From d125f383aced7e2d53978cb34b44d6bfaeffc6d5 Mon Sep 17 00:00:00 2001 From: Guyot Date: Tue, 26 Feb 2019 23:58:18 +0100 Subject: [PATCH] Avancement de l'annexe Incertitudes. --- Annexe-ChuteLune/Annexe-ChuteLune.aux | 1 + Annexe-Energies/Annexe-Energies.aux | 1 + Annexe-Exercices/Annexe-Exercices.aux | 1 + Annexe-Incertitudes/Annexe-Incertitudes.aux | 21 +- Annexe-Incertitudes/Annexe-Incertitudes.tex | 193 +- .../Annexe-Incertitudes.tex.bak | 272 + Annexe-Incertitudes/Images/baguettesgauss.dat | 9 + Annexe-MRUA/Annexe-MRUA.aux | 1 + Annexe-Maree/Annexe-Maree.aux | 1 + .../Annexe-MesuresDistances.aux | 1 + Annexe-Relativite/Annexe-Relativite.aux | 1 + Annexe-Rotations/Annexe-Rotations.aux | 1 + Annexe-Satellites/Annexe-Satellites.aux | 1 + .../Annexe-SystemeCoordonnees.aux | 1 + .../Annexe-TravauxPratiques.aux | 1 + .../Annexe-UnitesInternationales.aux | 1 + Cinematique/Cinematique.aux | 1 + CoursMecaniqueOSDF-gnuplottex-fig1.gnuplot | 25 + CoursMecaniqueOSDF-gnuplottex-fig1.tex | 104 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ThermodynamiqueOS/ThermodynamiqueOS.aux | 1 + 48 files changed, 3953 insertions(+), 2175 deletions(-) create mode 100644 Annexe-Incertitudes/Annexe-Incertitudes.tex.bak create mode 100644 Annexe-Incertitudes/Images/baguettesgauss.dat create mode 100644 CoursMecaniqueOSDF-gnuplottex-fig1.gnuplot create mode 100644 CoursMecaniqueOSDF-gnuplottex-fig1.tex create mode 100644 CoursMecaniqueOSDF.exp.gnuplot create mode 100644 CoursMecaniqueOSDF.exp.table create mode 100644 CoursMecaniqueOSDF.gnuploterrors create mode 100644 CoursMecaniqueOSDF.pgf-plot.gnuplot create mode 100644 CoursMecaniqueOSDF.pgf-plot.table create mode 100644 CoursMecaniqueOSDF.sin.gnuplot create mode 100644 CoursMecaniqueOSDF.sin.table create mode 100644 CoursMecaniqueOSDF.x.gnuplot create mode 100644 CoursMecaniqueOSDF.x.table create mode 100644 Preambule.tex.bak diff --git a/Annexe-ChuteLune/Annexe-ChuteLune.aux b/Annexe-ChuteLune/Annexe-ChuteLune.aux index 13a6700..2bfb880 100644 --- a/Annexe-ChuteLune/Annexe-ChuteLune.aux +++ b/Annexe-ChuteLune/Annexe-ChuteLune.aux @@ -31,6 +31,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{4} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Annexe-Energies/Annexe-Energies.aux b/Annexe-Energies/Annexe-Energies.aux index b1ce031..b1c11fd 100644 --- a/Annexe-Energies/Annexe-Energies.aux +++ b/Annexe-Energies/Annexe-Energies.aux @@ -37,6 +37,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{4} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Annexe-Exercices/Annexe-Exercices.aux b/Annexe-Exercices/Annexe-Exercices.aux index f5530ed..fca1019 100644 --- a/Annexe-Exercices/Annexe-Exercices.aux +++ b/Annexe-Exercices/Annexe-Exercices.aux @@ -85,6 +85,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{4} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Annexe-Incertitudes/Annexe-Incertitudes.aux b/Annexe-Incertitudes/Annexe-Incertitudes.aux index b538fdd..e05b941 100644 --- a/Annexe-Incertitudes/Annexe-Incertitudes.aux +++ b/Annexe-Incertitudes/Annexe-Incertitudes.aux @@ -8,27 +8,34 @@ \newlabel{baguettepain}{{M.1}{241}} \@writefile{lot}{\contentsline {table}{\numberline {M.2}{\ignorespaces La longueur d'autres baguettes de pain\relax }}{242}} \newlabel{baguettepain2}{{M.2}{242}} -\@writefile{toc}{\contentsline {section}{\numberline {M.3}Incertitude}{242}} +\@writefile{lot}{\contentsline {table}{\numberline {M.3}{\ignorespaces Des centaines de baguettes de pain\relax }}{242}} +\newlabel{enclassementbaguettes}{{M.3}{242}} +\@writefile{lof}{\contentsline {figure}{\numberline {M.1}{\ignorespaces Baguettes d'une ann\IeC {\'e}e\relax }}{243}} +\newlabel{baguettesgauss}{{M.1}{243}} +\@writefile{toc}{\contentsline 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\setcounter{lofdepth}{1} diff --git a/Annexe-Incertitudes/Annexe-Incertitudes.tex b/Annexe-Incertitudes/Annexe-Incertitudes.tex index eb52c18..e1ebe99 100644 --- a/Annexe-Incertitudes/Annexe-Incertitudes.tex +++ b/Annexe-Incertitudes/Annexe-Incertitudes.tex @@ -43,6 +43,8 @@ Par contre, déterminer quel est l'écart à la moyenne des baguettes est plus i Par ailleurs, si on sait que le boulanger avait décidé de faire des baguettes de \unit{60}{\centi\metre}, on peut se poser une autre question : quel est l'écart à cette valeur ? La quatrième colonne y apporte une réponse. Comme on utilise ce qu'on peut appeler une référence, on parlera d'\emph{erreur}, plutôt que d'écart. Ce qui est alors intéressant, c'est qu'on voit que le boulanger à tendance à faire des baguettes trop grandes. Cela peut avoir une importance pour lui s'il a prévu un budget précis de matières premières pour des baguettes de \unit{60}{\centi\metre}. Cela permet aussi de s'interroger sur la règle utilisée par le boulanger pour estimer la longueur des baguettes. Comme ses baguettes sont trop longues, on peut penser que sa règle est aussi trop longue, ce qui peut avoir pour conséquence une mauvaise estimation de la longueur de la baguette par le boulanger. On parlera alors d'une \emph{erreur systèmatique}\index{erreur systématique} induite par un matériel mal calibré. Ce type d'erreur se détecte par la présence d'une importante quantité de signes systématiquement positifs ou systématiquement négatifs dans les écarts. En effet, si la règle avait une longueur correcte, l'erreur faite par le boulanger devrait être aléatoirement répartie autour de la valeur de \unit{60}{\centi\metre} et les signes positifs et négatifs des écarts devraient être en nombres à peu près identiques. +Évidemment, la moyenne des écarts à la moyenne est nulle, cela par définition, alors que la moyenne des erreurs ne l'est pas en présence d'une erreur systématique. + Figurent aussi dans le tableau \ref{baguettepain} les écarts et erreurs relatifs en pourcents. Il s'agit du rapport entre l'écart et la valeur de référence : la moyenne pour l'écart et \unit{60}{\centi\metre} pour l'erreur. Autant pour l'écart que pour l'erreur, on a donc : \begin{equation} e=\frac{val-val_{ref}}{val_{ref}}\cdot 100 @@ -78,6 +80,193 @@ e=\frac{val-val_{ref}}{val_{ref}}\cdot 100 \end{tabular} \end{table} -Ces indications sont importantes si on désire comparer la production de deux boulangers dont la longueur de la baguette de référence n'est pas la même. Considérons le tableau \ref{baguettepain2} qui décrit la production d'un boulanger dont la baguette de référence est de \unit{40}{\centi\metre}. On voit que la moyenne des écarts est nulle comme pour le boulanger précédent. Ce qui est normal en raison du choix de la valeur moyenne comme référence. On voit aussi que la moyenne des erreurs est la même et qu'il y a une grande systématique dans celle-ci, puisqu'elles sont pratiquement toutes positives. Cela signifie probablement, comme précédemment, que l'appareil de mesure à un bied, que la règle utilisée est trop longue. Par contre, on voit grâce à la dernière colonne donnant l'erreur relative que celle-ci est plus importante pour le second boulanger. Cela s'explique facilement. En effet, l'erreur moyenne est la même, mais la baguette de référence du second boulanger est plus courte (\unit{40}{\centi\metre}). Ainsi, malgré la différence de longueur de la baguette de référence, l'erreur relative permet de comparer les erreurs des deux boulangers. +Ces indications sont importantes si on désire comparer la production de deux boulangers dont la longueur de la baguette de référence n'est pas la même. Considérons le tableau \ref{baguettepain2} qui décrit la production d'un boulanger dont la baguette de référence est de \unit{40}{\centi\metre}. On voit que la moyenne des écarts est nulle comme pour le boulanger précédent. Ce qui est normal en raison du choix de la valeur moyenne comme référence. On voit aussi que la moyenne des erreurs est la même et qu'il y a une grande systématique dans celle-ci, puisqu'elles sont pratiquement toutes positives. Cela signifie probablement, comme précédemment, que l'appareil de mesure à un biais, que la règle utilisée est trop longue. Par contre, on voit grâce à la dernière colonne donnant l'erreur relative que celle-ci est plus importante pour le second boulanger. Cela s'explique facilement. En effet, l'erreur moyenne est la même, mais la baguette de référence du second boulanger est plus courte (\unit{40}{\centi\metre}). Ainsi, malgré la différence de longueur de la baguette de référence, l'erreur relative permet de comparer les erreurs des deux boulangers. -\section{Incertitude} \ No newline at end of file +\bigskip +Imaginons maintenant qu'on s'intéresse à la production annuelle de baguettes d'un boulanger, soit des centaines de baguettes. Il devient difficile de les représenter dans un tableau, surtout si on fait des mesures d'une précision supérieures au centimètre. On peut alors réaliser des classes de mesures en mettant par exemple, toutes les baguettes entre \unit{42,5}{\centi\metre} et \unit{43,4}{\centi\metre} dans la classe des baguettes de \unit{43}{\centi\metre}. En procédant de la même manière pour les autres valeurs, on peut alors obtenir des mesures comme celles présentées dans le tableau \ref{enclassementbaguettes} où L est la longueur des baguettes et n le nombre de baguettes dans la classe associée à cette longueur, soit la fréquence d'apparition de la longueur. + +\begin{table}[ht] +\centering +\caption{Des centaines de baguettes de pain}\label{enclassementbaguettes} +\begin{tabular}{|c|c|} +\hline +\textbf{L} & n\\ +\centi\metre & -\\ +\hline +38 & 53\\ +39 & 72\\ +40 & 95\\ +41 & 121\\ +42 & 130\\ +43 & 118\\ +44 & 90\\ +45 & 67\\ +46 & 44\\ +\hline +\multicolumn{2}{|c|}{Moyennes}\\ +\hline +41,9 & -\\ +\hline +\end{tabular} +\end{table} +La moyenne \(\mu\) de la longueur des baguettes est alors calculée ainsi~: +\[\mu=\frac{\sum\limits_i L_i\cdot n_i}{\sum\limits_i N_i}=\frac{33'116}{790}=\unit{41,9}{\centi\metre}\] +La moyenne des fréquences n n'a que peu de sens. + +\smallskip +On peut aussi représenter ces mesures graphiquement, comme le montre la figure \ref{baguettesgauss}. + +\begin{figure} +\centering +\caption{Baguettes d'une année}\label{baguettesgauss} +\begin{gnuplot}[terminal=latex,terminaloptions=rotate,scale=0.6] + # domaine de définition + set xrange [36:48] + set yrange [0:140] + # flèches + set arrow 1 from 39.7,120 to 39.7,0 head filled + #size screen 0.5,30 + set label "Écart type" at 39.4,100 rotate by 90 + set arrow 2 from 44.1,120 to 44.1,0 head filled + set label "Écart type" at 44.4,138 rotate by -90 + set arrow 3 from 41.9,120 to 41.9,0 head filled + set label "Moyenne" at 42.2,80 rotate by -90 + #set grid + #set title "Baguettes d'une année" + # suppression de la légende + set key off + # légendes des axes + set xlabel "Longueur des baguettes (en cm)" + set ylabel "Nombre de baguettes" rotate by 90 + # largeur des colonnes (boxes) + #set boxwidth 0.1 + + # tracé du graphe + plot "Annexe-Incertitudes/Images/baguettesgauss.dat" with boxes +\end{gnuplot} +\end{figure} +Sur un grand nombre de mesures, la notion d'écart devient difficile à gérer. On utilise alors celle d'écart type\footnote{Le -1 apparaissant au dénominateur de la fraction de l'écart type vient d'une estimation de l'écart type (ou variance) de la population des mesures basée sur un échantillon de celle-ci. Pour bien le comprendre, il faut évoquer le calcul des probabilités et la notion de variable aléatoire, ce qui dépasse le cadre de ce document. Vous trouverez des informations sur internet.} définie par~: +\[\sigma=\sqrt{\frac{\sum\limits_i n_i\cdot (L_i-\mu)^2}{\sum\limits_i n_i-1}}=\sqrt{\frac{3'777.1}{789}}=\unit{2,2}{\centi\metre}\] +Il s'agit de la moyenne des écarts quadratiques, soit de la racine de la moyenne des écarts au carrés. L'élévation au carré permet de ne pas tenir compte du signe des écarts. + +On peut alors affirmer qu'une forte proportion de la longueur L des baguettes se trouve entre \(\mu-\sigma\) et \(\mu+\sigma\), soit entre \unit{39,7}{} et \unit{44,1}{\centi\metre}. + +\bigskip +Pour aller plus loin, il faut voir l'ensemble complet des mesures qui pourraient être faites dans le cadre d'une loi physique donnée comme une population qu'on ne pourrait étudier qu'à partir d'échantillons partiels extraits de celle-ci. Imaginez la population de tous les éléphants existants sur la Terre. Pour l'étudier, il faut en prélever un ou plusieurs échantillons dont les propriétés peuvent être différentes de celles de la population. Par exemple, la masse moyenne de la population des éléphants peut être différente de celle d'un échantillon d'éléphants pris par hasard obèses. La science des probabilités et statistiques dispose d'outils pour caractériser les échantillons, mais aussi pour en déduire les caractéristiques des populations. Mais cette science, qui repose sur les probabilités, est malheureusement hors du cadre de cet ouvrage. + +\section{Incertitude} +Avec les écarts, on étudie la répartition de mesures réalisées. Du point de vue des probabilités et statistiques, on parlera des propriété d'un échantillon de mesures relatives à une loi donnée. + +\medskip +Avec les incertitudes, on va s'intéresser à l'évaluation des capacités des instruments de mesures. Ces deux domaines n'ont rien à voir l'un avec l'autre, si ce n'est qu'ils sont les deux nécessaires à la réalisation des expériences de physique. Le problème est que l'évaluation de la qualité de mesure des instruments nécessite des méthodes de probabilités et statistiques. Cette évaluation peut donc devenir complexe si on entre réellement dans les détails. + +Nous en resterons ici à un niveau aussi simple que possible en admettant qu'il soit possible d'évaluer la précision d'un instrument de mesure sans avoir recours aux statistique pour le calibrer ou que cette calibration ait été réalisée par ailleurs et soit disponible. + +\medskip +Imaginons donc la mesure de la longueur d'une feuille A4 à l'aide d'une règle. La présentation de cette mesure est la suivante~: +\[L=L_m\pm I(L_m)\] +où, \(L\) est la grandeur, \(L_m\) sa valeur et \(I(L_m)\) son incertitude absolue. Par exemple, une mesure pourrait donner~: +\[L=\unit{29,0}\pm \unit{0,2}{\centi\metre}\] +L'origine de l'incertitude absolue peut être de diverses nature~: +\begin{enumerate} +\item une estimation de la précision suite à la lecture visuelle sur la règle selon sa graduation, la distance à laquelle ou l'angle sous lequel on la regarde, \dots, +\item une information du fabriquant qui a réalisé des tests approfondis, +\item une étude statistique à travers un ou plusieurs échantillons sur une mesure identique, etc. +\end{enumerate} +Conformément à ce que nous avons dit plus haut, on ne considérera pas le dernier point. + +\medskip +Admettons maintenant qu'on réalise aussi une mesure de la largeur de la feuille, mais avec une règle différente, et obtienne~: +\[l=\unit{21,1}\pm \unit{0,3}{\centi\metre}\] + +\subsection{Addition/soustraction} +Que pouvons-nous dire alors de la circonférence de la feuille ? + +\smallskip +Évidemment, elle est facile à calculer~: +\[C=2\cdot L_m+2\cdot l_m=2\cdot (29,0+21,1)=\unit{100,2}{\centi\metre}\] + +\medskip +Mais qu'en est-il de son incertitude ? Pour la connaître, on va calculer l'incertitude absolue de la somme de deux grandeurs en déterminant les maximum et minimum de celle-ci. Formellement, si~: +\begin{equation*} +s=l_1+l_2\;;\;l_1=l_{1m}\pm I(l_{1m})\;;\;l_2=l_{2m}\pm I(l_{2m}) +\end{equation*} +alors, on peut calculer les extrèmums~: +\begin{align*} +s_{max}&=l_{1\,max}+l_{2\,max}\\ +&=l_{1m}+I(l_{1m})+l_{2m}+I(l_{2m})\\ +&=l_{1m}+l_{2m}+I(l_{1m})+I(l_{2m})\\ +&=s_m+I(s_m)\\ +\text{et}\\ +s_{min}&=l_{1\,min}+l_{2\,min}\\ +&=l_{1m}-I(l_{1m})+l_{2m}-I(l_{2m})\\ +&=l_{1m}+l_{2m}-I(l_{1m})-I(l_{2m})\\ +&=s_m-I(s_m) +\end{align*} +et en déduire que~: + +\smallskip +\fbox{ +\(\text{Si : }s=l_1\pm l_2\;\Rightarrow\;I(s_m)=I(l_{1m})+I(l_{2m})\) +} + +\medskip +On peut alors calculer l'incertitude sur la circonférence~: +\begin{align*} +I(C_m)&=2\cdot I(L_m)+2\cdot I(l_m)\\ +&=2\cdot 0,2+2\cdot 0,3=\unit{1,0}{\centi\metre} +\end{align*} +et donner le résultat final sous la forme~: +\[C=\unit{100,2}\pm \unit{1,0}{\centi\metre}\] + +\subsection{Multiplication/division} +Que pouvons-nous dire de la surface de la feuille ? + +\smallskip +Évidemment, elle est facile à calculer~: +\[S=l_1\cdot l_2=29,0\cdot 21,1=\unit{611,9}{\centi\metre\squared}\] + +\medskip +Mais qu'en est-il de son incertitude ? Pour la connaître, on va calculer l'incertitude absolue de la multiplication de deux grandeurs en déterminant les maximum et minimum de celle-ci. Formellement, si~: +\begin{equation*} +S=l_1\cdot l_2\;;\;l_1=l_{1m}\pm I(l_{1m})\;;\;l_2=l_{2m}\pm I(l_{2m}) +\end{equation*} +alors, on peut calculer le maximum~: +\begin{align*} +S_{max}&=l_{1\,max}\cdot l_{2\,max}\\ +&=(l_{1m}+I(l_{1m}))\cdot (l_{2m}+I(l_{2m}))\\ +&=l_{1m}\cdot l_{2m}+l_{1m}\cdot I(l_{2m})\\ +&\;+l_{2m}\cdot I(l_{1m})+I(l_{1m})\cdot I(l_{2m})\\ +&=S_m+I(S_m) +\end{align*} +L'expression de l'incertitude s'avère alors pour le moins complexe~: +\[I(S_m)=l_{1m}\cdot I(l_{2m})+l_{2m}\cdot I(l_{1m})+I(l_{1m})\cdot I(l_{2m})\] +Mais, comme \(S=l_1\cdot l_2\), on peut diviser cette équation par \(S\)~: +\begin{align*} +\frac{I(S_m)}{S_m}&=\frac{l_{1m}\cdot I(l_{2m})}{l_{1m}\cdot l_{2m}}+\frac{l_{2m}\cdot I(l_{1m})}{l_{1m}\cdot l_{2m}}+\frac{I(l_{1m})\cdot I(l_{2m})}{l_{1m}\cdot l_{2m}}\\ +&=\frac{I(l_{1m})}{l_{1m}}+\frac{I(l_{2m})}{l_{2m}}+\frac{I(l_{1m})\cdot I(l_{2m})}{l_{1m}\cdot l_{2m}} +\end{align*} +Pour autant que l'incertitude soit nettement plus petite que la grandeur qui lui correspond, le terme \(I(l_{1m})\cdot I(l_{2m})\) est très petit devant \(l_{1m}\cdot l_{2m}\) et donc il est possible de négliger le dernier terme. Dans ces conditions~: +\[\frac{I(l_{1m})\cdot I(l_{2m})}{l_{1m}\cdot l_{2m}}<<1\] +et on peut écrire~: +\begin{align*} +\frac{I(S_m)}{S_m}&\cong\frac{I(l_{1m})}{l_{1m}}+\frac{I(l_{2m})}{l_{2m}}\;\Rightarrow\\ +i(S_m)&=i(l_{1m})+i(l_{2m}) +\end{align*} +en définissant l'incertitude relative par~: + +\smallskip +\fbox{ +\(i(G_m)=\frac{I(G_m)}{G_m}\) +} + +\medskip +L'incertitude relative, comme expression de l'incertitude absolue rapportée à la valeur de sa grandeur, représente un incertitude sans unités qu'on peut exprimer en pourcents. En utilisant l'incertitude relative, l'expression de l'incertitude d'une multiplication de deux grandeurs devient alors bien plus simple~: + +\smallskip +\fbox{ +\(\text{Si : }S=l_1\cdot l_2\;\Rightarrow\;i(S_m)=i(l_{1m})+i(l_{2m})\) +} + +\medskip +On peut aussi faire ... \ No newline at end of file diff --git a/Annexe-Incertitudes/Annexe-Incertitudes.tex.bak b/Annexe-Incertitudes/Annexe-Incertitudes.tex.bak new file mode 100644 index 0000000..56989ab --- /dev/null +++ b/Annexe-Incertitudes/Annexe-Incertitudes.tex.bak @@ -0,0 +1,272 @@ +\myclearpage + +\chapter{Ordre de grandeur, erreur et incertitudes} + +\section{Ordre de grandeur} + +\section{Écart et erreur} +On peut facilement déterminer l'écart entre deux valeurs $a$ et $b$ par leur différence $a-b$. On peut, par exemple, mesurer la longueur $L$ des baguettes de pain vendues par un boulanger et déterminer les différents écarts entre elles. Par exemple, on pourrait avoir une série de mesures telles que celle données dans le tableau \ref{baguettepain}. + +\begin{table}[ht] +\centering +\caption{La longueur des baguettes de pain}\label{baguettepain} +\begin{tabular}{|c|c|c|c|c|} +\hline +\textbf{L} & \textbf{Écart} & \textbf{Écart} & \textbf{Erreur} & \textbf{Erreur}\\ +\centi\metre &\centi\metre &\% &\centi\metre &\%\\ +\hline +60 &-2 &-3.2 &0 &0.0\\ +65 &3 &4.8 &5 &8.3\\ +64 &2 &3.2 &4 &6.7\\ +58 &-4 &-6.5 &-2 &-3.3\\ +61 &-1 &-1.6 &1 &1.7\\ +57 &-5 &-8.1 &-3 &-5.0\\ +60 &-2 &-3.2 &0 &0.0\\ +64 &2 &3.2 &4 &6.7\\ +62 &0 &0.0 &2 &3.3\\ +65 &3 &4.8 &5 &8.3\\ +63 &1 &1.6 &3 &5.0\\ +60 &-2 &-3.2 &0 &0.0\\ +64 &2 &3.2 &4 &6.7\\ +65 &3 &4.8 &5 &8.3\\ +\hline +\multicolumn{5}{|c|}{Moyennes}\\ +\hline +62 &0.0 &0.0 &2.0 &3.3\\ +\hline +\end{tabular} +\end{table} + +On voit immédiatement que le calcul des écarts pose un problème : il faut déterminer les écarts entre chaque baguettes deux par deux. On peut le faire. Mais quel sens cela a-t-il ? + +Par contre, déterminer quel est l'écart à la moyenne des baguettes est plus instructif. La moyenne vaut \unit{62}{\centi\metre} et la seconde colonne du tableau \ref{baguettepain} présente les écarts. On voit alors facilement que l'écart ne dépasse pas \unit{5}{\centi\metre}. Ce qui peut avoir de l'importance si on a faim. + +Par ailleurs, si on sait que le boulanger avait décidé de faire des baguettes de \unit{60}{\centi\metre}, on peut se poser une autre question : quel est l'écart à cette valeur ? La quatrième colonne y apporte une réponse. Comme on utilise ce qu'on peut appeler une référence, on parlera d'\emph{erreur}, plutôt que d'écart. Ce qui est alors intéressant, c'est qu'on voit que le boulanger à tendance à faire des baguettes trop grandes. Cela peut avoir une importance pour lui s'il a prévu un budget précis de matières premières pour des baguettes de \unit{60}{\centi\metre}. Cela permet aussi de s'interroger sur la règle utilisée par le boulanger pour estimer la longueur des baguettes. Comme ses baguettes sont trop longues, on peut penser que sa règle est aussi trop longue, ce qui peut avoir pour conséquence une mauvaise estimation de la longueur de la baguette par le boulanger. On parlera alors d'une \emph{erreur systèmatique}\index{erreur systématique} induite par un matériel mal calibré. Ce type d'erreur se détecte par la présence d'une importante quantité de signes systématiquement positifs ou systématiquement négatifs dans les écarts. En effet, si la règle avait une longueur correcte, l'erreur faite par le boulanger devrait être aléatoirement répartie autour de la valeur de \unit{60}{\centi\metre} et les signes positifs et négatifs des écarts devraient être en nombres à peu près identiques. + +Évidemment, la moyenne des écarts à la moyenne est nulle, cela par définition, alors que la moyenne des erreurs ne l'est pas en présence d'une erreur systématique. + +Figurent aussi dans le tableau \ref{baguettepain} les écarts et erreurs relatifs en pourcents. Il s'agit du rapport entre l'écart et la valeur de référence : la moyenne pour l'écart et \unit{60}{\centi\metre} pour l'erreur. Autant pour l'écart que pour l'erreur, on a donc : +\begin{equation} +e=\frac{val-val_{ref}}{val_{ref}}\cdot 100 +\end{equation} + +\begin{table}[ht] +\centering +\caption{La longueur d'autres baguettes de pain}\label{baguettepain2} +\begin{tabular}{|c|c|c|c|c|} +\hline +\textbf{L} & \textbf{Écart} & \textbf{Écart} & \textbf{Erreur} & \textbf{Erreur}\\ +\centi\metre &\centi\metre &\% &\centi\metre &\%\\ +\hline +40 &-2 &-4.8 &0 &0.0\\ +44 &2 &4.8 &4 &10.0\\ +41 &-1 &-2.4 &1 &2.5\\ +44 &2 &4.8 &4 &10.0\\ +43 &1 &2.4 &3 &7.5\\ +42 &0 &0.0 &2 &5.0\\ +44 &2 &4.8 &4 &10.0\\ +42 &0 &0.0 &2 &5.0\\ +42 &0 &0.0 &2 &5.0\\ +43 &1 &2.4 &3 &7.5\\ +43 &1 &2.4 &3 &7.5\\ +41 &-1 &-2.4 &1 &2.5\\ +40 &-2 &-4.8 &0 &0.0\\ +39 &-3 &-7.1 &-1 &-2.5\\ +\hline +\multicolumn{5}{|c|}{Moyennes}\\ +\hline +42 &0.0 &0.0 &2.0 &5.0\\ +\hline +\end{tabular} +\end{table} + +Ces indications sont importantes si on désire comparer la production de deux boulangers dont la longueur de la baguette de référence n'est pas la même. Considérons le tableau \ref{baguettepain2} qui décrit la production d'un boulanger dont la baguette de référence est de \unit{40}{\centi\metre}. On voit que la moyenne des écarts est nulle comme pour le boulanger précédent. Ce qui est normal en raison du choix de la valeur moyenne comme référence. On voit aussi que la moyenne des erreurs est la même et qu'il y a une grande systématique dans celle-ci, puisqu'elles sont pratiquement toutes positives. Cela signifie probablement, comme précédemment, que l'appareil de mesure à un biais, que la règle utilisée est trop longue. Par contre, on voit grâce à la dernière colonne donnant l'erreur relative que celle-ci est plus importante pour le second boulanger. Cela s'explique facilement. En effet, l'erreur moyenne est la même, mais la baguette de référence du second boulanger est plus courte (\unit{40}{\centi\metre}). Ainsi, malgré la différence de longueur de la baguette de référence, l'erreur relative permet de comparer les erreurs des deux boulangers. + +\bigskip +Imaginons maintenant qu'on s'intéresse à la production annuelle de baguettes d'un boulanger, soit des centaines de baguettes. Il devient difficile de les représenter dans un tableau, surtout si on fait des mesures d'une précision supérieures au centimètre. On peut alors réaliser des classes de mesures en mettant par exemple, toutes les baguettes entre \unit{42,5}{\centi\metre} et \unit{43,4}{\centi\metre} dans la classe des baguettes de \unit{43}{\centi\metre}. En procédant de la même manière pour les autres valeurs, on peut alors obtenir des mesures comme celles présentées dans le tableau \ref{enclassementbaguettes} où L est la longueur des baguettes et n le nombre de baguettes dans la classe associée à cette longueur, soit la fréquence d'apparition de la longueur. + +\begin{table}[ht] +\centering +\caption{Des centaines de baguettes de pain}\label{enclassementbaguettes} +\begin{tabular}{|c|c|} +\hline +\textbf{L} & n\\ +\centi\metre & -\\ +\hline +38 & 53\\ +39 & 72\\ +40 & 95\\ +41 & 121\\ +42 & 130\\ +43 & 118\\ +44 & 90\\ +45 & 67\\ +46 & 44\\ +\hline +\multicolumn{2}{|c|}{Moyennes}\\ +\hline +41,9 & -\\ +\hline +\end{tabular} +\end{table} +La moyenne \(\mu\) de la longueur des baguettes est alors calculée ainsi~: +\[\mu=\frac{\sum\limits_i L_i\cdot n_i}{\sum\limits_i N_i}=\frac{33'116}{790}=\unit{41,9}{\centi\metre}\] +La moyenne des fréquences n n'a que peu de sens. + +\smallskip +On peut aussi représenter ces mesures graphiquement, comme le montre la figure \ref{baguettesgauss}. + +\begin{figure} +\centering +\caption{Baguettes d'une année}\label{baguettesgauss} +\begin{gnuplot}[terminal=latex,terminaloptions=rotate,scale=0.6] + # domaine de définition + set xrange [36:48] + set yrange [0:140] + # flèches + set arrow 1 from 39.7,120 to 39.7,0 head filled + #size screen 0.5,30 + set label "Écart type" at 39.4,100 rotate by 90 + set arrow 2 from 44.1,120 to 44.1,0 head filled + set label "Écart type" at 44.4,138 rotate by -90 + set arrow 3 from 41.9,120 to 41.9,0 head filled + set label "Moyenne" at 42.2,80 rotate by -90 + #set grid + #set title "Baguettes d'une année" + # suppression de la légende + set key off + # légendes des axes + set xlabel "Longueur des baguettes (en cm)" + set ylabel "Nombre de baguettes" rotate by 90 + # largeur des colonnes (boxes) + #set boxwidth 0.1 + + # tracé du graphe + plot "Annexe-Incertitudes/Images/baguettesgauss.dat" with boxes +\end{gnuplot} +\end{figure} +Sur un grand nombre de mesures, la notion d'écart devient difficile à gérer. On utilise alors celle d'écart type\footnote{Le -1 apparaissant au dénominateur de la fraction de l'écart type vient d'une estimation de l'écart type (ou variance) de la population des mesures basée sur un échantillon de celle-ci. Pour bien le comprendre, il faut évoquer le calcul des probabilités et la notion de variable aléatoire, ce qui dépasse le cadre de ce document. Vous trouverez des informations sur internet.} définie par~: +\[\sigma=\sqrt{\frac{\sum\limits_i n_i\cdot (L_i-\mu)^2}{\sum\limits_i n_i-1}}=\sqrt{\frac{3'777.1}{789}}=\unit{2,2}{\centi\metre}\] +Il s'agit de la moyenne des écarts quadratiques, soit de la racine de la moyenne des écarts au carrés. L'élévation au carré permet de ne pas tenir compte du signe des écarts. + +On peut alors affirmer qu'une forte proportion de la longueur L des baguettes se trouve entre \(\mu-\sigma\) et \(\mu+\sigma\), soit entre \unit{39,7}{} et \unit{44,1}{\centi\metre}. + +\bigskip +Pour aller plus loin, il faut voir l'ensemble complet des mesures qui pourraient être faites dans le cadre d'une loi physique donnée comme une population qu'on ne pourrait étudier qu'à partir d'échantillons partiels extraits de celle-ci. Imaginez la population de tous les éléphants existants sur la Terre. Pour l'étudier, il faut en prélever un ou plusieurs échantillons dont les propriétés peuvent être différentes de celles de la population. Par exemple, la masse moyenne de la population des éléphants peut être différente de celle d'un échantillon d'éléphants pris par hasard obèses. La science des probabilités et statistiques dispose d'outils pour caractériser les échantillons, mais aussi pour en déduire les caractéristiques des populations. Mais cette science, qui repose sur les probabilités, est malheureusement hors du cadre de cet ouvrage. + +\section{Incertitude} +Avec les écarts, on étudie la répartition de mesures réalisées. Du point de vue des probabilités et statistiques, on parlera des propriété d'un échantillon de mesures relatives à une loi donnée. + +\medskip +Avec les incertitudes, on va s'intéresser à l'évaluation des capacités des instruments de mesures. Ces deux domaines n'ont rien à voir l'un avec l'autre, si ce n'est qu'ils sont les deux nécessaires à la réalisation des expériences de physique. Le problème est que l'évaluation de la qualité de mesure des instruments nécessite des méthodes de probabilités et statistiques. Cette évaluation peut donc devenir complexe si on entre réellement dans les détails. + +Nous en resterons ici à un niveau aussi simple que possible en admettant qu'il soit possible d'évaluer la précision d'un instrument de mesure sans avoir recours aux statistique pour le calibrer ou que cette calibration ait été réalisée par ailleurs et soit disponible. + +\medskip +Imaginons donc la mesure de la longueur d'une feuille A4 à l'aide d'une règle. La présentation de cette mesure est la suivante~: +\[L=L_m\pm I(L_m)\] +où, \(L\) est la grandeur, \(L_m\) sa valeur et \(I(L_m)\) son incertitude absolue. Par exemple, une mesure pourrait donner~: +\[L=\unit{29,0}\pm \unit{0,2}{\centi\metre}\] +L'origine de l'incertitude absolue peut être de diverses nature~: +\begin{enumerate} +\item une estimation de la précision suite à la lecture visuelle sur la règle selon sa graduation, la distance à laquelle ou l'angle sous lequel on la regarde, \dots, +\item une information du fabriquant qui a réalisé des tests approfondis, +\item une étude statistique à travers un ou plusieurs échantillons sur une mesure identique, etc. +\end{enumerate} +Conformément à ce que nous avons dit plus haut, on ne considérera pas le dernier point. + +\medskip +Admettons maintenant qu'on réalise aussi une mesure de la largeur de la feuille, mais avec une règle différente, et obtienne~: +\[l=\unit{21,1}\pm \unit{0,3}{\centi\metre}\] + +\subsection{Addition/soustraction} +Que pouvons-nous dire alors de la circonférence de la feuille ? + +\smallskip +Évidemment, elle est facile à calculer~: +\[C=2\cdot L_m+2\cdot l_m=2\cdot (29,0+21,1)=\unit{100,2}{\centi\metre}\] + +\medskip +Mais qu'en est-il de son incertitude ? Pour la connaître, on va calculer l'incertitude absolue de la somme de deux grandeurs en déterminant les maximum et minimum de celle-ci. Formellement, si~: +\begin{equation*} +s=l_1+l_2\;;\;l_1=l_{1m}\pm I(l_{1m})\;;\;l_2=l_{2m}\pm I(l_{2m}) +\end{equation*} +alors, on peut calculer les extrèmums~: +\begin{align*} +s_{max}&=l_{1\,max}+l_{2\,max}\\ +&=l_{1m}+I(l_{1m})+l_{2m}+I(l_{2m})\\ +&=l_{1m}+l_{2m}+I(l_{1m})+I(l_{2m})\\ +&=s_m+I(s_m)\\ +\text{et}\\ +s_{min}&=l_{1\,min}+l_{2\,min}\\ +&=l_{1m}-I(l_{1m})+l_{2m}-I(l_{2m})\\ +&=l_{1m}+l_{2m}-I(l_{1m})-I(l_{2m})\\ +&=s_m-I(s_m) +\end{align*} +et en déduire que~: + +\smallskip +\fbox{ +\(\text{Si : }s=l_1\pm l_2\;\Rightarrow\;I(s_m)=I(l_{1m})+I(l_{2m})\) +} + +\medskip +On peut alors calculer l'incertitude sur la circonférence~: +\begin{align*} +I(C_m)&=2\cdot I(L_m)+2\cdot I(l_m)\\ +&=2\cdot 0,2+2\cdot 0,3=\unit{1,0}{\centi\metre} +\end{align*} +et donner le résultat final sous la forme~: +\[C=\unit{100,2}\pm \unit{1,0}{\centi\metre}\] + +\subsection{Multiplication/division} +Que pouvons-nous dire de la surface de la feuille ? + +\smallskip +Évidemment, elle est facile à calculer~: +\[S=l_1\cdot l_2=29,0\cdot 21,1=\unit{611,9}{\centi\metre\squared}\] + +\medskip +Mais qu'en est-il de son incertitude ? Pour la connaître, on va calculer l'incertitude absolue de la multiplication de deux grandeurs en déterminant les maximum et minimum de celle-ci. Formellement, si~: +\begin{equation*} +S=l_1\cdot l_2\;;\;l_1=l_{1m}\pm I(l_{1m})\;;\;l_2=l_{2m}\pm I(l_{2m}) +\end{equation*} +alors, on peut calculer le maximum~: +\begin{align*} +S_{max}&=l_{1\,max}\cdot l_{2\,max}\\ +&=(l_{1m}+I(l_{1m}))\cdot (l_{2m}+I(l_{2m}))\\ +&=l_{1m}\cdot l_{2m}+l_{1m}\cdot I(l_{2m})\\ +&\;+l_{2m}\cdot I(l_{1m})+I(l_{1m})\cdot I(l_{2m})\\ +&=S_m+I(S_m) +\end{align*} +L'expression de l'incertitude s'avère alors pour le moins complexe~: +\[I(S_m)=l_{1m}\cdot I(l_{2m})+l_{2m}\cdot I(l_{1m})+I(l_{1m})\cdot I(l_{2m})\] +Mais, comme \(S=l_1\cdot l_2\), on peut diviser cette équation par \(S\)~: +\begin{align*} +\frac{I(S_m)}{S_m}&=\frac{l_{1m}\cdot I(l_{2m})}{l_{1m}\cdot l_{2m}}+\frac{l_{2m}\cdot I(l_{1m})}{l_{1m}\cdot l_{2m}}+\frac{I(l_{1m})\cdot I(l_{2m})}{l_{1m}\cdot l_{2m}}\\ +&=\frac{I(l_{1m})}{l_{1m}}+\frac{I(l_{2m})}{l_{2m}}+\frac{I(l_{1m})\cdot I(l_{2m})}{l_{1m}\cdot l_{2m}} +\end{align*} +Pour autant que l'incertitude soit nettement plus petite que la grandeur qui lui correspond, le terme \(I(l_{1m})\cdot I(l_{2m})\) est très petit devant \(l_{1m}\cdot l_{2m}\) et donc il est possible de négliger le dernier terme. Dans ces conditions~: +\[\frac{I(l_{1m})\cdot I(l_{2m})}{l_{1m}\cdot l_{2m}}<<1\] +et on peut écrire~: +\begin{align*} +\frac{I(S_m)}{S_m}&\cong\frac{I(l_{1m})}{l_{1m}}+\frac{I(l_{2m})}{l_{2m}}\;\Rightarrow\\ +i(S_m)&=i(l_{1m}+i(l_{2m}) +\end{align*} +en définissant l'incertitude relative par~: + +\smallskip +\fbox{ +\(i(G_m)=\frac{I(G_m)}{G_m}\) +} + +\medskip +L'incertitude relative, comme expression de l'incertitude absolue rapportée à la valeur de sa grandeur, représente un incertitude sans unités qu'on peut exprimer en pourcents. En utilisant l'incertitude relative, l'expression de l'incertitude d'une multiplication de deux grandeurs devient alors bien plus simple~: + +\smallskip +\fbox{ +\(\text{Si : }S=l_1\cdot l_2\;\Rightarrow\;i(S_m)=i(l_{1m}+i(l_{2m})\) +} + +\medskip +On peut aussi faire ... \ No newline at end of file diff --git a/Annexe-Incertitudes/Images/baguettesgauss.dat b/Annexe-Incertitudes/Images/baguettesgauss.dat new file mode 100644 index 0000000..122c332 --- /dev/null +++ b/Annexe-Incertitudes/Images/baguettesgauss.dat @@ -0,0 +1,9 @@ +38 53 +39 72 +40 95 +41 121 +42 130 +43 118 +44 90 +45 67 +46 44 \ No newline at end of file diff --git a/Annexe-MRUA/Annexe-MRUA.aux b/Annexe-MRUA/Annexe-MRUA.aux index 24e3601..f1f3e97 100644 --- a/Annexe-MRUA/Annexe-MRUA.aux +++ b/Annexe-MRUA/Annexe-MRUA.aux @@ -32,6 +32,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{4} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff 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\setcounter{r@tfl@t}{4} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Annexe-SystemeCoordonnees/Annexe-SystemeCoordonnees.aux b/Annexe-SystemeCoordonnees/Annexe-SystemeCoordonnees.aux index c5761e2..444df50 100644 --- a/Annexe-SystemeCoordonnees/Annexe-SystemeCoordonnees.aux +++ b/Annexe-SystemeCoordonnees/Annexe-SystemeCoordonnees.aux @@ -36,6 +36,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{4} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Annexe-TravauxPratiques/Annexe-TravauxPratiques.aux b/Annexe-TravauxPratiques/Annexe-TravauxPratiques.aux index 3aa09fc..ab376ce 100644 --- a/Annexe-TravauxPratiques/Annexe-TravauxPratiques.aux +++ b/Annexe-TravauxPratiques/Annexe-TravauxPratiques.aux @@ -66,6 +66,7 @@ \setcounter{table}{1} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{4} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Annexe-UnitesInternationales/Annexe-UnitesInternationales.aux b/Annexe-UnitesInternationales/Annexe-UnitesInternationales.aux index a6108b7..6abfa8e 100644 --- a/Annexe-UnitesInternationales/Annexe-UnitesInternationales.aux +++ b/Annexe-UnitesInternationales/Annexe-UnitesInternationales.aux @@ -45,6 +45,7 @@ \setcounter{table}{4} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{4} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Cinematique/Cinematique.aux b/Cinematique/Cinematique.aux index 83c748e..ae17d62 100644 --- a/Cinematique/Cinematique.aux +++ b/Cinematique/Cinematique.aux @@ -111,6 +111,7 @@ \setcounter{table}{3} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{1} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/CoursMecaniqueOSDF-gnuplottex-fig1.gnuplot b/CoursMecaniqueOSDF-gnuplottex-fig1.gnuplot new file mode 100644 index 0000000..d10c9e6 --- /dev/null +++ b/CoursMecaniqueOSDF-gnuplottex-fig1.gnuplot @@ -0,0 +1,25 @@ +set terminal latex rotate +set output 'CoursMecaniqueOSDF-gnuplottex-fig1.tex' +# domaine de définition + set xrange [36:48] + set yrange [0:140] + # flèches + set arrow 1 from 39.7,120 to 39.7,0 head filled + #size screen 0.5,30 + set label "Écart type" at 39.4,100 rotate by 90 + set arrow 2 from 44.1,120 to 44.1,0 head filled + set label "Écart type" at 44.4,138 rotate by -90 + set arrow 3 from 41.9,120 to 41.9,0 head filled + set label "Moyenne" at 42.2,80 rotate by -90 + #set grid + #set title "Baguettes d'une année" + # suppression de la légende + set key off + # légendes des axes + set xlabel "Longueur des baguettes (en cm)" + set ylabel "Nombre de baguettes" rotate by 90 + # largeur des colonnes (boxes) + #set boxwidth 0.1 + + # tracé du graphe + plot "Annexe-Incertitudes/Images/baguettesgauss.dat" with boxes diff --git a/CoursMecaniqueOSDF-gnuplottex-fig1.tex 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b/CoursMecaniqueOSDF.exp.table new file mode 100644 index 0000000..1275afb --- /dev/null +++ b/CoursMecaniqueOSDF.exp.table @@ -0,0 +1,30 @@ + +# Curve 0 of 1, 25 points +# Curve title: "0.05*exp(x)" +# x y type +0.00000 0.05000 i +2.08333 0.40156 i +4.16667 3.22500 i +6.25000 25.90064 i +8.33333 208.01310 i +10.41667 1670.59377 i +12.50000 13416.86433 i +14.58333 107753.45364 i +16.66667 865388.99767 i +18.75000 6950107.78773 i +20.83333 55817670.88691 i +22.91667 448282599.11909 i +25.00000 3600244966.86929 i +27.08333 28914269362.53708 i +29.16667 232216135419.34827 i +31.25000 1864973064785.95093 i +33.33333 14977962345709.12891 i +35.41667 120290936242143.25000 i +37.50000 966079965220141.87500 i +39.58333 7758776582476852.00000 i +41.66667 62312247664792064.00000 i +43.75000 500441811639179264.00000 i +45.83333 4019145773459419136.00000 i +47.91667 32278543424272236544.00000 i +50.00000 259235276429353615360.00000 i + diff --git a/CoursMecaniqueOSDF.gnuploterrors b/CoursMecaniqueOSDF.gnuploterrors new file mode 100644 index 0000000..e69de29 diff --git a/CoursMecaniqueOSDF.lof b/CoursMecaniqueOSDF.lof index 0601d4d..2658e22 100644 --- a/CoursMecaniqueOSDF.lof +++ b/CoursMecaniqueOSDF.lof @@ -187,3 +187,4 @@ \contentsline {figure}{\numberline {L.17}{\ignorespaces Frottements}}{238} \contentsline {figure}{\numberline {L.18}{\ignorespaces Forces sur bloc suspendu}}{239} \addvspace {10\p@ } +\contentsline {figure}{\numberline {M.1}{\ignorespaces Baguettes d'une ann\IeC {\'e}e\relax }}{243} diff --git a/CoursMecaniqueOSDF.log b/CoursMecaniqueOSDF.log index 183996e..fca58b7 100644 --- a/CoursMecaniqueOSDF.log +++ b/CoursMecaniqueOSDF.log @@ -1,6 +1,6 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex 2018.12.20) 19 FEB 2019 16:57 +This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex 2018.12.20) 26 FEB 2019 23:52 entering extended mode - restricted \write18 enabled. 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Package caption Info: rotating package is loaded. Package caption Info: subfigure package is loaded. Package caption Info: supertabular package is loaded. 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(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty Package: hhline 2014/10/28 v2.03 Table rule package (DPC) ) +ABD: EveryShipout initializing macros Package caption Info: Begin \AtBeginDocument code. Package caption Info: floatflt package is loaded. Package caption Info: listings package is loaded. @@ -1412,9 +1680,9 @@ Option `squaren' provided! Option `Gray' provided! Command \Gray defined by SIunits package! -\c@lstlisting=\count280 +\c@lstlisting=\count302 Using optional text marked with OS. -\AtBeginShipoutBox=\box79 +\AtBeginShipoutBox=\box89 \openout2 = `Prefaces/Prefaces.aux'. (./Prefaces/Prefaces.tex @@ -1427,6 +1695,11 @@ File: ./Images/TrouNoir.eps Graphic file (type eps) ] +LaTeX Font Info: Try loading font information for U+lasy on input line 33. + +(/usr/share/texlive/texmf-dist/tex/latex/base/ulasy.fd +File: ulasy.fd 1998/08/17 v2.2e LaTeX symbol font definitions +) LaTeX Font Info: Try loading font information for U+msa on input line 33. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd @@ -1455,8 +1728,8 @@ File: t1cmtt.fd 2014/09/29 v2.5h Standard LaTeX font definitions ] [6] [7] [8] [9] [10] [11] [12]) -\tf@toc=\write9 -\openout9 = `CoursMecaniqueOSDF.toc'. +\tf@toc=\write11 +\openout11 = `CoursMecaniqueOSDF.toc'. [13] [14 @@ -1464,12 +1737,12 @@ File: t1cmtt.fd 2014/09/29 v2.5h Standard LaTeX font definitions ] (./CoursMecaniqueOSDF.lof [15] [16] [17]) -\tf@lof=\write10 -\openout10 = `CoursMecaniqueOSDF.lof'. +\tf@lof=\write12 +\openout12 = `CoursMecaniqueOSDF.lof'. [18] (./CoursMecaniqueOSDF.lot) -\tf@lot=\write11 -\openout11 = `CoursMecaniqueOSDF.lot'. +\tf@lot=\write13 +\openout13 = `CoursMecaniqueOSDF.lot'. [19 @@ -1502,7 +1775,7 @@ Underfull \hbox (badness 2469) in paragraph at lines 17--17 []\T1/cmr/bx/n/14.4 De l'infiniment grand à [] -\openout7 = `CoursMecaniqueOSDF.ent'. +\openout9 = `CoursMecaniqueOSDF.ent'. File: ./Introduction/Images/UniversProfond.eps Graphic file (type eps) <./Introduction/Images/UniversProfond.eps> [21 @@ -3081,12 +3354,12 @@ Underfull \hbox (badness 1077) in paragraph at lines 34--35 (./Annexe-Exercices/Annexe-Exercices.tex Appendix L. Output from handle ans going to Solutions.tex -\ans@file=\write12 -\openout12 = `Solutions.tex'. +\ans@file=\write14 +\openout14 = `Solutions.tex'. Output from handle ansos going to SolutionsOS.tex -\ansos@file=\write13 -\openout13 = `SolutionsOS.tex'. +\ansos@file=\write15 +\openout15 = `SolutionsOS.tex'. Underfull \hbox (badness 10000) in paragraph at lines 9--9 @@ -3464,8 +3737,47 @@ Underfull \vbox (badness 4429) has occurred while \output is active [] [241 -]) -[242] (./CoursMecaniqueOSDF.ent +] +[242] +\openout7 = `CoursMecaniqueOSDF-gnuplottex-fig1.gnuplot'. + + Opening gnuplot stream CoursMecaniqueOSDF-gnuplottex-fig1.gnuplot +runsystem(gnuplot CoursMecaniqueOSDF-gnuplottex-fig1.gnuplot 2>>CoursMecaniqueO +SDF.gnuploterrors)...executed. + +Package gnuplottex Info: CoursMecaniqueOSDF-gnuplottex-fig1.gnuplot converted o +n input line 146. + +(./CoursMecaniqueOSDF-gnuplottex-fig1.tex +\plotpoint=\box90 +) +Underfull \hbox (badness 3068) in paragraph at lines 161--162 +[]\T1/cmr/m/n/10 Avec les in-cer-ti-tudes, on va s'intéresser à + [] + + +Underfull \hbox (badness 2753) in paragraph at lines 161--162 +\T1/cmr/m/n/10 l'évaluation des ca-pac-ités des in-stru-ments de + [] + +[243] +Overfull \hbox (1.48897pt too wide) in paragraph at lines 211--212 +[][] + [] + + +Overfull \hbox (10.62454pt too wide) in paragraph at lines 248--248 +[] + [] + + +Underfull \hbox (badness 2600) in paragraph at lines 264--265 +[]\T1/cmr/m/n/10 L'incertitude rel-a-tive, comme ex-pres-sion de + [] + +[244]) [245 + +] (./CoursMecaniqueOSDF.ent Underfull \hbox (badness 7415) in paragraph at lines 2--22 \T1/cmtt/m/n/8 hubble . nasa . gov / multimedia / astronomy . php$ \T1/cmr/m/n/ 8 no-tam-ment @@ -3673,7 +3985,7 @@ Underfull \hbox (badness 2057) in paragraph at lines 616--624 ommons . [] -[243 +[246 ] Underfull \hbox (badness 10000) in paragraph at lines 626--650 @@ -3808,7 +4120,10 @@ Underfull \hbox (badness 5077) in paragraph at lines 868--883 cmr/m/n/8 ou [] -) [244 +) [247 + +] [248 + ] (./CoursMecaniqueOSDF.bbl Underfull \hbox (badness 5954) in paragraph at lines 37--43 @@ -3831,19 +4146,20 @@ Underfull \hbox (badness 1325) in paragraph at lines 140--152 \T1/cmr/m/n/10 Ou-vrage très com-plet et très math-é-ma-tique [] -[245 +[249 -]) (./CoursMecaniqueOSDF.ind [246 +]) (./CoursMecaniqueOSDF.ind [250 -] [247 +] [251 -] [248] [249] [250] [251] [252] -[253] [254 +] [252] [253] [254] [255] [256] +[257] [258 -]) (./CoursMecaniqueOSDF.aux (./Prefaces/Prefaces.aux) -(./Introduction/Introduction.aux) (./Cinematique/Cinematique.aux) -(./Dynamique/Dynamique.aux) (./MecaniqueDim/MecaniqueDim.aux) +]) (./CoursMecaniqueOSDF.gnuploterrors) (./CoursMecaniqueOSDF.aux +(./Prefaces/Prefaces.aux) (./Introduction/Introduction.aux) +(./Cinematique/Cinematique.aux) (./Dynamique/Dynamique.aux) +(./MecaniqueDim/MecaniqueDim.aux) (./MecaniqueDifferentielle/MecaniqueDifferentielle.aux) (./QtiteMvt/QtiteMvt.aux) (./Energie/Energie.aux) (./EnergieOS/EnergieOS.aux) (./Thermodynamique/Thermodynamique.aux) @@ -3871,12 +4187,12 @@ LaTeX Warning: There were multiply-defined labels. ) Here is how much of TeX's memory you used: - 19512 strings out of 494830 - 309195 string characters out of 6176634 - 446355 words of memory out of 5000000 - 21967 multiletter control sequences out of 15000+600000 - 36314 words of font info for 89 fonts, out of 8000000 for 9000 + 24162 strings out of 494830 + 416166 string characters out of 6176634 + 555496 words of memory out of 5000000 + 26427 multiletter control sequences out of 15000+600000 + 36950 words of font info for 95 fonts, out of 8000000 for 9000 36 hyphenation exceptions out of 8191 - 55i,29n,92p,2650b,544s stack positions out of 5000i,500n,10000p,200000b,80000s + 56i,29n,92p,2650b,544s stack positions out of 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/end-hook known{end-hook}if diff --git a/CoursMecaniqueOSDF.sin.gnuplot b/CoursMecaniqueOSDF.sin.gnuplot new file mode 100644 index 0000000..00db0c8 --- /dev/null +++ b/CoursMecaniqueOSDF.sin.gnuplot @@ -0,0 +1,2 @@ +set table "CoursMecaniqueOSDF.sin.table"; set format "%.5f" +set samples 25; plot [x=0:50] sin(x) diff --git a/CoursMecaniqueOSDF.sin.table b/CoursMecaniqueOSDF.sin.table new file mode 100644 index 0000000..fcaee20 --- /dev/null +++ b/CoursMecaniqueOSDF.sin.table @@ -0,0 +1,30 @@ + +# Curve 0 of 1, 25 points +# Curve title: "sin(x)" +# x y type +0.00000 0.00000 i +2.08333 0.87150 i +4.16667 -0.85475 i +6.25000 -0.03318 i +8.33333 0.88729 i +10.41667 -0.83706 i +12.50000 -0.06632 i +14.58333 0.90211 i +16.66667 -0.81845 i +18.75000 -0.09939 i +20.83333 0.91593 i +22.91667 -0.79893 i +25.00000 -0.13235 i +27.08333 0.92874 i +29.16667 -0.77854 i +31.25000 -0.16517 i +33.33333 0.94053 i +35.41667 -0.75729 i +37.50000 -0.19780 i +39.58333 0.95128 i +41.66667 -0.73520 i +43.75000 -0.23021 i +45.83333 0.96099 i +47.91667 -0.71231 i +50.00000 -0.26237 i + diff --git a/CoursMecaniqueOSDF.toc b/CoursMecaniqueOSDF.toc index dba4810..5f1c272 100644 --- a/CoursMecaniqueOSDF.toc +++ b/CoursMecaniqueOSDF.toc @@ -399,4 +399,6 @@ \contentsline {chapter}{\numberline {M}Ordre de grandeur, erreur et incertitudes}{241} \contentsline {section}{\numberline {M.1}Ordre de grandeur}{241} \contentsline {section}{\numberline {M.2}\IeC {\'E}cart et erreur}{241} -\contentsline {section}{\numberline {M.3}Incertitude}{242} +\contentsline {section}{\numberline {M.3}Incertitude}{243} +\contentsline {subsection}{\numberline {M.3.1}Addition/soustraction}{244} +\contentsline {subsection}{\numberline {M.3.2}Multiplication/division}{244} diff --git a/CoursMecaniqueOSDF.x.gnuplot b/CoursMecaniqueOSDF.x.gnuplot new file mode 100644 index 0000000..e4d782e --- /dev/null +++ b/CoursMecaniqueOSDF.x.gnuplot @@ -0,0 +1,2 @@ +set table "CoursMecaniqueOSDF.x.table"; set format "%.5f" +set samples 25; plot [x=0:50] x diff --git a/CoursMecaniqueOSDF.x.table b/CoursMecaniqueOSDF.x.table new file mode 100644 index 0000000..65f9af4 --- /dev/null +++ b/CoursMecaniqueOSDF.x.table @@ -0,0 +1,30 @@ + +# Curve 0 of 1, 25 points +# Curve title: "x" +# x y type +0.00000 0.00000 i +2.08333 2.08333 i +4.16667 4.16667 i +6.25000 6.25000 i +8.33333 8.33333 i +10.41667 10.41667 i +12.50000 12.50000 i +14.58333 14.58333 i +16.66667 16.66667 i +18.75000 18.75000 i +20.83333 20.83333 i +22.91667 22.91667 i +25.00000 25.00000 i +27.08333 27.08333 i +29.16667 29.16667 i +31.25000 31.25000 i +33.33333 33.33333 i +35.41667 35.41667 i +37.50000 37.50000 i +39.58333 39.58333 i +41.66667 41.66667 i +43.75000 43.75000 i +45.83333 45.83333 i +47.91667 47.91667 i +50.00000 50.00000 i + diff --git a/Dynamique/Dynamique.aux b/Dynamique/Dynamique.aux index a60184a..dc4723f 100644 --- a/Dynamique/Dynamique.aux +++ b/Dynamique/Dynamique.aux @@ -118,6 +118,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{2} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Energie/Energie.aux b/Energie/Energie.aux index 189fb87..8f3eac4 100644 --- a/Energie/Energie.aux +++ b/Energie/Energie.aux @@ -98,6 +98,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{3} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/EnergieOS/EnergieOS.aux b/EnergieOS/EnergieOS.aux index 4afec51..dcabd05 100644 --- a/EnergieOS/EnergieOS.aux +++ b/EnergieOS/EnergieOS.aux @@ -58,6 +58,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{3} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Introduction/Introduction.aux b/Introduction/Introduction.aux index 96357ec..6e09ee5 100644 --- a/Introduction/Introduction.aux +++ b/Introduction/Introduction.aux @@ -97,6 +97,7 @@ \setcounter{table}{1} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{0} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Macros.aux b/Macros.aux index c500659..522e4d1 100644 --- a/Macros.aux +++ b/Macros.aux @@ -19,6 +19,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{4} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{0} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/MecaniqueDifferentielle/MecaniqueDifferentielle.aux b/MecaniqueDifferentielle/MecaniqueDifferentielle.aux index 609b31c..51c95cf 100644 --- a/MecaniqueDifferentielle/MecaniqueDifferentielle.aux +++ b/MecaniqueDifferentielle/MecaniqueDifferentielle.aux @@ -55,6 +55,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{2} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/MecaniqueDim/MecaniqueDim.aux b/MecaniqueDim/MecaniqueDim.aux index 5f68e38..f6ae793 100644 --- a/MecaniqueDim/MecaniqueDim.aux +++ b/MecaniqueDim/MecaniqueDim.aux @@ -90,6 +90,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{2} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Preambule.tex b/Preambule.tex index e5c2729..ec1f07c 100644 --- a/Preambule.tex +++ b/Preambule.tex @@ -37,6 +37,15 @@ \usepackage{pstricks} % pour étendre les possiblités de graphisme des figures de latex construite avec jpicedt %\usepackage{pstricks-add} % plein de bug semble-t-il, mais aurait été utile %\usepackage{pstcol} +%% pour utiliser gnuplot avec --shell-escape dans la compilation latex ! +\usepackage{gnuplottex} +%\usepackage{epstopdf} +\usepackage{latexsym} +\usepackage{keyval} +\usepackage{ifthen} +\usepackage{moreverb} +\usepackage{tikz} + \usepackage{multido} % pour répéter des figures en pstricks %\usepackage[figuresright ou figuresleft]{rotating} % permet de faire tourner des figures \sidewaysfigure ou des textes \rotatebox dans un sens ou un autre \usepackage{rotating} % permet de faire tourner des figures \sidewaysfigure ou des textes \rotatebox diff --git a/Preambule.tex.bak b/Preambule.tex.bak new file mode 100644 index 0000000..6bb73b5 --- /dev/null +++ b/Preambule.tex.bak @@ -0,0 +1,100 @@ +% Ce fichier constitue le préambule des différents cours de physique. Les définitions de modules propres à chaque cours se trouvent dans le préambule de ces cours. On trouve ici les définitions des modules communs. +% Ce fichier est appelé par chaque cours à travers l'instruction \input{../Preambule.tex} +\usepackage{etex} % nécessaire au package xy (pourquoi ????) +\usepackage[T1]{fontenc} % écrire avec les accents +\usepackage[utf8]{inputenc} % écrire avec les accents +%\usepackage{babel} +\usepackage[english,french]{babel} % pour le support des documents multilingues. Il vaut mieux mettre french là plutôt que dans le documentclass + % pour pouvoir gérer l'anglais ou autres langues. +%\usepackage{lmodern} +%\usepackage{geometry} % pour gérer les marges de la page +%\geometry{verbose,a4paper,lmargin=2cm,rmargin=2cm} % règle les marges de la page + +\setcounter{tocdepth}{5} % pour définir la profondeur de la table des matières +\usepackage{makeidx} % pour permettre de construire un index +\makeindex % pour construire effectivement l'index + +\makeatletter % pour faire de @ une lettre simple (et non un caractère associé à une macro interne +\usepackage{path} % pour mettre du code non interprété +\usepackage{fancybox} % pour faire des boites entourée de différents types de cadres +\usepackage{framed} % pour mettre des sortes de minipages encadrées sur plusieures pages, sur fond coloré +\usepackage[dvips]{color} % pour mettre du gris ou de la couleur dans les boites +\usepackage{pst-grad} % pour faire des dégradés en pstricks (! il faut le mettre juste ici pour que cela fonctionne !) +\definecolor{shadecolor}{rgb}{.9,.9,.9} % pour définir la couleur de fond du package framed +\usepackage{boxedminipage} % pour mettre des bordures aux minipages +\usepackage{relsize,fancyvrb} % pour mettre des textes non interprétés encadrés et changer la taille des caractères à l'intérieur +\usepackage{array} % pour étendre les possiblités de tabular par défaut pour les tableaux +\usepackage{multirow} % pour faire une cellule de tableau sur plusieurs lignes +\usepackage{colortbl} % pour mettre de la couleur dans un tableau +\usepackage{hhline} % pour faire des filets spéciaux dans les tableaux +\usepackage{supertabular} % pour faire des tableaux sur plusieurs pages +\usepackage{float} % permet un placement obligatoire +\usepackage{url} % permet une bonne gestion de la césure des url +\usepackage{graphicx} % pour charger le module étendu graphicx du module graphics. Pour différents effets graphiques dont l'orientation et le redimentionnement des eps +%\usepackage{graphics} % idem que graphicx, mais sans orientation et redimentionnement +\usepackage{wrapfig} % pour mettre des figures dans le flot du texte +\usepackage{psfrag} % pour mettre des équations latex dans des figures (ne fonctionne qu'avec ps) +\usepackage{pstricks} % pour étendre les possiblités de graphisme des figures de latex construite avec jpicedt +%\usepackage{pstricks-add} % plein de bug semble-t-il, mais aurait été utile +%\usepackage{pstcol} +%% pour utiliser gnuplot avec --shell-escape dans la compilation latex ! +%% il faut aussi utiliser l'environnement tikz car gnuplottex semble ne pas fonctionner. +\usepackage{gnuplottex} +%\usepackage{epstopdf} +\usepackage{latexsym} +\usepackage{keyval} +\usepackage{ifthen} +\usepackage{moreverb} +\usepackage{tikz} + +\usepackage{multido} % pour répéter des figures en pstricks +%\usepackage[figuresright ou figuresleft]{rotating} % permet de faire tourner des figures \sidewaysfigure ou des textes \rotatebox dans un sens ou un autre +\usepackage{rotating} % permet de faire tourner des figures \sidewaysfigure ou des textes \rotatebox +\usepackage{subfigure} % pour faire des figures comprenant plusieures figures +\usepackage{fancyhdr} % pour des entêtes et pieds de page entièrement personnalisés +\usepackage{fancyvrb} % pour mettre des \verb dans les minipages +\usepackage[Lenny]{fncychap} % pour des titres de chapitres différents ... +\usepackage{multicol} % pour mettre un \begin{multicols}...\end{multicols} et faire du multicolonnage localement dans une page +\usepackage{amsmath} % pour avoir beaucoup de symboles mathématiques et des alignements complexes +\usepackage{amssymb} % nécessaire pour avoir certains symboles mathématiques +\usepackage{amsfonts} % nécessaire pour avoir certains symboles mathématiques +%\usepackage{chappg} % numérotation par chapitre +\usepackage{epsfig} % pour inclure des fichiers .eps. Mais il semble que les modules graphics ou graphicx devraient le remplacer +\makeatother % pour refaire de @ une lettre différente des autres (exploitée dans des macros) +\usepackage{caption} % pour gérer encore mieux les légendes +\renewcommand{\captionfont}{\it \small} % pour avoir de l'italique pour le texte de légende +\renewcommand{\captionlabelfont}{\it \bf \small} +\DeclareCaptionLabelSeparator{endash}{ -- } % pour définir un tiret après le numéro des légentes +\captionsetup{labelsep=endash,justification=centering,belowskip=10pt} % pour avoir un petit trait après le no de légende +\usepackage{verbatim} % pour faire des commentaires longs \begin{comment} +\usepackage{endnotes} % pour faire des notes de fin +%\renewcommand{\theendnote}{\Roman{endnote}} % pour mettre les notes de fin en romain +\renewcommand{\thefootnote}{\alph{footnote}} % pour mettre les notes de bas de page en alphabétique +%\usepackage{apalike} % nécessaire pour que le style de bibliographie apalike soit correctement appliqué + % attention, le module apalike ne doit être utilisé qu'en anglais. Le titre bibliography est codé en dur à l'intérieur du module +\usepackage{natbib} % à utiliser impérativement avec apalike-fr pour parfaire les entrées +%\usepackage{textcomp} % pour mettre de beaux degrés +\usepackage{textcomp} % pour le symbole pourmille : \textperthousand (! pas en mode math) et pour le symbole de copyleft \textcopyleft +\usepackage{boites,boites_exemples} % pour faire des paragraphes encadrés sur plusieurs colonnes +\usepackage{textpos} % pour positionner du texte ou des images à une place très précise de la page. +\usepackage{floatflt} % pour placer des tableaux entourés de texte +\usepackage{nextpage} % pour retirer les entêtes de la dernière page vide d'un chapitre ; +\newcommand\myclearpage{\cleartooddpage[\thispagestyle{plain}]} % il faut ensuite placer \myclearpage à la fin de chaque chapitre. +\def\siecle#1{\textsc{\romannumeral #1}\textsuperscript{e}~siècle} % macro pour écrire les siècles : \siecle{19} par exemple + +%\usepackage{showidx} % pour voir les entrées d'index dans la marge. ATTENTION : commenter avant la compilation. + +%\usepackage{aeguill} % fontes utilisées par pdflatex (pour un pdf avec liens cliquables ; pas encore testé) +%\usepackage{hyperref} % pour faire des hyperréférences (pas encore testé) +%\hypersetup{colorlinks,citecolor=black,filecolor=black,linkcolor=black,urlcolor=black,pdftex} % pour définir la couleur des liens en noir, pour une meilleure impression (pas encore testé) +%\usepackage{unitsdef} % pour l'écriture des unités +\usepackage[derivedinbase,squaren,cdot,Gray]{SIunits} % pour utiliser des commandes relatives au unités du système internationnal ; il y a aussi les préfixes m M G ... + % le squaren est là pour éviter une incompatibilité avec le symbole ams de la racine. + % le derivedinbase est là pour fournir une traduction des unités SI dérivées en unités SI de base +%\usepackage{hepunits} % unités telles que Hz, eV, ... +%\usepackage{siunitx} % meilleure gestion des unités que siunits, mais incompatible avec celui-ci ! +%\usepackage[fixlanguage]{babelbib} % pour avoir une bibliographie en français ou différentes entrées de langues différentes (pour cela il faut renseigner un champ de langue) +%\selectbiblanguage{french} % sélection de la langue (attention, pour apalike, il faut sélectionner apalike-fr) +\usepackage{lettrine} % pour mettre des lettrines +\usepackage{listings} % pour mettre du code informatique non interprété +\usepackage{xspace} % pour gérer les expaces nott. entre X et e pour les siècles \ No newline at end of file diff --git a/Prefaces/Prefaces.aux b/Prefaces/Prefaces.aux index 7f213a2..e07630c 100644 --- a/Prefaces/Prefaces.aux +++ b/Prefaces/Prefaces.aux @@ -20,6 +20,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{0} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/QtiteMvt/QtiteMvt.aux b/QtiteMvt/QtiteMvt.aux index c05beb3..3c42a22 100644 --- a/QtiteMvt/QtiteMvt.aux +++ b/QtiteMvt/QtiteMvt.aux @@ -51,6 +51,7 @@ \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{2} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/Thermodynamique/Thermodynamique.aux b/Thermodynamique/Thermodynamique.aux index 21cf056..24e6420 100644 --- a/Thermodynamique/Thermodynamique.aux +++ b/Thermodynamique/Thermodynamique.aux @@ -56,6 +56,7 @@ \setcounter{table}{3} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{4} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} diff --git a/ThermodynamiqueOS/ThermodynamiqueOS.aux b/ThermodynamiqueOS/ThermodynamiqueOS.aux index 1c0b216..908f740 100644 --- a/ThermodynamiqueOS/ThermodynamiqueOS.aux +++ b/ThermodynamiqueOS/ThermodynamiqueOS.aux @@ -181,6 +181,7 @@ \setcounter{table}{6} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} +\setcounter{fignum}{0} \setcounter{r@tfl@t}{4} \setcounter{subfigure}{0} \setcounter{lofdepth}{1}

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