From 03743ba7663abcde83b951154427b5f31a5cb701 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Nathana=C3=ABl=20Restori?= Date: Sun, 14 Feb 2016 17:04:17 +0100 Subject: [PATCH] Add BA3 formula sheets --- .../BA3 - Analyse numérique - 2 pages.tex | 13 + .../BA3 - Analyse numérique.tex | 1059 +++++++++++++++++ BA3 - Physique III/BA3 - Physique III.tex | 327 +++++ BA3 - Physique III/Polarisation.png | Bin 0 -> 569238 bytes 4 files changed, 1399 insertions(+) create mode 100644 BA3 - Analyse numérique/BA3 - Analyse numérique - 2 pages.tex create mode 100644 BA3 - Analyse numérique/BA3 - Analyse numérique.tex create mode 100644 BA3 - Physique III/BA3 - Physique III.tex create mode 100644 BA3 - Physique III/Polarisation.png diff --git a/BA3 - Analyse numérique/BA3 - Analyse numérique - 2 pages.tex b/BA3 - Analyse numérique/BA3 - Analyse numérique - 2 pages.tex new file mode 100644 index 0000000..9818a5d --- /dev/null +++ b/BA3 - Analyse numérique/BA3 - Analyse numérique - 2 pages.tex @@ -0,0 +1,13 @@ +\documentclass[a4paper,10pt]{article} +%\documentclass[a4paper,10pt]{scrartcl} + +\input{../Common.tex} + +\geometry{top=6.5pt, bottom=6pt, left=6.5pt, right=6pt} + +\begin{document} + +% \pagestyle{plain} +\includepdf[width=0.5\textwidth,pages={-},nup=2x2]{BA3 - Analyse numérique.pdf} + +\end{document} \ No newline at end of file diff --git a/BA3 - Analyse numérique/BA3 - Analyse numérique.tex b/BA3 - Analyse numérique/BA3 - Analyse numérique.tex new file mode 100644 index 0000000..c7a5d89 --- /dev/null +++ b/BA3 - Analyse numérique/BA3 - Analyse numérique.tex @@ -0,0 +1,1059 @@ +\documentclass[a4paper,10pt]{article} +%\documentclass[a4paper,10pt]{scrartcl} + +\input{../Common.tex} + +\begin{document} + +\begin{multicols}{2}[\centering{\large\textbf{Définitions}}] + +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline +\textbf{Méthodes numériques :} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &F(x,d) = 0 \text{ un problème } \\ + &F_n(x_n,d_n) = 0 \text{, } n > 1 \text{ une suite de problèmes } \\ + &x = G(d) \text{ t.q } F(G(d),d) = 0 \text{ une application résolvante } \\ + &x_n \rightarrow x \text{ pour } n \rightarrow \infty \\ + &d_n \rightarrow d \text{ pour } n \rightarrow \infty \\ + &F_n \text{ approche } F \text{ pour } n \rightarrow \infty + \end{aligned} $ \newline + + \underline{Consistance :} \newline + $ \begin{aligned} + &\text{La méthode numérique (} P_n \text{) est consistante si} \\ + &F_n(x_n,d_n) - F(x,d) \rightarrow 0 \text{ pour } n \rightarrow \infty \\ + &\text{où } x \text{ est la solution du problème (} P \text{) correspondant} \\ &\text{à la donnée } d \\ + \end{aligned} $ \newline + + \underline{Stabilité :} \newline + $ \begin{aligned} + &\text{Une méthode numérique est bien posée (ou stable) s’il existe,} \\ &\text{pour tout } n \text{ une unique solution } x_n \text{ correspondant à la donnée } \\ &d_n \text{ et si } x_n \text{ dépend continûment des données, i.e.} \\ + &\forall d_n, \exists \eta_0 = \eta_0(d_n) > 0 \text{, } \exists K_0 = K_0(\eta_0,d_n) \text { t.q} \\ + &\forall \delta d_n : \|\delta d_n\| \leq \eta_0 \rightarrow \|\delta x_n\| \leq K_0 \cdot \|\delta d_n\| + \end{aligned} $ \newline + + \underline{Convergence :} \newline + $ \begin{aligned} + &\text{Si le problème numérique (} P_n \text{) est consistant avec le problème} \\ &\text{(} P \text{), alors il est convergent si, et seulement si, il est stable} + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} + +\end{multicols} +\newpage +\begin{multicols}{2}[\centering{\large\textbf{Équations non-linéaires}}] + +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline + +\textbf{Dichotomie ou bissection} \newline + \underline{Conditions :} \newline + $ \begin{aligned} + &f(a) \cdot f(b) < 0 \text{ et } f(x) \text{ continue sur } [a;b] \\ + &k \in \mathbb{N} + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &x^{(k)} = \frac{a^{(k)} + b^{(k)}}{2} \\ + &\text{Si } f(x^{(k)}) = 0 \text{ fin} \\ + &\text{Si } f(x^{(k)}) \cdot f(a) < 0 \text{, } a^{(k+1)} = a^{(k)} \text{ et } b^{(k+1)} = x^{(k)} \\ + &\text{Si } f(x^{(k)}) \cdot f(b) < 0 \text{, } a^{(k+1)} = x^{(k)} \text{ et } b^{(k+1)} = b^{(k)} + \end{aligned} $ \newline + + \underline{Erreur :} \newline + $ \begin{aligned} + &|e^{(k)}| = |x^{(k)} - \alpha| \leq \frac{b - a}{2^{k+1}} + \end{aligned} $ \newline +\\ \hline +\textbf{Méthode de Newton} \newline + \underline{Conditions :} \newline + $ \begin{aligned} + &f(x) \text{ dérivable } \\ + &k \in \mathbb{N} + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &x^{(k+1)} = x^{(k)} - \frac{f(x^{(k)})}{f'(x^{(k)})} + \end{aligned} $ \newline + + \underline{Convergence locale :} \newline + $ \begin{aligned} + &\text{Si } f(x) \text{ continue et deux fois dérivable sur } [a;b] \\ + &\text{Si } f(\alpha) = 0 \text{ et } f'(\alpha) \neq 0 \\ + &\text{Alors } \exists \delta > 0 \text{ t.q. si } |x^{(0)} - \alpha| \leq \delta \text{, la méthode converge} \\ + &\text{Alors } \lim_{k \to \infty}{\frac{x^{(k+1)} - \alpha}{(x^{(k)} - \alpha)^2}} = \frac{f''(\alpha)}{2 \cdot f'(\alpha)} \text{ (ordre 2)} + \end{aligned} $ \newline +\\ \hline + +\textbf{Point fixe} \newline + \underline{Conditions :} \newline + $ \begin{aligned} + &\phi(\alpha) = \alpha \Leftrightarrow f(\alpha) = 0 \text{, } x^{(k)} \rightarrow \alpha \text{ et } \phi(x) \text{ continue sur } [a;b] \\ + &k \in \mathbb{N} + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &x^{(k+1)} = \phi(x^{(k)}) + \end{aligned} $ \newline + + \underline{Convergence globale :} \newline + $ \begin{aligned} + &\text{Si } \phi(x) \text{ continue sur } [a;b] \\ + &\text{Si } \phi(x) \in [a;b] \text{ } \forall x \in [a;b] \\ + &\text{Si } \exists L < 1 \text{ t.q. } |\phi(x_1) - \phi(x_2)| \leq L \cdot |x_1 - x_2| \text{ } \forall x_1,x_2 \in [a;b] + \end{aligned} $ \newline + + \underline{Convergence globale (2) :} \newline + $ \begin{aligned} + &\text{Si } \phi(x) \text{ continue et dérivable sur } [a;b] \\ + &\text{Si } \phi(x) \in [a;b] \text{ } \forall x \in [a;b] \\ + &\text{Si } \exists K < 1 \text{ t.q. } |\phi'(x)| \leq K \text{ } \forall x \in [a;b] + \end{aligned} $ \newline + + \underline{Convergence locale :} \newline + $ \begin{aligned} + &\text{Si } \phi(x) \text{ continue et dérivable sur } [a;b] \\ + &\text{Si } |\phi'(\alpha)| < 1 \\ + &\text{Alors } \exists \delta > 0 \text{ t.q. si } |x^{(0)} - \alpha| \leq \delta \text{, la méthode converge} \\ + &\text{Alors } \lim_{k \to \infty}{\frac{x^{(k+1)} - \alpha}{x^{(k)} - \alpha}} = \phi'(\alpha) &&\hspace{-7em} \text{ (ordre 1)} + \end{aligned} $ \newline + + \underline{Convergence locale (2) :} \newline + $ \begin{aligned} + &\text{Si } \phi(x) \text{ continue et deux fois dérivable sur } [a;b] \\ + &\text{Si } \phi'(\alpha) = 0 \text{ et } \phi''(\alpha) \neq 0 \\ + &\text{Alors } \lim_{k \to \infty}{\frac{x^{(k+1)} - \alpha}{(x^{(k)} - \alpha)^2}} = \frac{\phi''(\alpha)}{2} &\hspace{-4em} \text{ (ordre 2)} + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} +\begin{tabularx}{0.5\textwidth}{ X| } +\hline + +\textbf{Méthode de Newton modifiée} \newline + \underline{Conditions :} \newline + $ \begin{aligned} + &f(x) \text{ dérivable et } \alpha \text{ de multiplicité de } m \\ + &k \in \mathbb{N} + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &x^{(k+1)} = x^{(k)} - m \cdot \frac{f(x^{(k)})}{f'(x^{(k)})} + \end{aligned} $ \newline + + \underline{Convergence locale :} \newline + $ \begin{aligned} + &\text{Si } f(x) \text{ continue et dérivable sur } [a;b] \\ + &\text{Si } f(\alpha) = 0 \text{ et } f'(\alpha) = 0 \\ + &\text{Alors } \exists \delta > 0 \text{ t.q. si } |x^{(0)} - \alpha| \leq \delta \text{, la méthode converge} + \end{aligned} $ \newline +\\ \hline + +\textbf{Méthode de la corde} \newline + \underline{Méthode :} \newline + $ \begin{aligned} + &x^{(k+1)} = x^{(k)} - \frac{f(x^{(k)})}{q(x^{(k)})} \\ + &q = \frac{f(b)-f(a)}{b-a} &&\text{ ou } &q = \frac{f(b)-f(x^{(k)})}{b-x^{(k)}} + \end{aligned} $ \newline + + \underline{Convergence :} \newline + $ \begin{aligned} + \left| 1 - \frac{f'(\alpha)}{q(\alpha)} \right| < 1 + \end{aligned} $ \newline +\\ \hline + +\textbf{Critères d'arrêt} \newline + \underline{Contrôle de l'incrément :} \newline + $ \begin{aligned} + &|x^{(k+1)} - x^{(k)}| < \epsilon + \end{aligned} $ \newline + + \underline{Contrôle du résidu :} \newline + $ \begin{aligned} + &|f(x^{(k)})| < \epsilon + \end{aligned} $ \newline + + \underline{Cas du point fixe :} \newline + $ \begin{aligned} + &\text{Le contrôle de l'incrément est optimal si } \phi'(\alpha) = 0 \text{, satisfaisant} \\ &\text{si } -1 < \phi'(\alpha) < 0 \text{ et n'est pas satisfaisant si } \phi'(\alpha) \text{ est proche} \\ &\text{de 1.} \\ + &\text{Le contrôle du résidu est satisfaisant si } |f'| \simeq 1 \text{ au voisinage de} \\ &\text{la racine } \alpha \text{.} + \end{aligned} $ \newline +\\ \hline + +\textbf{Multiplicité} \newline + \underline{Définition :} \newline + $ \begin{aligned} + &\text{On dit qu'un zéro } \alpha \text{ de } f \text{ est de multiplicité } m \in \mathbb{N} \text{ si} \\ + &f(\alpha) = ... = f^{m-1}(\alpha) = 0 \text{ et } f^m(\alpha) \neq 0 \\ + &\text{Un zéro de multiplicité } m = 1 \text{ est appelé zéro simple.} + \end{aligned} $ \newline +\\ \hline + +\textbf{Ordre de convergence} \newline + \underline{Définition :} \newline + $ \begin{aligned} + &\text{On dit que la convergence est d'ordre } p \geq 1 \text{ s'il existe une} \\ &\text{constante } C > 0 \text{ (avec } C < 1 \text{ lorsque } p = 1 \text {) telle que l'inégalité} \\ &\text{suivante soit satisfaite} \\ + &|x^{(k+1)} - \alpha| \leq C \cdot |x^{(k)} - \alpha|^p \\ + &\text{Lorsque } p = 1 \text{, la convergence est dite linéaire.} \\ + &\text{Lorsque } p = 2 \text{, la convergence est dite quadratique.} + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} + +\end{multicols} +\newpage +\begin{multicols}{2}[\centering{\large\textbf{Interpolation}}] + +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline + +\textbf{Polynôme d'interpolation (de degré n)} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &\text{Soient } n + 1 \text{ noeuds distincts } x_0, x_1, ... x_n \\ &\text{et } n + 1 \text{ valeurs } y_0, y_1, ..., y_n \\ + &p(x_j) = y_j \text{ } 0 \leq j \leq n \\ + &p(x) = \Pi_n(x) = \Pi_nf(x) \\ + &I = [a;b] \\ + &h = \frac{b-a}{n} + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &\varphi_k(x) = \prod_{j=0,j \neq k}^n \frac{x - x_j}{x_k - x_j} &\text{ (Base de Lagrange)} \\ + &\Pi_n(x) = \sum_{k=0}^n y_k \cdot \varphi_k(x) &\text{ (Polynôme d'interpolation)} \\ + &\Pi_nf(x) = \sum_{k=0}^n f(x_k) \cdot \varphi_k(x) &\text{ (Interpolant de } f \text{)} \\ + \end{aligned} $ \newline + + \underline{Erreur :} \newline + $ \begin{aligned} + &\max_{x \in I} |E_n f(x)| \leq \frac{1}{4 \cdot (n+1)} \cdot h^{n+1} \cdot \max_{x \in I} |f^{(n+1)}(x)| + \end{aligned} $ \newline +\\ \hline + +\textbf{Interpolation par morceaux :} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &a = x_0 < x_1 < ... < x_N = b \text{ les points qui divisent } I \text{ en} \\ &\text{réunion d'intervalles } I_i = [x_i, x_{i+1}] \\ + &H = \frac{b-a}{N} + \end{aligned} $ \newline + \underline{Méthode :} \newline + $ \begin{aligned} + &\Pi_1^Hf(x) = f(x_i) + \frac{f(x_{i+1}) - f(x_i)}{x_{i+1} - x_i} \cdot (x - x_i) \text{ pour } x \in I_i + \end{aligned} $ \newline + + \underline{Erreurs :} \newline + $ \begin{aligned} + &\max_{x \in I} |E_1^H f(x)| \leq \frac{H^2}{8} \cdot \max_{x \in I} |f''(x)| \\ + &\max_{x \in I} |E_r^H f(x)| \leq \frac{H^{r+1}}{4 \cdot (r+1)} \cdot \max_{x \in I} |f^{(r+1)}(x)| + \end{aligned} $ \newline +\\ \hline + +\textbf{Méthode des moindres carrés (de degré m)} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &\sum_{i=0}^n |y_i - \tilde{f}_m(x_i)|^2 \leq \sum_{i=0}^n |y_i - p_m(x_i)|^2 &\forall p_m(x) \in \mathbb{P}_m \\ + &\sum_{i=0}^n |f(x_i) - \tilde{f}_m(x_i)|^2 \leq \sum_{i=0}^n |f(x_i) - p_m(x_i)|^2 &\forall p_m(x) \in \mathbb{P}_m \\ + &\tilde{f}_m(x) = a_0 + a_1 \cdot x + a_2 \cdot x^2 + ... + a_m \cdot x^m + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &\Phi(b_0, b_1, ..., b_m) = \sum_{i=0}^n (y_i - (b_0 + b_1 \cdot x_i \\ + &\hspace{14em} + b_2 \cdot x_i^2 + ... + b_m \cdot x_i^m))^2 \\ + &\Phi(a_0, a_1, ..., a_m) = \min_{b_i, i = 0,...,m} \Phi(b_0, b_1, ..., b_m) \\ + &\frac{\partial \Phi}{\partial b_i}(a_0, a_1, ..., a_m) = 0 \text{, } 0 \leq i \leq m \text{ (Système de degré } m+1 \text{)} + \end{aligned} $ \newline + + \underline{Méthode (2) :} \newline + $ \begin{aligned} + &B = \begin{pmatrix} + 1 & x_0 & \ldots & x_0^m \\ + 1 & x_1 & \ldots & x_1^m \\ + \vdots & \vdots & \ddots & \vdots \\ + 1 & x_n & \ldots & x_n^m \\ + \end{pmatrix} &&\vec{y} = \begin{pmatrix} + y_0 \\ + y_1 \\ + \vdots \\ + y_n \\ + \end{pmatrix} \\ + &B^T \cdot B \cdot \vec{a} = B^T \cdot \vec{y} + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} +\begin{tabularx}{0.5\textwidth}{ X| } +\hline + +\textbf{Interpolation par fonctions splines} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &I = [a;b] \\ + &a = x_0 < x_1 < ... < x_n = b \text{ les points qui divisent } I \text{ en réunion} \\ &\text{d'intervalles } I_i = [x_i, x_{i+1}] + \end{aligned} $ \newline + + \underline{Spline :} \newline + $ \begin{aligned} + &s_{3|I_i} \in \mathbb{P}_3 &&\forall i = 0,...,n-1 \\ + &s_3(x_i) = f(x_i) &&\forall i = 0,...,n \\ + &s_3 \in C^2(I) + \end{aligned} $ \newline + + \underline{Conditions aux bords :} \newline + $ \begin{aligned} + &s_3(x_i^-) = s_3(x_i^+) \\ + &s_3'(x_i^-) = s_3'(x_i^+) \\ + &s_3''(x_i^-) = s_3''(x_i^+) + \end{aligned} $ \newline + + \underline{Spline naturelle :} \newline + $ \begin{aligned} + &s_3''(x_0^+) = 0 \\ + &s_3''(x_n^-) = 0 + \end{aligned} $ \newline + + \underline{Spline not-a-knot :} \newline + $ \begin{aligned} + &s_3'''(x_1^-) = s_3'''(x_1^+) \\ + &s_3'''(x_{n-1}^-) = s_3'''(x_{n-1}^+) + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} + +\end{multicols} +\newpage +\begin{multicols}{2}[\centering{\large\textbf{Intégration}}] + +\begin{tabularx}{0.5\textwidth}{ |X } +\hline + +\textbf{Formules d'intégration simples} \newline + \underline{Définition :} \newline + $ \begin{aligned} + &I(f) = \int_a^b f(x) \cdot \dif x + \end{aligned} $ \newline + + \underline{Méthodes :} \newline + $ \begin{aligned} + &I_{pm}(f) = (b - a) \cdot f\left( \frac{a+b}{2} \right) &\text{ (Point millieu)} \\ + &I_t(f) = (b - a) \cdot \frac{f(a) + f(b)}{2} &\text{ (Trapèze)} \\ + &I_s(f) = \frac{b - a}{6} \cdot \left[ f(a) + 4 \cdot f\left( \frac{a+b}{2} \right) + f(b) \right] &\text{ (Simpson)} + \end{aligned} $ \newline + + \underline{Formule générale :} \newline + $ \begin{aligned} + &I_{appr}(f) = \sum_{k=0}^n \alpha_k \cdot f(x_k) \\ + &x_k \text{ sont les noeuds et } \alpha_k \text{ sont les poids} + \end{aligned} $ \newline + + \underline{Noeuds et poids :} \newline + $ \begin{aligned} + &\text{Point milieu} &&x_0 = \frac{1}{2} \cdot (a+b) && \alpha_0 = b-a \\ + &\text{Trapèze} &&x_0 = a, x_1 = b && \alpha_0 = \alpha_1 = \frac{1}{2} \cdot (b-a) \\ + &\text{Simpson} &&\begin{array}{l} x_0 = a \\ x_1 = \frac{1}{2} \cdot (a+b) \\ x_2 = b \\ \end{array} &&\begin{array}{l} \alpha_0 = \alpha_2 = \frac{1}{6} \cdot (b-a) \\ \alpha_1 = \frac{2}{3} \cdot (b-a) \\ \end{array} + \end{aligned} $ \newline + + \underline{Erreur :} \newline + $ \begin{aligned} + &|I(f) - I_{appr}(f)| \leq \max_{x \in [a;b]} |f(x) - \Pi_nf(x)| \cdot (b-a) + \end{aligned} $ \newline +\\ \hline + +\textbf{Degré d'exactitude} \newline + \underline{Défintion :} \newline + $ \begin{aligned} + &\text{Une formule de quadrature } I_{appr}(f) \text{ sur l'intervalle } [a;b] \\ &\text{est exacte de degré } r \text{ pour une fonction } f \text{ si} \\ + &I_{appr}(f) = \int_a^b f(x) \cdot \dif x &&\hspace{-12em} \forall f \in \mathbb{P}_r \\ + &\text{mais pas pour } r+1 + \end{aligned} $ \newline +\\ \hline + +\textbf{Ordre} \newline + \underline{Défintion :} \newline + $ \begin{aligned} + &\text{On définit l'ordre d'une formule d'intégration par l'ordre de} \\ &\text{son erreur par rapport à } H + \end{aligned} $ \newline +\\ \hline + + +\end{tabularx} +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline + +\textbf{Formules d'intégration composites} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &\text{Soient } M \text{ sous-intervalles } I_k = [x_{k-1};x_k] \text{, } &&k = 1,...,M \\ + &x_k = a + k \cdot H \\ + &H = \frac{b-a}{M} \\ + &I(f) = \sum_{k=1}^M \int_{I_k} f(x) \cdot \dif x \approx \sum_{k=1}^M \int_{I_k} \Pi_nf(x) \cdot \dif x \\& \hspace{10em} \approx \int_a^b \sum_{k=1}^M \Pi_n^Hf(x) \cdot \dif x + \end{aligned} $ \newline + + \underline{Méthodes :} \newline + $ \begin{aligned} + &I_{pm}^c(f) = H \cdot \sum_{k=1}^M f\left( \frac{x_{k-1} + x_k}{2} \right) \\ + &I_t^c(f) = \frac{H}{2} \cdot \sum_{k=1}^M [f(x_{k-1}) + f(x_k)] \\ + &\hspace{2.5em} = \frac{H}{2} \cdot [f(a) + f(b)] + H \cdot \sum_{k=1}^{M-1} f(x_k) \\ + &I_s^c(f) = \frac{H}{6} \cdot \sum_{k=1}^{M} \left[ f(x_{k-1}) + 4 \cdot f\left( \frac{x_{k-1} + x_k}{2} \right) + f(x_k) \right] + \end{aligned} $ \newline + + \underline{Erreurs :} \newline + $ \begin{aligned} + &|I(f) - I_{pm}^c(f)| \leq \frac{b-a}{24} \cdot H^2 \cdot \max_{x \in [a;b]} |f''(x)| \\ + &|I(f) - I_t^c(f)| \leq \frac{b-a}{12} \cdot H^2 \cdot \max_{x \in [a;b]} |f''(x)| \\ + &|I(f) - I_s^c(f)| \leq \frac{b-a}{180 \cdot 16} \cdot H^4 \cdot \max_{x \in [a;b]} |f''''(x)| + \end{aligned} $ \newline + + \underline{Degrés d'exactitude :} \newline + $ \begin{aligned} + &\text{Point milieu} &&\text{1} \\ + &\text{Trapèze} &&\text{1} \\ + &\text{Simpson} &&\text{3} + \end{aligned} $ \newline + + \underline{Ordres par rapport à H:} \newline + $ \begin{aligned} + &\text{Point milieu} &&\text{2} \\ + &\text{Trapèze} &&\text{2} \\ + &\text{Simpson} &&\text{4} + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} + +\end{multicols} +\newpage +\begin{multicols}{2}[\centering{\large\textbf{Systèmes linéaires - Méthodes directes}}] + +\begin{tabularx}{0.5\textwidth}{ |X } +\hline + +\textbf{Énoncé du problème} \newline + \underline{Problème :} \newline + $ \begin{aligned} + &A \cdot \vec{x} = \vec{b} &\text{Système linéaire d'ordre } n \\ + &\sum_{j=1}^n a_{ij} \cdot x_j = b_i \text{, } & i = 1,...,n + \end{aligned} $ \newline + + \underline{Régularité :} \newline + $ \begin{aligned} + &A \text{ est dite régulière (non signulière) } \Leftrightarrow \det(A) \neq 0 \\ + &\text{Si } A \text{ est régulière, la solution } \vec{x} \text{ du système est unique} + \end{aligned} $ \newline +\\ \hline + +\textbf{Substitution progressive} \newline + \underline{Condition :} \newline + $ \begin{aligned} + &L \text{ matrice triangulaire inférieur} + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &y_1 = b_1/l_{11} \\ + &y_i = \frac{1}{l_{ii}} \cdot \left( b_i - \sum_{j=1}^{i-1} l_{ij} \cdot y_j \right) \text{, } i = 2,3,...,n + \end{aligned} $ \newline + + \underline{Coût :} \newline + $ \begin{aligned} + &n^2 \text{ opérations} + \end{aligned} $ \newline +\\ \hline + +\textbf{Substitution rétrograde} \newline + \underline{Condition :} \newline + $ \begin{aligned} + &U \text{ matrice triangulaire supérieure} + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &x_n = y_n/u_{nn} \\ + &x_i = \frac{1}{u_{ii}} \cdot \left( y_i - \sum_{j=i+1}^{n} u_{ij} \cdot x_j \right) \text{, } i = n-1,n-2,...,1 + \end{aligned} $ \newline + + \underline{Coût :} \newline + $ \begin{aligned} + &n^2 \text{ opérations} + \end{aligned} $ \newline +\\ \hline + +\textbf{Factorisation LU} \newline + \underline{Condition :} \newline + $ \begin{aligned} + &A \text{ matrice carrée non singulière} + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &A \cdot \vec{x} = \vec{b} \Leftrightarrow L \cdot U \cdot \vec{x} = \vec{b} \Leftrightarrow \left \{ \begin{array}{r c l} L \cdot \vec{y} & = & \vec{b} \\ U \cdot \vec{x} & = & \vec{y} \\ \end{array} \right . + \end{aligned} $ \newline + + \underline{Coût :} \newline + $ \begin{aligned} + &\frac{2 \cdot n^3}{3} \text{ opérations (pour la factorisation)} + \end{aligned} $ \newline + + \underline{Remarque :} \newline + $ \begin{aligned} + &\det(A) = \det(L) \cdot \det(U) = \det(U) = \prod_{k=1}^n u_{kk} + \end{aligned} $ \newline +\\ \hline + +\textbf{Changement de pivot} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &P \cdot A \cdot Q = L \cdot U \\ + &P = P_{n-1} \cdot ... \cdot P_1 \\ + &Q = Q_1 \cdot ... \cdot Q_{n-1} + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &A \cdot \vec{x} = \vec{b} \Leftrightarrow P \cdot A \cdot Q \cdot Q^{-1} \cdot \vec{x} = P \cdot \vec{b} \Leftrightarrow \left \{ \begin{array}{r c l} L \cdot \vec{y} & = & P \cdot \vec{b} \\ U \cdot \vec{x^{*}} & = & \vec{y} \\ x & = & Q \cdot \vec{x^{*}} \\ \end{array} \right . + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline + +\textbf{Méthode d'élimination de Gauss} \newline + \underline{Conditions :} \newline + $ \begin{aligned} + &A \text{ matrice carrée non singulière} \\ + &k = 1,...,n-1 + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &l_{ik} = \frac{a_{ik}^{(k)}}{a_{kk}^{(k)}} \text{, } &&i = k+1,...,n \\ + &a_{ij}^{(k+1)} = a_{ij}^{(k)} - l_{ik} \cdot a_{kj}^{(k)} \text{, } &&i,j = k+1,...,n \\ + &b_{i}^{(k+1)} = b_{i}^{(k)} - l_{ik} \cdot b_{k}^{(k)} \text{, } &&i = k+1,...,n \\ + &l_{ii} = 1 \text{, } U=A^{(n)} + \end{aligned} $ \newline + + \underline{Coût :} \newline + $ \begin{aligned} + &\frac{2 \cdot n^3}{3} \text{ opérations} + \end{aligned} $ \newline + + \underline{Existence :} \newline + $ \begin{aligned} + &\text{La factorisation existe et est unique} \\ &\Leftrightarrow \text{Les sous-matrices } A_i (i=1,...,n-1) \text{ ne sont pas singulières} \\ + &\text{Ou si la matrice A est à diagonale dominante par ligne} \\ &|a_{ii}| \geq \sum_{j=1,...,n;j \neq i} |a_{ij}| \text{, } && \hspace{-10em} i = 1,...,n \\ + &\text{Ou si la matrice A est à diagonale dominante par colonne} \\ &|a_{jj}| \geq \sum_{i=1,...,n;i \neq j} |a_{ij}| \text{, } && \hspace{-10em} j = 1,...,n \\ + &\text{Ou si la matrice A est symétrique définie positive} \\ &A = A^T \text{ et } \lambda_i(A) > 0 \text{, } && \hspace{-10em} i = 1,...,n \\ + &\text{La factorisation existe et est infinie} \\ &\Leftrightarrow \text{Les sous-matrices } A_i (i=1,...,n) \text{ sont singulières} \\ + \end{aligned} $ \newline +\\ \hline + +\textbf{Factorisation de Cholesky} \newline + \underline{Condition :} \newline + $ \begin{aligned} + &A \text{ matrice symétrique définie positive} + \end{aligned} $ \newline + + \underline{Définition :} \newline + $ \begin{aligned} + &A = R^T \cdot R \\ + &i = 2,...,n + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &r_{11} = \sqrt{a_{11}} \\ + &r_{ji} = \frac{1}{r_{jj}} \cdot \left( a_{ij} - \sum_{k=1}^{j-1} r_{ki} \cdot r_{kj} \right) \text{, } &&j = 1,...,i-1 \\ + &r_{ii} = \sqrt{a_{ii} - \sum_{k=1}^{i-1} r_{ki}^2} \\ + \end{aligned} $ \newline + + \underline{Coût :} \newline + $ \begin{aligned} + &\frac{n^3}{3} \text{ opérations} + \end{aligned} $ \newline +\\ \hline + +\textbf{Divers} \newline + \underline{Matrice de Hilbert :} \newline + $ \begin{aligned} + &a_{ij} = \frac{1}{i+j-1} \text{, } &&i,j = 1,...,n \\ + \end{aligned} $ \newline + + \underline{Conditionnment :} \newline + $ \begin{aligned} + &K(A) = \frac{\lambda_{max}(A)}{\lambda_{min}(A)} + \end{aligned} $ \newline + + \underline{Erreur :} \newline + $ \begin{aligned} + &\frac{\|\vec{x} - \vec{\hat{x}}\|}{\|x\|} \leq K(A) \cdot \frac{\|\vec{r}\|}{\|\vec{b}\|} \text{, } &&\vec{r} = \vec{b} - A \cdot \vec{\hat{x}} + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} + +\end{multicols} +\newpage +\begin{multicols}{2}[\centering{\large\textbf{Systèmes linéaires - Méthodes itératives}}] + +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline + +\textbf{Méthode de Jacobi, Gauss-Seidel et SOR} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &\vec{e}^{(k)} = \vec{x} - \vec{x}^{(k)} \\ + &\vec{e}^{(k+1)} = B^{k+1} \cdot \vec{e}^{(0)} \\ + &\rho(B) = \max|\lambda_i(B)| < 1 \\ + &\vec{r}^{(k)} = \vec{b} - A \cdot \vec{x}^{(k)} \\ + &B = P^{-1} \cdot (P - A) = I - P^{-1} \cdot A \\ + &\vec{x}^{(k+1)} = B \cdot \vec{x}^{(k)} + \vec{g} + \end{aligned} $ \newline + + \underline{Méthode de Jacobi :} \newline + $ \begin{aligned} + &x_i^{(k+1)} = \frac{1}{a_{ii}} \cdot \left(b_i - \sum_{j=1,j \neq i}^n a_{ij} \cdot x_j^{(k)} \right) \\ %\text{, } &&j = 1,...,n \\ + &B_J = D^{-1} \cdot (D - A) = I - D^{-1} \cdot A \\ + &\vec{g}_J = D^{-1} \cdot \vec{b} + \end{aligned} $ \newline + + \underline{Méthode de Gauss-Seidel :} \newline + $ \begin{aligned} + &x_i^{(k+1)} = \frac{1}{a_{ii}} \cdot \left(b_i - \sum_{j=1}^{i-1} a_{ij} \cdot x_j^{(k+1)} - \sum_{j=i+1}^n a_{ij} \cdot x_j^{(k)} \right) \\ %\text{, } &&j = 1,...,n \\ + &B_{GS} = (D - E)^{-1} \cdot (D - E - A) = (D - E)^{-1} \cdot F \\ + &\vec{g}_{GS} = (D - E)^{-1} \cdot \vec{b} + \end{aligned} $ \newline + + \underline{Méthode de sur-relaxation successive :} \newline + $ \begin{aligned} + &x_i^{(k+1)} = \frac{\omega}{a_{ii}} \cdot \left(b_i - \sum_{j=1}^{i-1} a_{ij} \cdot x_j^{(k+1)} - \sum_{j=i+1}^n a_{ij} \cdot x_j^{(k)} \right) \\ + &\hspace{6em} + (1 - \omega) \cdot x_i^{(k)} \\ %\text{, } &&j = 1,...,n \\ + &B_{SOR} = (\frac{1}{\omega} \cdot D - E)^{-1} \cdot (\frac{1}{\omega} \cdot D - E - A) \\ + &\hspace{3em} = (I - \omega \cdot D^{-1} \cdot E)^{-1} \cdot [(1-\omega) \cdot I + \omega \cdot D^{-1} \cdot F] \\ + &\vec{g}_{SOR} = (\frac{1}{\omega} \cdot D - E)^{-1} \cdot \vec{b} + \end{aligned} $ \newline + + \underline{Convergence :} \newline + $ \begin{aligned} + &\text{Si } A \text{ à diagonale dominante stricte par ligne, Jacobi et} \\ &\text{Gauss-Seidel sont convergentes} \\ + &\text{Si } A \text{ régulière, tridiagonale et dont les coefficients diagonaux} \\ &\text{sont tous non-nuls, Jacobi et Gauss-Seidel sont toutes les deux} \\ &\text{soit divergentes, soit convergentes, dans ce dernier cas, } \\ & + \rho(B_{GS}) = \rho(B_J)^2 \\ + &\text{Si } A \text{ symétrique définie positive, Gauss-Seidel converge,} \\ &\text{Jacobi pas forcément} \\ + &\text{Si } A \text{ symétrique définie positive, SOR converge } \Leftrightarrow 0 < \omega < 2 \\ + &\text{Si } A \text{ à diagonale dominante stricte, SOR converge si } 0 < \omega < 1 \\ + \end{aligned} $ \newline +\\ \hline + +\textbf{Méthode de Richardson stationnaire préconditionné} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &P \cdot (\vec{x}^{(k+1)} - \vec{x}^{(k)}) = \alpha \cdot \vec{r}^{(k)} \\ + &\lambda_i = \text{ valeurs propres de } P^{-1} \cdot A \text{ strictement positives} \\ + &\lambda_{max} = \lambda_1 \geq \lambda_2 \geq ... \geq \lambda_N = \lambda_{min} \\ + &\alpha_{opt} = \frac{2}{\lambda_{max} + \lambda_{min}} \\ + &\rho_{opt} = \frac{\lambda_{max} - \lambda_{min}}{\lambda_{max} + \lambda_{min}} + \end{aligned} $ \newline + + \underline{Convergence :} \newline + $ \begin{aligned} + &A \text{ et } P \text{ symétriques définies positives} \\ + &0 < \alpha < 2/\lambda_{max} + \end{aligned} $ \newline + + \underline{Erreur :} \newline + $ \begin{aligned} + &\|\vec{x}^{(k)} - \vec{x}\|_A \leq \left( \frac{K(P^{-1} \cdot A) - 1}{K(P^{-1} \cdot A) + 1} \right)^k \cdot \|\vec{x}^{(0)} - \vec{x}\|_A \text{, } &&k \geq 0 + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} +\begin{tabularx}{0.5\textwidth}{ X| } +\hline + +\textbf{Méthode de Richardson dynamique précondionné} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &P \cdot (\vec{x}^{(k+1)} - \vec{x}^{(k)}) = \alpha_k \cdot \vec{r}^{(k)} \\ + &\alpha_k = \frac{(\vec{r}^{(k)},\vec{z}^{(k)})}{(A \cdot \vec{z}^{(k)},\vec{z}^{(k)})} \\ + &\vec{z}^{(k)} = P^{-1} \cdot \vec{r}^{(k)} + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &\vec{r}^{(0)} = \vec{b} - A \cdot \vec{x}^{(0)} \\ + &P \cdot z^{(k)} = r^{(k)} \\ + &\alpha_k = \frac{(\vec{r}^{(k)},\vec{z}^{(k)})}{(A \cdot \vec{z}^{(k)},\vec{z}^{(k)})} \\ + &\vec{x}^{(k+1)} = \vec{x}^{(k)} + \alpha_k \cdot \vec{z}^{(k)} \\ + &\vec{r}^{(k+1)} = \vec{r}^{(k)} - \alpha_k \cdot A \cdot \vec{z}^{(k)} + \end{aligned} $ \newline + + \underline{Erreur :} \newline + $ \begin{aligned} + &\|\vec{x}^{(k)} - \vec{x}\|_A \leq \left( \frac{K(P^{-1} \cdot A) - 1}{K(P^{-1} \cdot A) + 1} \right)^k \cdot \|\vec{x}^{(0)} - \vec{x}\|_A \text{, } &&k \geq 0 + \end{aligned} $ \newline + + \underline{Convergence :} \newline + $ \begin{aligned} + &\text{Si } A \text{ et } P \text{ symétriques définies positives} \\ + &\text{Si } P = I \text{, on a la méthode du gradient} + \end{aligned} $ \newline +\\ \hline + +\textbf{Méthode du gradient conjugué} \newline + \underline{Méthode :} \newline + $ \begin{aligned} + &\vec{r}^{(0)} = \vec{b} - A \cdot \vec{x}^{(0)} \text{, } \vec{p}^{(0)} = \vec{r}^{(0)} \\ + &\alpha_k = \frac{(\vec{r}^{(k)},\vec{p}^{(k)})}{(A \cdot \vec{p}^{(k)},\vec{p}^{(k)})} \\ + &\vec{x}^{(k+1)} = \vec{x}^{(k)} + \alpha_k \cdot \vec{p}^{(k)} \\ + &\vec{r}^{(k+1)} = \vec{r}^{(k)} - \alpha_k \cdot A \cdot \vec{p}^{(k)} \\ + &\beta_k = \frac{(\vec{r}^{(k+1)},A \cdot \vec{p}^{(k)})}{(\vec{p}^{(k)},A \cdot \vec{p}^{(k)})} \\ + &\vec{p}^{(k+1)} = \vec{r}^{(k+1)} - \beta_k \cdot \vec{p}^{(k)} + \end{aligned} $ \newline +\\ \hline + +\textbf{Méthode du gradient conjugué préconditionné} \newline + \underline{Méthode :} \newline + $ \begin{aligned} + &\vec{r}^{(0)} = \vec{b} - A \cdot \vec{x}^{(0)} \text{, } \vec{z}^{(0)} = P^{-1} \cdot \vec{r}^{(0)} \text{, } \vec{p}^{(0)} = \vec{z}^{(0)} \\ + &\alpha_k = \frac{(\vec{r}^{(k)},\vec{p}^{(k)})}{(A \cdot \vec{p}^{(k)},\vec{p}^{(k)})} \\ + &\vec{x}^{(k+1)} = \vec{x}^{(k)} + \alpha_k \cdot \vec{p}^{(k)} \\ + &\vec{r}^{(k+1)} = \vec{r}^{(k)} - \alpha_k \cdot A \cdot \vec{p}^{(k)} \\ + &P \cdot z^{(k+1)} = r^{(k+1)} \\ + &\beta_k = \frac{(\vec{z}^{(k+1)},A \cdot \vec{p}^{(k)})}{(\vec{p}^{(k)},A \cdot \vec{p}^{(k)})} \\ + &\vec{p}^{(k+1)} = \vec{z}^{(k+1)} - \beta_k \cdot \vec{p}^{(k)} + \end{aligned} $ \newline + + \underline{Erreur :} \newline + $ \begin{aligned} + &\|\vec{x}^{(k)} - \vec{x}\|_A \leq \frac{2 \cdot c^k}{1 + c^{2 \cdot k}} \cdot \|\vec{x}^{(0)} - \vec{x}\|_A \text{, } &&k \geq 0 \\ + &c = \frac{\sqrt{K(P^{-1} \cdot A)} - 1}{\sqrt{K(P^{-1} \cdot A)} + 1} + \end{aligned} $ \newline +\\ \hline + +\textbf{Critère de convergence} \newline + \underline{Convergence :} \newline + $ \begin{aligned} + &\text{Si } P^{-1} \cdot A \text{ symétrique définie positive} \\ + &\frac{\|\vec{x}^{(k)} - \vec{x}\|}{\|\vec{x}\|} \leq K(P^{-1} \cdot A) \cdot \frac{\|P^{-1} \cdot \vec{r^{(k)}}\|}{\|P^{-1} \cdot \vec{b}\|} + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} + +\end{multicols} +\newpage +\begin{multicols}{2}[\centering{\large\textbf{Systèmes linéaires - Méthodes directes (suite)}}] + +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline + +\textbf{Inverse d'un matrice} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &A^{-1} = (\vec{x}^{(1)},...,\vec{x}^{(n)}) \\ + &\vec{e}^{(k)} \text{ vecteur avec composantes nulle sauf la k\textsuperscript{ième} composante } \\ &\text{qui vaut } 1 + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &A \cdot \vec{x}^{(k)} = \vec{e}^{(k)} + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} +\begin{tabularx}{0.5\textwidth}{ X| } +\hline + +\textbf{Systèmes triangulaires} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &\text{Une matrice } U = (u_{ij}) \text{ est triangulaire supérieure si} \\ + &u_{ij} = 0 &&\hspace{-15em} \forall i,j : 1 \leq j < i \leq n \\ + &\text{Une matrice } L = (l_{ij}) \text{ est triangulaire inférieure si} \\ + &l_{ij} = 0 &&\hspace{-15em} \forall i,j : 1 \leq i < j \leq n + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} + +\end{multicols} + +\begin{multicols}{2}[\centering{\large\textbf{Systèmes linéaires - Méthodes itératives (suite)}}] + +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline + +\textbf{Splitting de A} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &A = D - E - F \\ + &\left \{ \begin{array}{l l} d_{ij} = a_{ij} &\text{si } i = j \\ d_{ij} = 0 & \text{si } i \neq j \\ \end{array} \right . && \text{Diagonale} \\ + &\left \{ \begin{array}{l l} e_{ij} = -a_{ij} & \text{si } i > j \\ e_{ij} = 0 & \text{si } i \leq j \\ \end{array} \right . && \text{Triangulaire inf.}\\ + &\left \{ \begin{array}{l l} f_{ij} = -a_{ij} & \text{si } i < j \\ f_{ij} = 0 & \text{si } i \geq j \\ \end{array} \right . && \text{Triangulaire sup.}\\ + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} +\begin{tabularx}{0.5\textwidth}{ X| } +\hline + +\textbf{Produit scalaire et normes} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &\|\vec{v}\| = \sqrt{\vec{v}^T \cdot \vec{v}} \\ + &\|\vec{v}\|_A = \sqrt{\vec{v}^T \cdot A \cdot \vec{v}} \\ + &(\vec{v},\vec{w}) = \vec{w}^T \cdot \vec{v} \\ + \end{aligned} $ \newline +\\ \hline +\end{tabularx} + +\end{multicols} + +\begin{multicols}{2}[\centering{\large\textbf{Équations différentielles ordinaires (suite)}}] + +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline + +\textbf{Stabilité absolue} \newline + \underline{Problème modèle :} \newline + $ \begin{aligned} + &\left \{ \begin{array}{l} + y'(t) = \lambda \cdot y(t)), \hspace{2em} \lambda < 0 \\ + y(t_0) = y_0 \\ + \end{array} \right . \\ + &0 = t_0 < t_1 < ... < t_n < t_{n+1} ... \text{ tels que } t_n = n \cdot h &&(h > 0) \\ + &\text{dont la solution est } y(t) = e^{\lambda \cdot t} \text{ et } \lim_{t \rightarrow \infty} y(t) = 0 + \end{aligned} $ \newline + + \underline{Définition :} \newline + $ \begin{aligned} + &\text{Un schéma de résolution est absolument stable si} \\ + &\lim_{n \rightarrow \infty} u_n = 0 + \end{aligned} $ \newline + + \underline{Stablilité :} \newline + $ \begin{aligned} + &\text{EP : si } 0 < h < 2/\lambda \\ + &\text{ER : inconditionnellement stable }\\ + &\text{PM : inconditionnellement instable }\\ + &\text{CN : inconditionnellement stable }\\ + \end{aligned} $ \newline +\\ \hline +\end{tabularx} +\begin{tabularx}{0.5\textwidth}{ X| } +\hline + +\textbf{Stabilité} \newline + \underline{Définition :} \newline + $ \begin{aligned} + &\text{Une méthode numérique absolument stable pour le problème} \\ &\text{modèle est stable pour un problème de Cauchy quelconque.} \\ + &\text{Il existe } 0 < \lambda_{min} < \lambda_{max} < \infty \text{ tel que} \\ + &-\lambda_{max} < \frac{\partial f(t,y)}{\partial y} < - \lambda_{min} &&\hspace{-10em} \forall t \geq 0, \forall y \in D_y \\ + \end{aligned} $ \newline + + \underline{Stablilité :} \newline + $ \begin{aligned} + &\text{EP : si } 0 < h < 2/\lambda_{max} \\ + &\text{ER : inconditionnellement stable }\\ + &\text{PM : inconditionnellement instable }\\ + &\text{CN : inconditionnellement stable }\\ + \end{aligned} $ \newline +\\ \hline +\end{tabularx} + +\end{multicols} + +\begin{multicols}{2}[\centering{\large\textbf{Systèmes non linéaires}}] + +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline + +\textbf{Méthode de Newton} \newline + \underline{Définition :} \newline + $ \begin{aligned} + &\vec{f}(\vec{x}) = 0 + \end{aligned} $ \newline + + \underline{Méthode :} \newline + $ \begin{aligned} + &[J_f(\vec{x}^{(k)})] \cdot (\vec{x}^{(k+1)} - \vec{x}^{(k)}) = -\vec{f}(\vec{x}^{(k)}) \text{, } &&k = 0,1,2,... \\ + &\Leftrightarrow \vec{x}^{(k+1)} = \vec{x}^{(k)} -[J_f(\vec{x}^{(k)})]^{-1} \cdot \vec{f}(\vec{x}^{(k)}) \text{, } &&k = 0,1,2,... + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} + +\end{multicols} +\newpage +\begin{multicols}{2}[\centering{\large\textbf{{Équations différentielles ordinaires}}}] + +\begin{tabularx}{0.5\textwidth}{ |X } +\hline + +\textbf{Énoncé du problème} \newline + \underline{Problème de Cauchy:} \newline + $ \begin{aligned} + &\left \{ \begin{array}{l} + y'(t) = f(t, y(t)) \\ + y(t_0) = y_0 \\ + \end{array} \right . + \end{aligned} $ \newline + + \underline{Solution :} \newline + $ \begin{aligned} + &\text{Si la fonction } f(t,y) \text{ est continue par rapport à ses deux} \\ & \text{variables et lipschitzienne par rapport à sa deuxième variable,} \\ &\text{alors la solution existe, est unique et appartient à } C^1(I) + \end{aligned} $ \newline +\\ \hline + +\textbf{Dérivée numérique} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &\text{Soient } t_0, t_1, ..., t_{N_h} N_h + 1 \text{ noeuds équirépartis dans } [t_0;t_{N_h}] \\ + &h = \frac{t_{N_h} - t_0}{N_h} \\ + &(Dy)_n \approx y'(t_n) + \end{aligned} $ \newline + + \underline{Méthodes :} \newline + $ \begin{aligned} + &(Dy)_n^P = \frac{y(t_{n+1}) - y(t_n)}{h} \text{, } &&n = 0,...,N_h-1 \\ + &(Dy)_n^R = \frac{y(t_n) - y(t_{n-1})}{h} \text{, } &&n = 1,...,N_h \\ + &(Dy)_n^C = \frac{y(t_{n+1}) - y(t_{n-1})}{2 \cdot h} \text{, } &&n = 1,...,N_h-1 + \end{aligned} $ \newline + + \underline{Erreurs :} \newline + $ \begin{aligned} + &|y'(t_n) - (Dy)_n^P| \leq \frac{h}{2} \cdot \max_{t \in [t_n;t_{n+1}]} |y''(t)| \\ + &|y'(t_n) - (Dy)_n^R| \leq \frac{h}{2} \cdot \max_{t \in [t_{n-1};t_n]} |y''(t)| \\ + &|y'(t_n) - (Dy)_n^C| \leq \frac{h^2}{6} \cdot \max_{t \in [t_{n-1};t_{n+1}]} |y'''(t)| + \end{aligned} $ \newline +\\ \hline + +\textbf{Résolution} \newline + \underline{Définition :} \newline + $ \begin{aligned} + &u_n \text{ une approximation de } y(t_n) + \end{aligned} $ \newline + + \underline{Méthodes :} \newline + $ \begin{aligned} + &u_{n+1} = u_n + h \cdot f(t_n, u_n) &\hspace{-0.5em} \text{EP} \\ + &u_{n+1} = u_n + h \cdot f(t_{n+1}, u_{n+1}) &\hspace{-0.5em} \text{ER} \\ + &u_{n+1} = u_{n-1} + 2 \cdot h \cdot f(t_n, u_n) &\hspace{-0.5em} \text{PM} \\ + &u_{n+1} = u_n + \frac{h}{2} \cdot [f(t_n, u_n) + f(t_{n+1}, u_{n+1})] &\hspace{-0.5em} \text{CN} \\ + &u_{n+1} = u_n + \frac{h}{2} \cdot [f(t_n, u_n) + f(t_{n+1}, u_n + h \cdot f(t_n, u_n))] &\hspace{-0.5em} \text{H} \\ + &u_{n+1} = u_n + h \cdot f\left(t_{n+\frac{1}{2}}, u_n + \frac{h}{2} \cdot f(t_n, u_n)\right) &\hspace{-0.5em} \text{EM} \\ + &\left \{ \begin{array}{l} + u_{n+1} = u_n + \frac{h}{6} \cdot (K_1 + 2 \cdot K_2 + 2 \cdot K_3 + K_4) \\ + K_1 = f(t_n, u_n) \\ + K_2 = f(t_n + \frac{h}{2}, u_n + \frac{h}{2} \cdot K_1) \\ + K_3 = f(t_n + \frac{h}{2}, u_n + \frac{h}{2} \cdot K_2) \\ + K_4 = f(t_{n+1}, u_n + h \cdot K_3) \\ + \end{array} \right . &\hspace{-0.5em} \text{RK} \\ + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} +\begin{tabularx}{0.5\textwidth}{ |X| } +\hline + +\textbf{Convergence} \newline + \underline{Définitions :} \newline + $ \begin{aligned} + &\text{Une méthode est dite convergente si} \\ + &\forall n = 0,..,N_h : |u_n - y(t_n)| \leq C(h) \\ + &\text{Où } C(h) \rightarrow 0 \text{ lorsque } h \rightarrow 0 \\ + &\text{Si en plus, il existe } p > 0 \text{ tel que } C(h) = \mathcal{O}(h^p) \text{ on dit que la} \\ &\text{méthode est convergente d'ordre } p + \end{aligned} $ \newline + + \underline{Convergence de EP :} \newline + $ \begin{aligned} + &\text{Si } y \in C^2([0;T]) \text{, } f \text{ est lipschitzienne et en plus} \\ + &-\lambda_{max} < \frac{\partial f(t,y)}{\partial y} < 0 &&\hspace{-6em} \forall t \in [0;T], \forall y \in [-\infty;\infty] \\ + &\text{Alors } |y(t_n) - u_n| \leq t_n \cdot \frac{h}{2} \cdot \max_t |y''(t)| \\ + \end{aligned} $ \newline + + \underline{Convergence de EP (2) :} \newline + $ \begin{aligned} + &\text{Si } y \in C^2([0;T]) \text{ et } f \text{ est lipschitzienne avec sa constante } L \\ + &\text{Alors } |y(t_n) - u_n| \leq h \cdot \frac{e^{L \cdot t_n} - 1}{2 \cdot L} \cdot \max_t |y''(t)| \\ + \end{aligned} $ \newline + + \underline{Convergence de ER :} \newline + $ \begin{aligned} + &\text{Le même type de résultat peut être établi pour ER} + \end{aligned} $ \newline +\\ \hline + +\textbf{Nb. pas/Ordre/Stabilité/Explicite-Implicite} \newline + $ \begin{aligned} + &\text{Euler progressif} &&\text{1} &&\text{1} &\text{Cond.} &&\text{Expl.} \\ + &\text{Euler rétrograde} &&\text{1} &&\text{1} &\text{Incond.} &&\text{Impl.} \\ + &\text{Point millieu} &&\text{2} &&\text{1} &\text{Instable} &&\text{Expl.} \\ + &\text{Crank-Nicolson} &&\text{1} &&\text{2} &\text{Incond.} &&\text{Impl.} \\ + &\text{Heun} &&\text{1} &&\text{2} &\text{Cond.} &&\text{Expl.} \\ + &\text{Euler modifiée} &&\text{1} &&\text{2} &\text{Cond.} &&\text{Expl.} \\ + &\text{Runge-Kutta} &&\text{1} &&\text{4} &\text{Cond.} &&\text{Expl.} + \end{aligned} $ \newline +\\ \hline + +\textbf{Systèmes d'équations} \newline + \underline{Problème :} \newline + $ \begin{aligned} + &\left \{ \begin{array}{l} + \vec{y}'(t) = A \cdot \vec{y}(t)) + \vec{b}(t) \\ + \vec{y}(t_0) = \vec{y}_0 \\ + \end{array} \right . + \end{aligned} $ \newline + + \underline{Méthodes :} \newline + $ \begin{aligned} + &\vec{u}_{n+1} = (I + h \cdot A) \cdot \vec{u}_n + h \cdot \vec{b}_n &\text{EP} \\ + &(I - h \cdot A) \cdot \vec{u}_{n+1} = \vec{u}_n + h \cdot \vec{b}_n &\text{ER} \\ + &(I - \frac{h}{2} \cdot A) \cdot \vec{u}_{n+1} = (I + \frac{h}{2} \cdot A) \cdot \vec{u}_n + \frac{h}{2} \cdot (\vec{b}_n + \vec{b}_{n+1}) &\text{CN} + \end{aligned} $ \newline + + \underline{Stabilité :} \newline + $ \begin{aligned} + &\lambda_i(A) < 0 &&\forall i\\ + &h < 2/\rho(A) &&\text{pour EP, ER et CN incond. stables} + \end{aligned} $ \newline +\\ \hline + +\textbf{Système d'équations non linéaires} \newline + \underline{Problème :} \newline + $ \begin{aligned} + &\left \{ \begin{array}{l} + \vec{y}'(t) = \vec{F}(t,\vec{y}(t)) \\ + \vec{y}(t_0) = \vec{y}_0 \\ + \end{array} \right . \\ + & J = \frac{\partial \vec{F}}{\partial \vec{y}} + \end{aligned} $ \newline + + \underline{Méthodes :} \newline + $ \begin{aligned} + &\vec{u}_{n+1} = \vec{u}_n + h \cdot \vec{F}(t_n, \vec{u}_n) &\text{EP} \\ + &\vec{u}_{n+1} = \vec{u}_n + h \cdot \vec{F}(t_{n+1}, \vec{u}_{n+1}) &\text{ER} \\ + &\vec{u}_{n+1} = \vec{u}_n + \frac{h}{2} \cdot [\vec{F}(t_n, \vec{u}_n) + \vec{F}(t_{n+1}, \vec{u}_{n+1})] &\text{CN} + \end{aligned} $ \newline + + \underline{Stabilité :} \newline + $ \begin{aligned} + &\lambda_i(J) < 0 &&\forall i\\ + &h < 2/\rho(J) &&\text{pour EP, ER et CN incond. stables} + \end{aligned} $ \newline +\\ \hline + +\end{tabularx} + +\end{multicols} + +\end{document} \ No newline at end of file diff --git a/BA3 - Physique III/BA3 - Physique III.tex b/BA3 - Physique III/BA3 - Physique III.tex new file mode 100644 index 0000000..b803767 --- /dev/null +++ b/BA3 - Physique III/BA3 - Physique III.tex @@ -0,0 +1,327 @@ +\documentclass[a4paper,10pt]{article} +%\documentclass[a4paper,10pt]{scrartcl} + +\input{../Common.tex} + +\begin{document} + +\begin{tabularx}{\textwidth}{ |X|X| } +\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 +& +\textbf{Formes intégrales} \newline + $ \oiint_\Sigma \vec{E} \bullet \dif\vec{\sigma} = \frac{Q_{int}}{\varepsilon_0} \hspace{25mm} = \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{15mm} 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 + +\end{tabularx} + +\offinterlineskip + +\begin{tabularx}{\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 + +\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 + +\end{tabularx} + +\offinterlineskip + +\begin{tabularx}{\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 +& +\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 + +\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 +& +\textbf{Aimantation} \newline + $ j_{lie} = \vec{M} \bullet \vec{e}_n $ \newline + $ <\vec{B}> = \mu_r \cdot B_{ext} $ \newline +\\ \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 +& +\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 + +\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 +& +\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 + +\textbf{Électrostatique} \newline + $ \vec{F} = q \cdot (\vec{E} + \vec{v} \times \vec{B}) $ \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 +& +\textbf{Magnétostatique} \newline + $ r = \frac{m \cdot v}{q \cdot B_0} $ \hfill Rayon de Larmor \newline + $ \vec{\omega} = - \frac{q}{m} \cdot \vec{B_0} $ \hfill Fréquence de cyclotron \newline + $ \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 + $ 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 + +\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 +& +\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 + +\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 + $ 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 + $ 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 +& +\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 = 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 + +\textbf{Onde stationnaire} \newline + $ \xi = 2 \cdot \xi_0 \cdot \sin(k \cdot x) \cdot \cos(\omega \cdot t) $ \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 + $ 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 +& +\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 + $ \delta = 2 \cdot m \cdot \pi $ \hfill Max \newline + $ \delta = (2 \cdot m + 1) \cdot \pi $ \hfill Min \newline +\\ \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 + $ \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 +& +\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 + $ 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 + +\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 +& +\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 + +\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{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 + + $ \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 +& +\textbf{} \newline +\\ \hline + +\end{tabularx} + +\end{document} \ No newline at end of file diff --git a/BA3 - Physique III/Polarisation.png b/BA3 - Physique III/Polarisation.png new file mode 100644 index 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