\infty: \symbb{R} \rightarrow \symbb{C} \tq \forall n, m \in \symbb{N} \comma \lim_{x \rightarrow \pm\infty} x^m \cdot \varphi^{m} \left( x \right) = 0 $ \\
Fonction CCL~: & Fonction $ f: \symbb{R} \rightarrow \symbb{C} \tq \exists n \in \symbb{N} \tq \lim_{x \rightarrow \pm\infty} \frac{f \left( x \right)}{x^n} = 0 $ \\
Fonctionelle~: & Application linéaire $ T: \symcal{S} \rightarrow \symbb{C} $ définie par $ \left\langle T_f, \varphi \right\rangle = \int_{-\infty}^{+\infty} f \left( x \right) \cdot \varphi \left( x \right) \cdot \dif x \in \symcal{S}' $ \\
Distribution tempérée~: & $ T_f^{\left( n \right)}: \symcal{S} \rightarrow \symbb{C} $ définie par $ \left\langle T^{\left( n \right)}_f, \varphi \right\rangle = \left( -1 \right)^n \cdot \int_{-\infty}^{+\infty} f \left( x \right) \cdot \varphi \left( x \right)^{\left( n \right)} \cdot \dif x \in \symcal{S}' $ \\
Linéarité~: & $ \left\langle a \cdot S + b \cdot T, \varphi \right\rangle = a \cdot \left\langle S, \varphi \right\rangle + b \cdot \left\langle T, \varphi \right\rangle $; $ \left\langle T_f, a \cdot \varphi_1 + b \cdot \varphi_2 \right\rangle = a \cdot \left\langle T_f, \varphi_1 \right\rangle + b \cdot \left\langle T_f, \varphi_2 \right\rangle $ \\
Dérivée~: & $ \left\langle T^{\left( k \right)}, \varphi \right\rangle = \left( -1 \right)^k \cdot \left\langle T, \varphi^{\left( k \right)} \right\rangle $ \\
T. de F.~: & $ \left\langle \symcal{F}T, \varphi \right\rangle = \left\langle T, \symcal{F}\varphi \right\rangle $ \\
Réflexion~: & $ \left\langle T^{\vee}, \varphi \right\rangle = \left\langle T, \varphi^{\vee} \right\rangle $ \\
Translation~: & $ \left\langle \symcal{T}_a T, \varphi \right\rangle = \left\langle T, \symcal{T}_{-a} \varphi \right\rangle $ \\
Changement d'échelle~: & $ \left\langle \symcal{S}_a T, \varphi \right\rangle = \left\langle T, \frac{1}{\abs{a}} \cdot \symcal{S}_{1/a} \varphi \right\rangle $ \\
Mult. par C\textsuperscript{\infty}CL~: & $ \left\langle g \cdot T, \varphi \right\rangle = \left\langle T, g \cdot \varphi \right\rangle $ \\
Distribution $ \delta $~: & $ \delta_a: \symcal{S} \rightarrow \symbb{R} $ définie par $ \left\langle \delta_a, \varphi \right\rangle = \varphi \left( a \right) $ \hspace{5em} $ \symcal{F}\delta_a \left( x \right) = \frac{1}{\sqrt{2 \cdot \pi}} \cdot \e^{-\im \cdot a \cdot x} $ \hspace{5em} $ \delta = \delta_0 $ \\
Convolution~: & $ \left( f \ast \varphi \right) \left( x \right) = \left\langle f, \symcal{T}_{-x} \left( \varphi^{\vee} \right) \right\rangle $ \hfill $ \int_{-\infty}^{+\infty} \left( f \ast g \right) \left( x \right) \cdot \varphi \left( x \right) \cdot \dif x = \int_{-\infty}^{+\infty} f \left( x \right) \cdot \left( g^{\vee} \ast \varphi \right) \left( x \right) \cdot \dif x $ \hfill $ \delta \ast \varphi = \varphi $ \\
& $ \left( T \ast \varphi \right) \left( x \right) = \left\langle T, \symcal{T}_{-x} \left( \varphi^{\vee} \right) \right\rangle $ \hfill $ \left\langle T_1 \ast T_2, \varphi \right\rangle = \left\langle T_1, T_2^{\vee} \ast \varphi \right\rangle $ \hfill $ T \ast \delta = T $ \\
Cohérence~: & $ T_f^{\left( 1 \right)} = T_{f'} \comma \symcal{F}T_f = T_{\symcal{F}f} \comma T_f^{\vee} = T_{f^{\vee}} \comma \symcal{T}_a T_f = T_{\symcal{T}_a f} \comma \symcal{S}_a T_f = T_{\symcal{S}_a f} \comma g \cdot T_f = T_{g \cdot f} $
\end{tabu}}
Les propriétés de la transformée de Fourier et du produit de convolution restent valables.
%TODO~: tableaux transformées de fourier de Distribution ?
\\\hline
\end{tabu}
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\hline
\textbf{Équations différentielles} \newline
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Problème de Cauchy~: & $ a_2 \cdot y'' \left( t \right) + a_1 \cdot y' \left( t \right) + a_0 \cdot y \left( t \right) = f \left( t \right) \comma t > 0 \comma y \left( 0 \right) = y_0 \comma y' \left( 0 \right) = y_1 $ \\
Résolution~: & $ \symcal{L} \left( a_2 \cdot y'' + a_1 \cdot y' + a_0 \cdot y \right) \left( z \right) = \symcal{L}f \left( z \right) \quad \Leftrightarrow \quad \dots \quad \Leftrightarrow \quad Y \left( z \right) = \frac{F \left( z \right) + a_2 \cdot y_0 \cdot z + a_1 \cdot y_0 + a_2 \cdot y_1}{a_2 \cdot z^2 + a_1 \cdot z + a_0} $ \hspace{5em} $ y \left( t \right) = \symcal{L}^{-1} \left( Y \right) \left( t \right) $ \\
Cas particulier~: & $ y'' \left( t \right) + \lambda \cdot y \left( t \right) = 0 \quad \Rightarrow \quad \left\{ \begin{array}{l} \lambda = 0: y \left( t \right) = y_0 + y_1 \cdot t \\
\lambda < 0: y \left( t \right) = y_0 \cdot \cosh \left( \sqrt{-\lambda} \cdot t \right) + \frac{y_1}{\sqrt{-\lambda}} \cdot \sinh \left( \sqrt{-\lambda} \cdot t \right) \\
\lambda > 0: y \left( t \right) = y_0 \cdot \cos \left( \sqrt{\lambda} \cdot t \right) + \frac{y_1}{\sqrt{\lambda}} \cdot \sin \left( \sqrt{\lambda} \cdot t \right) \end{array} \right. $ \\
Sturm-Liouville~: & $ y'' \left( t \right) + \lambda \cdot y \left( t \right) = 0 \comma t \in \left] 0;L \right[ \quad \Rightarrow \quad \left\{ \begin{array}{ll} \text{Si } y \left( 0 \right) = y \left( L \right) = 0 \comma & \lambda = \left( \frac{n \cdot \pi}{L} \right)^2 \comma y \left( t \right) = \alpha_n \cdot \sin \left( \frac{n \cdot \pi}{L} \cdot t \right) \comma n \in \symbb{N} \\
\text{Si } y' \left( 0 \right) = y' \left( L \right) = 0 \text{, }& \lambda = \left( \frac{n \cdot \pi}{L} \right)^2 \text{, } y \left( t \right) = \beta_n \cdot \cos \left( \frac{n \cdot \pi}{L} \cdot t \right) \text{, } n \in \symbb{N} \end{array} \right. $
\end{tabu}}
Équations sur $ \symbb{R}_+ $~: utiliser la transformée de Laplace. \newline
Équations sur $ \symbb{R} $~: utiliser la transformée de Fourier. \newline
Équations périodiques~: utiliser les séries de Fourier.
\\\hline
\textbf{Équations aux dérivées partielles} \newline
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Dans $ \symbb{R} $~: & On fixe une variable et on prend la transformée de Fourier en l'autre. On utilise les propriétés de la transformée pour obtenir une EDO en la variable fixée. On résout cette EDO et finalement, on prend la transformée inverse. \\
Dans un intervalle~: & On sépare les variables. On obtient deux EDO qu'on résout pour obtenir une solution. On superpose ces solutions puis on impose les conditions initiales pour obtenir la solution.\\
Dans un rectangle~: & Même démarche que pour un intervalle, mais faite deux fois (une fois dans chaque direction).
\end{tabu}}
\\\hline
\textbf{Équations différentielles avec des distributions} \newline
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Solution fondamentale~: & $ y $ est solution de $ a_2 \cdot y'' + a_1 \cdot y' + a_0 \cdot y = f \quad \Leftrightarrow \quad y = G \ast f \quad $ avec $ \quad G \quad $ solution de $ \quad a_2 \cdot G'' + a_1 \cdot G' + a_0 \cdot G = \delta $
\end{tabu}}
\\\hline
\textbf{Équation de la chaleur dans $ \symbf{\symbb{R}} $} \newline
$ \left\{ \begin{array}{ll} \frac{\partial}{\partial t} u \left( x, t \right) = a^2 \cdot \frac{\partial^2}{\partial x^2} u \left( x, t \right) \quad &\forall x \in \symbb{R} \text{, } t > 0 \\ u \left( x, 0 \right) = f \left( x \right) \quad &\forall x \in \symbb{R} \end{array} \right. $ \newline
$ \symcal{F} \left( \frac{\partial u}{\partial t} \right) \left( \alpha, t \right) = a^2 \cdot \symcal{F} \left( \frac{\partial^2 u}{\partial x^2} \right) \left( \alpha, t \right) $ avec $ \symcal{F}u \left( \alpha, 0 \right) = \symcal{F}f \left( \alpha \right) $ \hspace{5em} On pose $ \symcal{F}u \left( \alpha, t \right) = v \left( \alpha, t \right) $ . \newline
$ \left\{ \begin{array}{ll} \frac{\partial}{\partial t} v \left( \alpha, t \right) = -a^2 \cdot \alpha^2 \cdot v \left( \alpha, t \right) \\ v \left( \alpha, 0 \right) = \symcal{F}f \left( \alpha \right) \end{array} \right. \quad \Leftrightarrow \quad v \left( \alpha, t \right) = v \left( \alpha, 0 \right) \cdot \e^{-\alpha^2 \cdot a^2 \cdot t} = \symcal{F}f \left( \alpha \right) \cdot \e^{-\alpha^2 \cdot a^2 \cdot t} $ \newline
$ u \left( x, t \right) = \symcal{F}^{-1}v \left( x, t \right) = \frac{1}{\sqrt{2 \cdot \pi}} \cdot \int_{-\infty}^{+\infty} \symcal{F}f \left( \alpha \right) \cdot \e^{-\alpha^2 \cdot a^2 \cdot t} \cdot \e^{\im \cdot \alpha \cdot x} \cdot \dif \alpha $
\\\hline
\textbf{Équation des ondes sur un intervalle} \newline
$ \frac{\partial^2}{\partial t^2} u \left( x, t \right) = c^2 \cdot \frac{\partial^2}{\partial x^2} u \left( x, t \right) \quad x \in \left] 0;L \right[ \text{, } t > 0 \quad \text{ avec } \quad u \left( 0, t \right) = u \left( L, t \right) = 0 \text{, } u \left( x, 0 \right) = f \left( x \right) \text{, } \frac{\partial}{\partial t} u \left( x, 0 \right) = g \left( x \right) $ \newline
$ u \left( x, t \right) = v \left( x \right) \cdot w \left( t \right) \quad \Leftrightarrow \quad v \left( x \right) \cdot w'' \left( t \right) = c^2 \cdot v'' \left( x \right) \cdot w \left( t \right) \quad \Leftrightarrow \quad \frac{1}{c^2} \cdot \frac{w'' \left( t \right)}{w \left( t \right)} = \frac{v'' \left( x \right)}{v \left( x \right)} = -\lambda $ \newline
$ \left\{ \begin{array}{lll} v'' \left( x \right) + \lambda \cdot v \left( x \right) = 0 &x \in \left[ 0;L \right] \text{, } v \left( 0 \right) = v \left( L \right) = 0 &\text{Sturm-Liouville} \\ w'' \left( t \right) + \lambda \cdot c^2 \cdot w \left( t \right) = 0 &t > 0 &\text{Problème de Cauchy} \end{array} \right. $ \newline
$ \lambda = \left( \frac{n \cdot \pi}{L} \right)^2 \qquad v_n \left( x \right) = \alpha_n \cdot \sin \left( \frac{n \cdot \pi}{L} \cdot x \right) \qquad w_n \left( t \right) = a_n \cdot \cos \left( \frac{c \cdot n \cdot \pi}{L} \cdot t \right) + b_n \cdot \sin \left( \frac{c \cdot n \cdot \pi}{L} \cdot t \right) $ \newline
$ u_n \left( x, t \right) = v_n \left( x \right) \cdot w_n \left( t \right) = \sin \left( \frac{n \cdot \pi}{L} \cdot x \right) \cdot \left[ A_n \cdot \cos \left( \frac{c \cdot n \cdot \pi}{L} \cdot t \right) + B_n \cdot \sin \left( \frac{c \cdot n \cdot \pi}{L} \cdot t \right) \right] \qquad u \left( x, t \right) = \sum_{n = 1}^\infty u_n \left( x, t \right) $ \newline
$ u \left( x, 0 \right) = f \left( x \right) \quad \text{ et } \quad \frac{\partial}{\partial t} u \left( x, 0 \right) = g \left( x \right) \quad \text{ donnent } \quad A_n \quad \text{ et } \quad B_n $
\\\hline
%TODO copier crayon ?
%TODO arctan style
\end{tabu}
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% \textbf{Configurabilité} \newline
% $ a \oldcdot b $ ou $ a b $ \newline
% $ \frac{a}{b} $ ou $ a/b $ \newline
% $ \vec{a} \oldbullet \vec{b} $ ou $ \vec{a} \circ \vec{b} $ \newline
% $ \oldvec{a} $ ou $ \overrightarrow{a} $ ou $ \symbf{a} $ ou $ \oldvec{\symbf{a}} $ \newline
% $ \dot{x} $ ou $ \frac{\dif x}{\dif t} $ \newline
% $ \ddot{x} $ ou $ \frac{\symrm{d^2}x}{\dif t^2} $ \newline

43
Common.tex
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@ -1,43 +0,0 @@
\usepackage{xltxtra}
\usepackage{pbox}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{tabularx}
\usepackage{siunitx}
\usepackage{unicode-math}
\usepackage[space]{grffile}
\usepackage{pdfpages}
\usepackage{multicol}
\usepackage[european,siunitx,betterproportions]{circuitikz}
\usepackage[top=13pt, bottom=12pt, left=13pt, right=12pt]{geometry}
% \setromanfont[Mapping=tex-text]{Linux Libertine O}
% \setsansfont[Mapping=tex-text]{DejaVu Sans}
% \setmonofont[Mapping=tex-text]{DejaVu Sans Mono}
\title{}
\author{}
\date{}
\let\oldcdot\cdot
\let\oldbullet\bullet
\let\oldvec\vec
\let\olddot\dot
\let\oldddot\ddot
% \renewcommand{\cdot}{ }
\renewcommand{\bullet}{\circ}
% \renewcommand{\vec}{\mathbf}
% \renewcommand{\dot}[1]{\frac{\mathrm{d}#1}{\mathrm{d}t}}
% \renewcommand{\ddot}[1]{\frac{\mathrm{d^2}#1}{\mathrm{d}t^2}}
% \renewcommand{\frac}[2]{#1 / #2}
\newcommand{\cdotbis}{ }
\newcommand{\dif}{\mathrm{d}}
\newcommand{\ul}{\underline}
\setlength{\parindent}{0pt}
\setlength{\parskip}{0pt}
\setlength{\columnsep}{0pt}
% \everymath{\displaystyle}

2
Draft.tex
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@ -1,4 +1,4 @@
\usepackage{draftwatermark}
\SetWatermarkText{\textsc{Draft}}
\SetWatermarkScale{1}
\SetWatermarkColor[gray]{0.9}
\SetWatermarkColor[gray]{0.9}

211
Rules.py Executable file
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@ -0,0 +1,211 @@
#!/usr/bin/env python3
# MIT License
#
# Copyright (c) 2016 Nathanaël Restori
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
#
# The above copyright notice and this permission notice shall be included in all
# copies or substantial portions of the Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
# SOFTWARE.
import argparse
import re
parser = argparse.ArgumentParser()
parser.add_argument('infile', nargs='+', type=argparse.FileType('r'))
args = parser.parse_args()
#TODO: list forbidden characters (|)
# \\\!|\\\;|\\\:|\\\,
# hphantom ?, hspace ?
# e^
# cte, text{, }, t.q.
rules_bef = [
# Add space before and after $ unless at the beginning or the end of a line, after a { or a ( and before a } or a )
{'symbol': r'(?<!^)\$', '+<': ' ' },
{'symbol': r'\$(?!$)', '+>': ' ' },
{'symbol': r'(?<=\( |{ )\$', '-<': r' ', },
{'symbol': r'\$(?= \)| })', '->': r' ', },
# No space after \text
{'symbol': r'\\text', '->': r' ' },
]
rules_math = [
# Add space around
{'symbol': r'=', '+<': ' ', '+>': ' ', },
{'symbol': r'\\cdot', '+<': ' ', '+>': ' ', },
{'symbol': r'\\quad', '+<': ' ', '+>': ' ', },
{'symbol': r'\\leftrightarrow', '+<': '\\quad ', '+>': ' \\quad', },
{'symbol': r'\\Leftrightarrow', '+<': '\\quad ', '+>': ' \\quad', },
{'symbol': r'\\Leftarrow', '+<': '\\quad ', '+>': ' \\quad', },
{'symbol': r'\\Rightarrow', '+<': '\\quad ', '+>': ' \\quad', },
# Standard functions
{'symbol': '(arc)?sinh?', '+<': '\\', '+>': ' ', 'w!>': r'h? ?(?:\^(?:{.*}|.))? \\left',},
{'symbol': '(arc)?cosh?', '+<': '\\', '+>': ' ', 'w!>': r'h? ?(?:\^(?:{.*}|.))? \\left',},
{'symbol': '(arc)?tanh?', '+<': '\\', '+>': ' ', 'w!>': r'h? ?(?:\^(?:{.*}|.))? \\left',},
{'symbol': '(?<!{)min(?!})', '+<': '\\', '+>': ' ', },
{'symbol': '(?<!{)max(?!})', '+<': '\\', '+>': ' ', },
{'symbol': 'ln', '+<': '\\', '+>': ' ', 'w!>': r' ?(?:_(?:{.*}|.))? \\left| \\abs',},
{'symbol': 'log', '+<': '\\', '+>': ' ', 'w!>': r' ?(?:_(?:{.*}|.))? \\left| \\abs',},
{'symbol': 'lim(?!its)', '+<': '\\', '+>': ' ', },
# \left or \right before delimiter and space after
{'symbol': r'\(', '+>': ' ', 'w!<': r'\\left|right', },
{'symbol': r'\[', '+>': ' ', 'w!<': r'\\left|right', },
{'symbol': r'\\{', '+>': ' ', 'w!<': r'\\left|right', },
{'symbol': r'\\langle', '+>': ' ', 'w!<': r'\\left|right', },
{'symbol': r'\)', '+>': ' ', 'w!<': r'\\left|right', },
{'symbol': r'\]', '+>': ' ', 'w!<': r'\\left|right', },
{'symbol': r'\\}', '+>': ' ', 'w!<': r'\\left|right', },
{'symbol': r'\\rangle', '+>': ' ', 'w!<': r'\\left|right', },
{'symbol': r'\\\|', '+>': ' ', 'w!<': r'\\left|right', },
{'symbol': r'(?<!\\)\|', '+>': ' ', 'w!<': r'\\left|right', },
# Space before \left or \right but not after
{'symbol': r'\\left', '+<': ' ', '->': r' ', },
{'symbol': r'\\right', '+<': ' ', '->': r' ', },
# No space before ^, _ and !
{'symbol': r'\^', '-<': r' ', },
{'symbol': r'_', '-<': r' ', },
{'symbol': r'!', '-<': r' ', },
# No space after { and before } (but keep after \{ and after \}
{'symbol': r'(?<!\\){', '->': r' ', },
{'symbol': r'(?<!\\)}', '-<': r' ', },
]
rules_text = [
# Use non-breaking space before :
{'symbol': r':', '+<': '~', }, # Add space ?
{'symbol': r'~', '-<': r' ', '->': r' ', },
]
rules_end = [
# Correct spacing around punctuation.
{'symbol': r',', '+>': ' ', '-<': r' ', },
{'symbol': r';', '-<': r' ', }, # Do not add space, cause problems in [a;b]
# Remove trailing whitespaces
{'symbol': r'$', '-<': r'[ \t]*', },
]
# {} after ^ and _ ?
# \text{, } vs something else ?
# Ensure no cdot after partial frac ( frac{\partial U}{\partial \phi} \cdot)
def apply_rules(text, rules):
for s in rules:
if s.get('+<'):
regex = r'(?:' + re.escape(s.get('+<')) + r')?(' + s.get('symbol') + r')'
subst = s.get('+<').replace('\\', '\\\\') + r'\1'
text = re.sub(regex, subst, text, flags=re.MULTILINE | re.DOTALL | re.UNICODE)
if s.get('+>'):
regex = r'(' + s.get('symbol') + r')(?:' + re.escape(s.get('+>')) + r')?'
subst = r'\1' + s.get('+>').replace('\\', '\\\\')
text = re.sub(regex, subst, text, flags=re.MULTILINE | re.DOTALL | re.UNICODE)
if s.get('-<'):
regex = r'(?:' + s.get('-<') + r')(' + s.get('symbol') + r')'
subst = r'\1'
text = re.sub(regex, subst, text, flags=re.MULTILINE | re.DOTALL | re.UNICODE)
if s.get('->'):
regex = r'(' + s.get('symbol') + r')(?:' + s.get('->') + r')'
subst = r'\1'
text = re.sub(regex, subst, text, flags=re.MULTILINE | re.DOTALL | re.UNICODE)
if s.get('w!<'):
regex = r'(?<!' + s.get('w!<') + r')(' + s.get('symbol') + r')'
# use findall
result = re.search(regex, text, flags=re.MULTILINE | re.DOTALL | re.UNICODE)
if result:
print("In file " + file_current.name + ": missing " + s.get('w!<') + " before " + s.get('symbol') + " (regex: " + regex + ")")
# Print what's around match
#print(text[result.start()-250:result.end()+250])
#print(text[result.start():result.end()])
#print(text[result.start()-1:result.end()+1])
# Print something like:
#(?<!\\left)\[(.*?)(?<!\\right)\]
#\left[\1\right]
if s.get('w!>'):
regex = r'(' + s.get('symbol') + r')(?!' + s.get('w!>') + r')'
# use findall
result = re.search(regex, text, flags=re.MULTILINE | re.DOTALL | re.UNICODE)
if result:
print("In file " + file_current.name + ": missing " + s.get('w!>') + " after " + s.get('symbol') + " (regex: " + regex + ")")
# Print what's around match
print(text)
print(text[result.start()-250:result.end()+250])
print(text[result.start():result.end()])
print(text[result.start()-1:result.end()+1])
print(text[result.start()-10:result.end()+10])
return text
for file_current in args.infile:
file_content = file_current.read()
file_original = file_content
#TODO: add other cases (\$ for example)
## Check for $ in comments (we will have troubles if a comment contain an odd number of $)
#if re.search(r'%.*\$', file_content, flags=re.MULTILINE | re.UNICODE):
#print("Warning, file " + file_current.name + " contain $ in comments, ignoring file")
#continue
file_content = apply_rules(file_content, rules_bef)
splited = re.split(r'(\$.*?\$)', file_content, flags=re.MULTILINE | re.DOTALL | re.UNICODE) # Split file content in math parts and normal parts
for i in range(1, len(splited), 2):
splited_b = re.split(r'(\\text{.*?})', splited[i], flags=re.MULTILINE | re.DOTALL | re.UNICODE) # Split file content in math parts and normal parts
for j in range(0, len(splited_b), 2):
splited_b[j] = apply_rules(splited_b[j], rules_math)
splited[i] = ''.join(splited_b)
for i in range(0, len(splited), 2):
splited[i] = apply_rules(splited[i], rules_text)
file_content = ''.join(splited)
file_content = apply_rules(file_content, rules_end)
file_content = re.sub(r'\\left\\\| (.*?) \\right\\\|', r'\\norm{\1}', file_content, flags=re.MULTILINE)
file_content = re.sub(r'\\left\| (.*?) \\right\|', r'\\abs{\1}', file_content, flags=re.MULTILINE)
file_content = re.sub(r'\\left< (.*?) \\right>', r'\\left\\langle \1 \\right\\rangle}', file_content, flags=re.MULTILINE)
file_content = re.sub(r'\.\.\.', r'\\dots', file_content, flags=re.MULTILINE)
file_content = re.sub(r' \\newline\n&', r'\n&', file_content, flags=re.MULTILINE) # Ensure no newline at the end of a cell
file_content = re.sub(r' \\\\\n&', r'\n&', file_content, flags=re.MULTILINE) # Ensure no newline at the end of a cell
file_content = re.sub(r' \\newline\n\\\\', r'\n\\\\', file_content, flags=re.MULTILINE) # Ensure no newline at the end of a cell
file_content = re.sub(r' \\\\\n\\\\', r'\n\\\\', file_content, flags=re.MULTILINE) # Ensure no newline at the end of a cell
file_content = re.sub(r' \\\\\n( *)\\end\{tabu\}', r'\n\1\\end{tabu}', file_content, flags=re.MULTILINE) # Ensure no newline at the end of a cell
file_content = re.sub(r'\\\\ +\\hline', r'\\\\\\hline', file_content, flags=re.MULTILINE) # Remove spaces between \\ and \hline
file_content = apply_rules(file_content, rules_end)
# Save only if needed
if file_original == file_content:
print("File untouched: " + file_current.name)
else:
print("File modified: " + file_current.name)
with open(file_current.name, "w") as f:
f.seek(0)
f.truncate()
f.write(file_content)