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-{% block logo_title %} -Deutsch -{% endblock %} - -{% block header_text %} -back to {{ sage }} main page -{% endblock %} - -{% block menu %} -
  • Index
  • -{% endblock %} - -{% block header %} - -{% endblock %} - -{% block content %} - -

    {{ sage }} Deutsch

    - -
    - -
    -{{ sage }} ist ein freies Open Source Mathematiksystem das unter der GPL steht. -{{ sage }} vereint die Funktionalität diverser Open Source Pakete unter einer gemeinsamen auf Python basierenden Oberflaeche. -
    - -
    -Unser Ziel: Erstellen eines vollwertigen open-source Ersatzes für Magma, Maple, Mathematica und Matlab. -
    - - - -
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  • Index
  • -{% endblock %} \ No newline at end of file diff --git a/src/es/index.html b/src/es/index.html deleted file mode 100755 index bfe072b3..00000000 --- a/src/es/index.html +++ /dev/null @@ -1,16 +0,0 @@ -{% extends 'es/_es.html' %} -{% set title = "Español" %} - -{% block content %} - -

    {{ sage }} Español

    - - -{% endblock %} diff --git "a/src/fr/48/W_UE_Courbes_param\303\251tr\303\251es_v02.html" "b/src/fr/48/W_UE_Courbes_param\303\251tr\303\251es_v02.html" deleted file mode 100755 index 6d143aad..00000000 --- "a/src/fr/48/W_UE_Courbes_param\303\251tr\303\251es_v02.html" +++ /dev/null @@ -1,388 +0,0 @@ -{% extends 'fr/_fr.html' %} -{% set title = 'Tracer des courbes paramétrées avec {{ sage }}' %} - -{% block content %} -

    Tracer des courbes paramétrées avec {{ sage }} -

    -

    auteur : Philippe Saadé (2008), psaade@gmail.com -

    -

    - - -
    Nous présentons ici différentes méthodes permettant de tracer des courbes paramétrées dans le plan et dans l'espace (courbes gauches). -
    -
    -Nous détaillons autant que possible les paramètres controlant le tracé (couleur, épaisseur, type de trait, ...) ainsi que ceux contrôlant le graphique qui en résulte. Pour une présentation encore plus approfondie des réglages et opérations possibles avec les graphiques, on pourra se reporter à la feuille de calcul Graphique_Utilisation_avancée. -
    -
    -Remarque importante : nous avons fait en sorte de rendre chaque cellule indépendante des autres. Le code contenu dans une cellule peut donc être utilisé par copier-coller dans une autre feuille de calcul {{ sage }}. Le lecteur ayant une certaine habitude des langages informatiques voudra bien nous excuser pour la répétition de certaines déclarations de variable, inutile dans le contexte d'une utilisation continue de la feuille de calcul. -
    - -Plan : - - - - - -

    Courbes paramétrées dans le plan -

    - - - -

    Tracé simple, dans un repère orthonormé -

    - - - - - - - - - - - - - - - - -
    -
    - sage: t = var('t')
    - sage: p = parametric_plot( (sin(t), sin(2*t)),
    -    0, 2*pi, rgbcolor=hue(0.6) )
    - sage: p.show(aspect_ratio=1)
    -
    -
    -
    -
    - Il faut donner la fonction à tracer comme un couple (x(t),y(t)) ou bien comme une liste [x(t),y(t)]. Ensuite, il faut indiquer les valeurs minimale et maximale d'évolution du paramètre.
    -L'option rgbcolor permet bien sûr de choisir la couleur. - -
    -
    - -
    - -
    - - - -

    Suppression des axes du graphique -

    - - - - - - - - - - - - - - - -
    -
    - sage: t = var('t')
    - sage: p = parametric_plot( [sin(t), sin(4*t)],
    -    0, 2*pi, rgbcolor=hue(0.4) )
    - sage: p.show(aspect_ratio=1, axes=False)
    -
    -
    -
    -
    - La présence ou non des axes du repère est déterminée par la valeur booléenne de l'option axes. - -
    -
    - -
    - -
    - - - -

    Ajout de labels aux axes -

    - - - - - - - - - - - - - - - - -
    -
    - sage: t = var('t')
    - sage: p = parametric_plot( [t*cos(4*t), t*sin(
    -   4*t)], 0, pi, rgbcolor=hue(0.2) )
    - sage: p.show(aspect_ratio=1, axes_labels=['axe
    -    des x','axe des $y$'])
    -
    -
    -
    -
    - On utilise pour cela l'option axes_labels en lui affectant une liste de deux chaînes de caractères. On peut remarquer qu'il est possible d'utiliser du code LaTeX dans ces chaînes de caractères.
    -Cependant, si l'affectation de labels aux axes doit se faire selon une exigence typographique plus soutenue, il est préférable d'utiliser la commande text() qui permet un meilleur rendu du code mathématique et un placement libre. - -
    -
    - -
    - -
    - - - -

    Utiliser un cadre pour les unités d'axes -

    - - - - - - - - - - - - - - - -
    -
    - sage: t = var('t')
    - sage: p = parametric_plot( [t*cos(4*t), t*sin(
    -   4*t)], 0, pi, rgbcolor='red')
    - sage: p.show(aspect_ratio=1, frame=True)
    -
    -
    -
    -
    - -
    -
    - -
    - -
    - - - - - - - - - - - - - - - - -
    -
    - sage: t = var('t')
    - sage: p = parametric_plot( [-t*cos(4*t), t*sin
    -   (4*t)], 0, pi, rgbcolor='blue' )
    - sage: p.show(aspect_ratio=1, gridlines=True)
    -
    -
    -
    -
    - -
    -
    - -
    - -
    - - - -

    Contrôler la dimension du graphique final -

    - - - - - - - - - - - - - - - - -
    -
    - sage: t = var('t')
    - sage: p = parametric_plot( [cos(t)*cos(4*t), s
    -   in(t)*sin(4*t)], 0, pi, rgbcolor=hue(0.8
    -   ))
    - sage: p.show(figsize=[4,3], dpi=200)
    -
    -
    -
    -
    - Il faut distinguer ici le pixel (ou point graphique) du point typographique. Une convention ancienne veut que le point typographique occupe environ 1/72ème de pouce. La résolution par défaut des graphiques produits par {{ sage }} est de 100dpi (pixels par pouce). Le point typographique est donc généralement un peu plus grand qu'un pixel. Comme le paramètre figsize permet d'imposer la taille du graphique final en pouce, il est facile de conjuguer figsize et dpi pour fixer le taille de l'image en pixels.
    -
    -C'est ce que nous avons pour obtenir ici une image de 800x600 pixels.
    -
    -On remarque cependant que le trait utilisé pour le tracé est deux fois plus épais. - -
    -
    - -
    - -
    - - - - - - - - - - - - - - - - - -
    -
    - sage: t = var('t')
    - sage: fx = lambda t: t
    - sage: fy = lambda t: 0
    - sage: p = parametric_plot( [fx,fy], 0, pi, thi
    -   ckness=72)
    - sage: p.show(figsize=[4,3], dpi=100)
    -
    -
    -
    -
    - Pour être complet, on pourra rappeler que l'option thickness de la commande parametric_plot() utilise le point typographique comme unité. Ainsi, si thickness=72, le trait aura une épaisseur d'un pouce, et donc de 100 pixels si dpi=100
    -C'est ce que l'on peut constater dans cet exemple. - -
    -
    - -
    - -
    - - - -

    Courbes gauches -

    - - - - - - - - - - - - - - - - -
    -
    - sage: t = var('t')
    - sage: p = parametric_plot3d( [cos(4*t), sin(4*
    -   t),cos(10*t)], (0, pi), rgbcolor='orange
    -   ' )
    - sage: p.show(viewer='tachyon',aspect_ratio=[1,
    -   1,1], axes=True)
    -
    -
    -
    -
    - Il faut utiliser la commande parametric_plot3d au lieu de parametric_plot. Plusieurs des options précédentes fonctionnent encore avec cette commande. - -
    -
    - -
    - -
    - - - - - - - - - - - - - -
    -
    - sage: 
    -
    -
    -
    -
    - -
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zbq<<74Lb%gh)iLJif~KPugOJth#?9khS0K-{Wvaur#__aD!-wQh1l~Tmonxv{%Os0 z{}asgs==pIr=Cz|(vFlCxq>lI&!2-9VT?$;tD3cI&-cpiABiv^v=}4sPiRwwjW%dB zy+MdUp?j*tZw{une;*tX(4CDoFfFrwfWv|xGO|xWO)tzVzn#pivpb&ccx8@>Q27x& z)=553Ix0}|ajz+&cKUf{FWE`iX+)tSb!e@398b~A3z4K+CSitAQK16-3z217JL+r< z1OwXT7i7P~5RzmF$&6mAuLXaOdyxUtilisation avancée des objets graphiques - -

    auteur : Philippe Saadé (2008), psaade@gmail.com -

    -

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    Avec {{ sage }}, la plupart des commandes graphiques produisent d'abord un objet de type Graphics qui est ensuite rendu par la commande show(), et ce même si l'utilisateur ne la sollicite pas directement.
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    -Nous allons expliquer ici comment l'on peut modifier un objet graphique et également comment on peut associer plusieurs objets graphiques. -

    - -Plan : - - - -

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    Modification d'un objet graphique -

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    Création de l'objet graphique -

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    - sage: var('x')
    - sage: f(x) = sin(10*x)/(1+x^2)
    - sage: p = plot(f, (-3,3), plot_points=400)
    - sage: p
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    - Il est très facile de produire un objet graphique. La simple évaluation de cet objet entraine son rendu par la commande show() - -
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    Modification du rapport d'échelle entre abscisse et ordonnée -

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    - sage: p = plot(f, (-3,3), plot_points=300)
    - sage: p.set_aspect_ratio(1)
    - sage: p
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    - La méthode set_aspect_ratio() permet de fixer le rapport de l'échelle des abscisses sur l'échelle des ordonnées.
    -On obtient une représentation orthonormée en fixant ce rapport à 1. - -
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    Modification de la plage des x et y -

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    - sage: p = plot(f, (-3,3), plot_points=400)
    - sage: p.axes_range(-4,4,-2,2)  # ou p.xmin(-4)
    -   ; p.xmax(4); p.ymin(-2); p.ymax(2)
    - sage: p
    -
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    - Le moyen le plus direct est d'utiliser la commande axes_range() à laquelle on fournit les valeurs xmin, xmax, ymin, ymax. Un accès individuel à chacune de ces valeurs est également possible avec xmin(), etc. - -
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    Changer l'épaisseur des axes -

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    - sage: p = plot(f, (-3,3), plot_points=400)
    - sage: p.axes_width(2)
    - sage: p
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    - La valeur (en point typographique) est fixée par la méthode axes_width(). - -
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    Suppression des axes -

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    - sage: p = plot(f, (-3,3), plot_points=300)
    - sage: p.set_aspect_ratio(1)
    - sage: p.axes(False)
    - sage: p
    -
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    - Si l'on ne souhaite pas afficher les axes, on utilise la modificateur axes() avec la valeur False. L'opération inverse est possible avec la valeur True. - -
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    Réglages de couleurs pour les axes -

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    - sage: p = plot(f, (-3,3), plot_points=400)
    - sage: p.axes_color((1,0.5,0.2))
    - sage: p.tick_label_color('green')
    - sage: p
    -
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    - Il est possible d'influer sur la couleur des axes eux-mêmes avec la méthode axes_color et sur la couleur des chiffres de graduation avec la méthode tick_label_color(). - -
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    Fixer des labels pour les axes et ajuster leur couleur -

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    - sage: p = plot(f, (-3,3), plot_points=400)
    - sage: p.axes_labels(['ici x','ici y'])
    - sage: p.axes_label_color(hue(0.8))
    - sage: p
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    - Cela se fait très simplement avec les méthodes axes_label() et axes_label_color(). - -
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    Modifier la taille de police pour les labels d'axes et les graduations -

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    - sage: p = plot(f, (-3,3), plot_points=400)
    - sage: p.axes_labels(['ici x','ici y'])
    - sage: p.axes_label_color(hue(0.8))
    - sage: p.fontsize(15)
    - sage: p
    -
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    - C'est la méthode fontsize() qui permet de le faire. Le chiffre fourni a pour unité le point typographique et fixe la taille de la police pour les labels et les chiffres de graduation. - -
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    - -
    - -
    - - - -

    Sauvegarder un graphique dans un fichier informatique -

    - - - - - - - - - - - - - - - -
    -
    - sage: p = plot(f, (-3,3), plot_points=400)
    - sage: p.save('/tmp/fichier.png')  # ou .ps, .e
    -   ps, .svg
    -
    -
    -
    -
    - On fournit à la méthode save() une chaîne de caractère correspondant à un chemin valide sur le disque dur de l'ordinateur (système de fichier) en précisant le format souhaité grâce à l'extension. Quatre choix sont possibles : .png, .ps, .eps, .svg. - -
    - - - - - - - - - - - - - - - -
    -
    - sage: 
    -
    -
    -
    -
    - -
    - - - -

    Utiliser plusieurs objets graphiques -

    - - - -

    Superposer deux graphiques -

    - - - - - - - - - - - - - - - - -
    -
    - sage: x = var('x')
    - sage: g1 = plot(cos(20*x)*exp(-2*x), 0, 1, rgb
    -   color='red')
    - sage: g2 = plot(2*exp(-30*x) - exp(-3*x), 0, 1
    -   )
    - sage: g1+g2
    -
    -
    -
    -
    - {{ sage }} offre un moyen très simple de fusionner deux graphiques l'un sur l'autre pour obtenir un graphique unique : l'addition ! - -
    -
    - -
    - -
    - - - - - - - - - - - - - - - - - -
    -
    - sage: var('x')
    - sage: p1 = plot( cos(x), (-pi,pi), rgbcolor='r
    -   ed')
    - sage: p2 = plot( sin(x), (-pi,pi), linestyle='
    -   --')
    - sage: p1+p2
    -
    -
    -
    -
    - Et cela marche même si les types de traits sont différents - -
    -
    - -
    - -
    - - - -

    Superposer plusieurs graphiques -

    - - - - - - - - - - - - - - - - -
    -
    - sage: var('x')
    - sage: G = sum([plot(sin(n*x), (x,-2*pi, 2*pi),
    -    hue=n).plot() for n in [0,0.1,..1]])
    - sage: G
    -
    -
    -
    -
    - C'est avec la même simplicité que l'on peut superposer plusieurs tracés de courbes sur un même graphique.
    -C'est d'ailleurs une règle qui est valable pour tous les objets graphiques. - -
    -
    - -
    - -
    - - - -

    Agencer des objets graphiques pour n'en former qu'un -

    - - - - - - - - - - - - - - - - - -
    -
    - sage: x = var('x')
    - sage: g1 = plot(cos(20*x)*exp(-2*x), 0, 1)
    - sage: g2 = plot(2*exp(-30*x) - exp(-3*x), 0, 1
    -   )
    - sage: show(graphics_array([g1, g2], 2, 1), xmi
    -   n=0)
    -
    -
    -
    -
    - {{ sage }} permet de positionner des objets graphiques existant en ligne et/ou en colonne pour former un graphique unique structuré en tableau. On utilise pour cela la commande graphics_array() en plaçant tous les graphiques dans une liste, puis en indiquant le format du tableau souhaité (nombre de lignes puis nombre de colonnes). - -
    -
    - -
    - -
    - - - -

    Placer du texte sur un graphique -

    - - - - - - - - - - - - - - - - -
    -
    - sage: f = lambda x: (x-3)*(x-5)*(x-7)+40
    - sage: P = line([(2,0),(2,f(2))], rgbcolor=(0,0
    -   ,0))
    - sage: P += line([(8,0),(8,f(8))], rgbcolor=(0,
    -   0,0))
    - sage: P += polygon([(2,0),(2,f(2))] + [(x, f(x
    -   )) for x in [2,2.1,..,8]] + [(8,0),(2,0)
    -   ],  rgbcolor='#ffecc9')
    - sage: P += text("$\\int_{a}^b f(x) dx$", (5, 2
    -   0), fontsize=18, rgbcolor=(0,0,0))
    - sage: P += plot(f, 1, 8.5, thickness=3, rgbcol
    -   or='#a66c1f')
    - sage: P
    -
    -
    -
    -
    - Ceci peut se faire à l'aide de la commande text() qui prend en entrée une chaîne de caractères, contenant éventuellement des formules mathématiques simples. Les compositions LaTeX complexes ne seront malheureusement pas acceptées par cette commande. - -
    -
    - -
    - -
    - - - -

    Placer du texte au format LaTeX dans une feuille interactive -

    - - - - - - - - - - - - - - - -
    -
    - sage: jsmath(r'\left(\begin{array}{ll}1&2\\3&4
    -   \end{array}\right)')
    -
    -
       
    -   
    \left(\begin{arr - ay}{ll}1&2\\3&4\end{array}\right)
    < - /html> - -
    -
    - Ceci ne s'applique pas au rendu des objets de type Graphics et ne fonctionne que dans les cellules interactives (utilisant donc l'annotation @interact).
    -La commande jsmath() produit une image prise en charge par le navigateur web et qui s'affiche à la suite de la commande. Comme il ne s'agit pas d'un objet de type Graphics, on ne peut pas l'additionner à un graphe de courbe par exemple. Mais la commande jsmath() peut traiter beaucoup plus de code LaTeX que la commande text(). -
    -Il faut noter la présence du 'r' précédant la chaîne de caractère. Ce modificateur empêche Python de considérer le caractère \ comme un caractère d'échappement, laissant le code LaTeX intact. - -
    - - - - - - - - - - - - - - - - -
    -
    - sage: jsmath('\\left(\\begin{array}{ll}1&2\\\\
    -   3&4\\end{array}\\right)')
    -
    -
       
    -   
    \left(\begin{arr - ay}{ll}1&2\\3&4\end{array}\right)
    < - /html> - -
    -
    - Sans la présence du 'r' précédant la chaîne de caractère, il faut doubler tous les \. - -
    - - - - - - - - - - - - - - - - -
    -
    - sage: var('u')
    - sage: f(u) = sin(u)
    - sage: DLf = taylor(f, u, 0, 5)
    - sage: jsmath(r'%s'%latex(DLf))
    -
    -
       
    -   
    u \{\mapsto}\ u - \frac{ {u}^{3} }{6} + \frac{ {u}^{5} }{120}
    - -
    -
    - L'intérêt essentiel de la commande jsmath() est de permettre l'affichage du résultat d'un calcul effectué par {{ sage }} avec une mise en forme LaTeX. C'est d'autant plus intéressant que dans une cellule interactive, le résultat n'est pas connu à l'avance ! - -
    - - - - - - - - - - - - - - - - - -
    -
    - sage: text('$%s$'%latex(DLf),(0.5,0.5),fontsiz
    -   e=18)
    -
    -
    -
    -
    - Ici, la commande text() ne produit pas le bon résultat. - -
    -
    - -
    - -
    - - - -

    Placer du code LaTeX dans une cellule

    - -
    Il est possible de remplir une cellule de code LaTeX. Il suffit de commencer par %latex. -
    - -\{\{\{id=32| -%latex -The equation $y^2 = x^3 + x$ defines an elliptic curve. -We have $2006 = \sage{factor(2006)}$. -/// -\}\}\} - -\{\{\{id=35| - -/// -\}\}\} -
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a/src/fr/61/W_UE_Definir_une_fonction_v03.html +++ /dev/null @@ -1,639 +0,0 @@ -{% extends 'fr/_fr.html' %} -{% set title = 'Définir une fonction dans {{ sage }}' %} - -{% block content %} -

    Définir une fonction dans {{ sage }} -

    - -Plan : - - - - - - - - - -

    Désignation d'une fonction -

    - - -
    - Il existe plusieurs façons de définir une fonction dans {{ sage }}. Nous allons présenter chacune d'elles et exposer les avantages et inconvénients, selon qu'il s'agit d'une fonction d'une ou plusieurs variables, et que la fonction comporte ou non des paramètres. -
    - - -

    Le cas particulier des fonctions usuelles -

    - - - -

    Fonctions usuelles -

    - - - - - - - - - - - - - - - -
    -
    - sage: plot(atan,(-2,2))
    -
    -
    -
    -
    - - Les fonctions usuelles existent en tant qu'objet dans {{ sage }} et peuvent être utilisées sans indiquer de variable. - -
    -
    - -
    - -
    - - -
    Les fonctions usuelles se notent : -abs, -acos ou arccos, -acosh ou arccosh, -acot ou arccot, -acoth ou arccoth, -acsc ou arccsc, -acsch ou arccsch, -asec ou arcsec, -asech ou arcsech, -asin ou arcsin, -asinh ou arcsinh, -atan ou arctan, -atanh ou arctanh, -ceil, -cos, -cosh, -cot, -coth, -csc, -csch, -erf, -exp, -floor, -log, -sec, -sech, -sin, -sinh, -sqrt, -tan, -tanh. -
    - - - - -

    Fonction d'une variable, sans paramètre -

    - - -
    - Remarque : il faut déclarer au préalable la variable utilisée. -
    - - -

    Utilisation d'une expression pour manipuler la fonction -

    - - - - - - - - - - - - - - - -
    -
    -
    - sage: var('t')
    -
    - sage: f = 2^t
    -
    - sage: plot(f)
    -
    -
    -
    -
    - - La fonction peut être utilisée comme une expression. Dans ce cas, la variable ne figure pas dans le nom de la fonction. - -
    -
    - -
    - -
    - - - -

    Définition fonctionnelle abrégée -

    - - - - - - - - - - - - - - - -
    -
    - sage: var('x')
    - sage: f(x) = x^2 - 0.5
    - sage: plot(f)
    -
    -
    -
    -
    - - La fonction peut être définie de façon implicite comme application en faisant figurer dans son nom la variable choisie, entre parenthèse, comme dans la notation mathématique $$ f(x) $$. - -
    -
    - -
    - -
    - - - -

    Définition fonctionnelle explicite -

    - - - - - - - - - - - - - - - -
    -
    - sage: g = lambda x: -x^2 + 0.3*cos(20*x)
    - sage: plot(g)
    -
    -
    -
    -
    - - La fonction peut être définie de façon explicite comme un objet dont l'évaluation nécessite un élément en entrée. On utilise pour cela le mot clef lambda de Python. - -
    -
    - -
    - -
    - - - -

    Fonction d'une variable avec paramètre -

    - - - -

    Définition fonctionnelle explicite -

    - - - - - - - - - - - - - - - -
    -
    - sage: var('k, t')
    - sage: f = lambda t: k^t
    - sage: k = 0.5
    - sage: p1 = plot(f(t), (0.2,3), rgbcolor='red')
    -   
    - sage: k = 2
    - sage: p2 = plot(f(t), (0.2,3), rgbcolor='purpl
    -   e')
    - sage: show(p1 + p2)
    -
    -
    -
    -
    - - C'est la méthode la moins ambiguë pour distinguer la variable du paramètre. On constate que dans notre exemple, les modifications du paramètre sont bien prises en compte. - -
    -
    - -
    - -
    - - - -

    Définition par une expression -

    - - - - - - - - - - - - - - - -
    -
    - sage: k = 2  # DANGER
    - sage: var('t')
    - sage: f = k^t
    - sage: k = 0.5
    - sage: plot(f,(t,0.2,3))
    -
    -
    -
    -
    - - C'est une définition qui comporte des risques et que nous déconseillons.
    - Premier piège : le paramètre $$ k $$ a déjà une valeur lors de la définition de $$ f $$. Celle-ci n'est plus modifiable par la suite. - -
    -
    - -
    - -
    - - - - - - - - - - - - - - - - - -
    -
    - sage: k = 2  # DANGER
    - sage: var('t, k')
    - sage: f = k^t
    - sage: k = 0.5
    - sage: plot(f(t),(t,0.2,3))
    -
    -
    -
    -
    - - Deuxième piège : bien que le paramètre $$ k $$ soit explicitement déclaré comme étant une inconnue, la définition de $$ f $$ ne précise pas qui de $$ k $$ et de $$ t $$ est la variable. {{ sage }} considère alors qu'il s'agit de la première inconnue rencontrée. De ce fait, l'expression $$ f(t) $$ dans le plot devient $$ t^t $$, ce qui justifie le tracé mais surprendra certainement l'utilisateur. - -
    -
    - -
    - -
    - - - -

    Fonction de plusieurs variables -

    - - - -

    Définition fonctionnelle explicite -

    - - - - - - - - - - - - - - - -
    -
    - sage: var('u, v')
    - sage: h = lambda u,v: u^2 + 2*v^2
    - sage: plot3d(h, (u,-1,1), (v,-1,1)).show(viewe
    -   r='tachyon',figsize=[6,6])
    -
    -
    -
    -
    - - Cette notation généralise celle rencontrée pour une variable.
    - Remarque : les deux variables ne doivent pas être encadrées par des parenthèses ! - -
    -
    - -
    - -
    - - - - - - - - - - - - - - - - -
    -
    - sage: var('u, v')
    - sage: h = lambda u,v: u^2 + 2*v^2
    - sage: h(2,1)
    -
    -
       6
    -
    -
    -
    - - La distinction entre les deux variables est parfaite. - -
    - - - -

    Définition fonctionnelle abrégée -

    - - - - - - - - - - - - - - - -
    -
    - sage: var('u, v')
    - sage: h(u,v) = u^2 - v^2
    - sage: plot3d(h, (u,-1,1), (v,-1,1)).show(viewe
    -   r='tachyon',figsize=[6,6])
    -
    -
    -
    -
    - - Lorsque la fonction ne comporte pas de paramètres, ce mode de définition ne présente aucun danger. C'est certainement ce qui s'approche le plus des notations mathématiques standards. - -
    -
    - -
    - -
    - - - -

    Définition par une expression -

    - - - - - - - - - - - - - - - -
    -
    - sage: var('u, v')
    - sage: f1 = (4+(3+cos(v))*sin(u), 4+(3+cos(v))*
    -   cos(u), 4+sin(v))
    - sage: f2 = (8+(3+cos(v))*cos(u), 3+sin(v), 4+(
    -   3+cos(v))*sin(u))
    - sage: p1 = parametric_plot3d(f1, (u,0,2*pi), (
    -   v,0,2*pi), color="red", opacity=0.5, asp
    -   ect_ratio=[1,1,1])
    - sage: p2 = parametric_plot3d(f2, (u,0,2*pi), (
    -   v,0,2*pi), color="blue", opacity=0.5)
    - sage: (p1 + p2).show(viewer='tachyon',figsize=
    -   [6,6])
    -
    -
    -
    -
    - - Encore une fois, nous déconseillons vivement l'utilisation de ce mode de définition de fonctions.
    - Parfois, l'absence de paramètre permet l'utilisation de cette notation pour les tracés de surfaces, à condition de préciser la variable affectée à chaque intervalle. - -
    -
    - -
    - -
    - - - - - - - - - - - - - - - - -
    -
    - sage: var('u, v')
    - sage: f1 = (4+(3+cos(v))*sin(u), 4+(3+cos(v))*
    -   cos(u), 4+sin(v))
    - sage: f1(0,0)
    -
    -
       Traceback (most recent call last):
    -     File "", line 1, in 
    -     File "/home/phil/.sage/sage_notebook/w
    -   orksheets/admin/61/code/11.py", line 8, 
    -   in 
    -       exec compile(ur'f1(Integer(0),Intege
    -   r(0))' + '\n', '', 'single')
    -     File "/donnees/SAGE-MATH/sage-3.1.1-de
    -   bian32-intel-i686-Linux/local/lib/python
    -   2.5/site-packages/sympy/thirdparty/pygle
    -   t/", line 1, in 
    -       
    -   TypeError: 'tuple' object is not callabl
    -   e
    -
    -
    -
    - - Mais il devient impossible de distinguer une variable d'une autre et, à cause de cette ambiguïté, certains calculs simples sont interdits. - -
    - - - - - - - - - - - - - - - -
    -
    - sage: 
    -
    -
    -
    -
    - -
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z+P8bdK=-r=&Rd`w>q~^~MB#tazITh!h{y^p&e(Um;(1~87}TQ$zzc)Xe*ip1&wPQf zZ#$KbpIK#-{jS?|h}E6>?zh&I=YOQxz>LQn_`i(zS|_Z6>VH=NZg#q@fB{*idGlrf zSQ&4@kVsRBzvN0uinNX#19(@hg%rFkMTH$rn;Eylce{a16utYNnu(d&5>yo&9AgnN z4|1CBF(HX?y@Hah%;ukw)juPFnjre_);^&u_1bmjn>1KQj*EaBI63TcXOXr8N`cmq zJ5&E^Iajc!MU6h}=^f*!L7iqeVDwdHxzCt%*z7)JB1V{dV|+3){SUYmG~y=G($f_S z%vx|RE@fcc*hG$spnBl4;03GGJz#oOi-KYu`Q(3Vu+l@+eh33?Qh6-2%Cnp^B%r14 zjx1d(C~HpklS-!ZW8iEVgk&TaSPoc5U0;MT>hDn+)UOWb*V6g<#{UB_!^DqTdf7Ip z(u%q*S^Kj}s_#1@)E2IF$ATnc_K$oibp6JTi=8&GBt0UCsDMGG zp3m+BA@^{%PHoK}{g-nRoy1uKXM5UL_h0`iMcpdqx?nD-TYLi~y;cS;%K-`&mVKr7 z%RwTcOZzu2yD)HQ=f67)2@{Q;F3{B0PKCUp3q9L2M3ito0%uJynP9R*0gTracer des graphes de fonctions avec {{ sage }} - -

    auteur : Philippe Saadé (2008), psaade@gmail.com -

    -

    - - -
    - Nous allons présenter l'utilisation de {{ sage }} pour la représentation graphique de fonctions d'une variable. -
    -

    - - - -Plan : -
    - - - -

    - - - - - -

    Un tracé sans option -

    - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp)
    -
    -
    -
    -
    - - Voici un moyen direct de tracer une partie du graphe de la fonction exponentielle. On constate que l'intervalle par défaut utilisé pour le tracé est l'intervalle [-1,1].
    - On constate également la présence des axes et des marques sur les axes, qui se prolongent un peu en dehors de l'intervalle [-1,1], pour faciliter la lecture du graphique. - - -
    -
    - -
    - -
    - - -
    - Nous allons voir dans ce tutoriel comment ajuster le tracé en fonction de nos besoins. -
    - - - - -

    Options de bases -

    - - - -

    Choix de l'intervalle pour le tracé de la fonction -

    - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp,(-2,3))
    -
    -
    -
    -
    - - L'intervalle est fourni sous forme d'un couple : - -
    -
    - -
    - -
    - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp,-2,3)
    -
    -
    -
    -
    - - ou bien en indiquant les bornes à la suite de la fonction. - -
    -
    - -
    - -
    - - - -

    Affichage des options propres à plot -

    - - - - - - - - - - - - - - - -
    -
    - sage: plot.options
    -
    -
       {'plot_points': 200, 'thickness': 1, 'pl
    -   ot_division': 1000, 'rgbcolor': (0, 0, 1
    -   ), 'max_bend': 0.10000000000000001, 'alp
    -   ha': 1}
    -
    -
    -
    - - Il est facile d'obtenir la liste des options directement attachées à la commande plot ainsi que leurs valeurs par défaut.
    - Remarque : beaucoup d'autres options existes pour contrôler le rendu graphique, comme nous allons le voir ici, mais elles ne sont pas directement attachées à la commande plot. Celles-ci dépendent du type d'objet tracé. - -
    - - - -

    Réglage de l'épaisseur du trait lors du tracé -

    - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp,thickness=3.5)
    -
    -
    -
    -
    - Il suffit d'utiliser l'option thickness de la commande plot. - -
    -
    - -
    - -
    - - - -

    Réglage de la couleur du trait lors du tracé -

    - - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp, rgbcolor='red')  # ou (1,0,0) 
    -   ou '#FF0000'
    -
    -
    -
    -
    - Il suffit d'utiliser l'option rbgcolor de la commande plot.
    -Ici, il est possible de fournir la couleur par son nom (en anglais et entre '...'), par son code RGB (valeurs entre 0 et 1) ou bien par un code hexadécimal de type HTML. Les couleurs possibles sont : "red", "orange", "yellow", "green", "blue", "purple", "white", "black", "grey". - -
    -
    - -
    - -
    - - - -

    Réglage de la transparence du trait lors du tracé -

    - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp,alpha=0.5)
    -
    -
    -
    -
    - - Il suffit d'utiliser l'option alpha de la commande plot. Il s'agit d'une valeur entre 0 et 1 : pour 1, le trait est pleinement opaque. Pour 0, il est invisible. - -
    -
    - -
    - -
    - - - -

    Ajuster le nombre de point pour le tracé du graphe -

    - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp,plot_points=3,plot_division=4, 
    -   marker='+',markeredgecolor='red')
    -
    -
    -
    -
    - - Les options plot_points, plot_division et max_bend permettent, respectivement, de choisir le nombre de points calculés initialement sur la courbe, le nombre maximal de points qui peuvent être ajouter par {{ sage }} pour lisser le tracé et d'imposer une contrainte plus ou moins forte à la qualité du rendu.
    - Remarque : dans une prochaine version de {{ sage }}, ces deux dernières options seront remplacées par les options adaptive_recursion et adaptive_tolerance qui reposeront sur un algorithme récursif plus efficace.
    - Dans notre exemple, les points ont été indiqués par une croix rouge. - -
    -
    - -
    - -
    - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp, plot_points=3, plot_division=4
    -   , max_bend=10, marker='+', markeredgecol
    -   or='red')
    -
    -
    -
    -
    - - Une grande tolérance sur le paramètre max_bend fait en sorte que {{ sage }} ne rajoute aucun point pour affiner le tracé. - -
    -
    - -
    - -
    - - - -

    Options avancées contrôlant le tracé -

    - -
    - Nous allons présenter les options qui permettent de contrôler le type de trait produit et la nature des points. -
    - - -

    Choix de la couleur par la désignation d'une teinte -

    - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp, hue=0.3)
    -
    -
    -
    -
    - - Au lieu d'utiliser l'option rgbcolor de la commande plot, il est possible de choisir une couleur par sa teinte (hue en anglais). L'arc en ciel des couleurs est parcouru de façon circulaire, de rouge à rouge, par un paramètre réel allant de 0 à 1. - -
    -
    - -
    - -
    - - - -

    Choix d'un style de ligne -

    - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp, linestyle='-.')
    -
    -
    -
    -
    - - Par défaut, le tracé se fait à l'aide d'un trait continu. Il est possible de choisir d'autres type de traits en utilisant l'option linestyle. Voici la liste complète des valeurs possibles ainsi que le résultat obtenu : - - - - - - - - - - - - - - - - - - - - - - - - - -
    '--'pointillé
    '-.'mixte
    '-'continu
    'steps'en escalier
    ':'filet de points
    - -
    -
    - -
    - -
    - - - -

    Choix du type de marqueur de points -

    - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp, marker='>', plot_points=10, ma
    -   x_bend=10)
    -
    -
    -
    -
    - - Le tracé du graphe de la fonction se fait à partir d'une liste de points répartis sur l'intervalle parcouru par la variable. Même si ces points ont tendance à disparaître sous la ligne continue de segments qui les relie, il n'en est pas moins possible de les utiliser et, en particulier de les afficher en choisissant un marqueur.
    - Pour ce faire, on utilise l'option marker. La valeur que peut prendre cette option est à prendre parmi la liste suivante : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    '0'tickleft
    '1'tickright
    '2'tickup
    '3'tickdown
    ''nothing
    ' 'nothing
    '+'plus
    ','pixel
    '.'point
    '1'tri_down
    '3'tri_left
    '2'tri_up
    '4'tri_right
    '<'triangle_left
    '>'triangle_right
    'None'nothing
    'D'diamond
    'H'hexagon2
    '_'hline
    '^'triangle_up
    'd'thin_diamond
    'h'hexagon1
    'o'circle
    'p'pentagon
    's'square
    'v'triangle_down
    'x'x
    '|'vline
    - -
    -
    - -
    - -
    - - - -

    Choix de la couleur pour les marqueurs de points -

    - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp, marker='o', markeredgecolor='b
    -   lack', plot_points=10, max_bend=10, rgbc
    -   olor="yellow")
    -
    -
    -
    -
    - - L'option markeredgecolor permet de choisir une couleur pour le tracé des marqueurs. Il faut noter que cette couleur ne s'applique qu'au trait dessinant le marqueur et que lorsqu'un marqueur possède une surface de fond, celle-ci est de la même couleur que le trait de la courbe. - -
    -
    - -
    - -
    - - - -

    Choix de la taille des marqueurs de points -

    - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp, marker='^', markeredgecolor='r
    -   ed', markersize=15, plot_points=10, max_
    -   bend=10)
    -
    -
    -
    -
    - - L'option markersize permet de fixer la taille des marqueurs (en pixel). - -
    -
    - -
    - -
    - - - -

    Choix de l'épaisseur du trait des marqueurs de points -

    - - - - - - - - - - - - - - - - - -
    -
    - sage: plot(exp, marker='p', markeredgecolor='r
    -   ed', markeredgewidth=4, markersize=30, p
    -   lot_points=10, max_bend=10)
    -
    -
    -
    -
    - - L'option markeredgewidth permet de fixer l'épaisseur du trait dessinant les marqueurs (en pixel). - -
    -
    - -
    - -
    - - - - - - - - - - - - - - - -
    -
    - sage: 
    -
    -
    -
    -
    - -
    - - - - - - - - - - - - - - - -
    -
    - sage: 
    -
    -
    -
    -
    - -
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  • Index
  • -
  • Télécharger
  • -
  • Documentation
  • -
  • Components
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    Components

    - -
    - -
      -
    • ATLAS: Librairie d'algèbre linéaire configurée automatiquement
    • - -
    • BLAS: Routines d'algèbre linéaire en Fortran 77
    • - -
    • Bzip2: Compression de données
    • - -
    • Cddlib: Méthode de double description de Motzkin
    • - -
    • Common Lisp: Un langage de programmation multiparadigme et généraliste
    • - -
    • CVXOPT: Optimisation convexe, programmation linéaire, moindres carrés, etc.
    • - -
    • Cython: Extensions en C pour Python
    • - -
    • Docutils: un projet open-source pour convertir des formats textes dans des formats pratiques comme HTML ou LaTeX. Il inclut en particulier reStructuredText, le format facile à lire, facile à utiliser et wysiwyg.
    • - -
    • mwrank: Logiciel pour le calcul des groupes de Mordell-Weil groups des courbes elliptiques sur Q via 2-descente. Depuis Novembre 2007, mwrank fait parti du package eclib qui est inclus dans {{ sage }}.
    • - -
    • F2c: Converti le code Fortran 77 en C
    • - -
    • Flint: Librairie rapide pour la théorie des nombres (Fast Library for Number Theory)
    • - -
    • FpLLL: Réduction des réseaux euclidien
    • - -
    • FreeType: Un logiciel de police libre, de qualité et portable
    • - -
    • G95: Compilateur Fortran 95
    • - -
    • GAP: Un logiciel d'algèbre (Groups, Algorithms, Programming)
    • - -
    • GD: Une librairie de génération dynamique de graphique
    • - -
    • Genus2reduction: Librairie pour le calcul des réductions des courbes hyperelliptiques de genre 2.
    • - -
    • Gfan: Bases de Gröbner et variétés tropicales
    • - -
    • Givaro: Librairie C++ pour l'arithmétique et l'algèbre
    • - -
    • GMP-ECM: Factorisation des entiers par l'utilisation de courbes elliptiques
    • -
    • GNU TLS: Réseaux sécurisés
    • - -
    • GSL: La librairie scientifique Gnu
    • - -
    • Jinja: Un engin de template
    • - -
    • JsMath: Une implémentation JavaScript de LaTeX
    • - -
    • IML: Librairie pour les matrices à coefficients entiers
    • - -
    • IPython: Un shell Python interactif
    • - -
    • LAPACK: Algèbre linéaire en Fortran 77
    • - -
    • Lcalc: calculs des fonctions-L
    • - -
    • Libgcrypt: librairie de cryptographie généraliste
    • - -
    • Libgpg-error: Gestions des erreurs pour les composants GnuPG
    • - -
    • libpng: Support des images bitmap
    • - -
    • Linbox: Librairie d'algèbre linéaire en C++
    • - -
    • M4RI: Algèbre linéaire sur le corps fini GF(2)
    • - -
    • Matplotlib: Librairie Python pour le graphisme
    • - -
    • Maxima: Logiciel de calcul formel
    • - -
    • Mercurial: Un gestionnaire de version libre
    • - -
    • MoinMoin Wiki: Un wiki écrit en python
    • - -
    • MPFI: Librairie de précisions multiples pour les réels.
    • - -
    • MPFR: Librairie C pour les flottants en précision multiple avec correction des arrondis.
    • - -
    • MPIR: Librairie pour les entiers et les rationnels en précision arbitraire.
    • - -
    • ECLib:Cremona's Programme pour l'énumération des courbes elliptiques
    • - -
    • NetworkX: Une librairie Python pour la théorie des graphes.
    • - -
    • NTL: Une librairie C++ de théorie des nombres
    • - -
    • Numpy: Librairie python d'algèbre linéaire numérique
    • - -
    • OpenCDK: Un logiciel de cryptographie (Open Crypto Development Kit)
    • - -
    • OpenOpt: Intégrations de solveurs pour l'optimisation numérique en Python.
    • - -
    • PALP: Une librairie pour l'analyse des polytopes des réseaux
    • - -
    • PARI/GP: Un logiciel de théorie des nombres
    • - -
    • Pexpect: Contrôle pseudo-tty pour python (émulation de terminaux)
    • - -
    • PolyBoRi: Librairie pour les polynômes sur les anneaux booléens
    • - -
    • PyCrypto: Une librairie python de cryptographie (Python Cryptography Toolkit)
    • - -
    • Python: Un langage interprété
    • - -
    • Pynac: Manipulation symbolique avec des objets Python basé sur GiNaC
    • - -
    • Qd: Librairie pour le calcul Quad-double/Double-double
    • - -
    • R: Un logiciel de statistiques
    • - -
    • Readline: Librairie Gnu pour l'édition
    • - -
    • Rpy: Interface Python pour le logiciel de statistiques R
    • - -
    • Scipy: Librairie Python pour le calcul scientifique
    • - -
    • Singular: Logiciel pour l'algèbre commutative et non-commutative.
    • - -
    • Scons: Un logiciel pour construire d'autres logiciels (Software CONstruction Tool)
    • - -
    • Sphinx: Générateur de documentation pour les fichiers python
    • - -
    • SQLAlchemy: Librairie Python pour les bases de données SQL et pour le Mapping objet-relationnel
    • - -
    • SQLite: Base de données relationnelle
    • - -
    • Sympow: Logiciel pour le calcul des fonctions-L
    • - -
    • Symmetrica: Théorie des représentations des groupes
    • - -
    • Sympy: Librairie Python pour le calcul symbolique
    • - -
    • mpmath: Une librairie Python pour les flotants en précision arbitraire
    • - - -
    • Tachyon: logiciel léger pour le dessin 3d
    • - -
    • Termcap: Simplification de l'écriture de solutions portables en mode texte
    • - -
    • Twisted: Librairie Python pour les réseaux
    • - -
    • Weave: Des outils pour inclure du C/C++ dans du code Python
    • - -
    • Zlib: Compression de données
    • - -
    • ZODB: Une base de donnée orientée objet
    • -
    - - -
    -{% endblock %} diff --git a/src/fr/documentation.html b/src/fr/documentation.html deleted file mode 100755 index aa1f09af..00000000 --- a/src/fr/documentation.html +++ /dev/null @@ -1,35 +0,0 @@ -{% extends 'fr/_fr.html' %} -{% set title = 'Documentation' %} - -{% block content %} -

    Documentation

    - -
    - -
    L'intégralité de la documentation de {{ sage }} est accessible ici. - Vous pouvez aussi aller faire un tour sur le wiki {{ sage }} pour - l'éducation et participer à la liste de - discussion {{ sage }} et enseignement (sur laquelle toute les questions en - français sont les bienvenues).
    - -
    Quelques tutoriels en français sont également disponibles :
    - -
    - - -{% endblock %} diff --git a/src/fr/index.html b/src/fr/index.html deleted file mode 100755 index 6c626b03..00000000 --- a/src/fr/index.html +++ /dev/null @@ -1,67 +0,0 @@ -{% extends 'fr/_fr.html' %} -{% set title = 'Français' %} - -{% block content %} -

    {{ sage }} Français

    - -
    - -
    -{{ sage }} est un logiciel libre de mathématiques sous licence GPL. Il -combine la puissance de nombreux programmes libres dans une interface commune -basée sur le langage de programmation Python.
    - -
    -Mission: Création d'une alternative viable libre et open source à Magma, - Maple, Mathematica et Matlab. -
    - -
    -{{ sage }} permet de faire des mathématiques générales et avancées, pures et -appliquées. Il couvre une vaste gamme de mathématiques, dont l'algèbre, -l'analyse, la théorie des nombres, la cryptographie, l'analyse numérique, -l'algèbre commutative, la théorie des groupes, la combinatoire, la théorie des -graphes, l'algèbre linéaire formelle, etc ...

    -Il permet l'utilisation simultanée et transparente de -dizaines de logiciels spécialisés. Il est -conçu pour l'éducation ou les études autant que pour la recherche.

    -{{ sage }} a deux modes d'utilisation, un mode bloc-note (ou notebook) dont -l'interface est un navigateur web et un mode ligne de commande. Avec le -mode bloc-note, vous n'êtes pas obligé d'installer {{ sage }} sur votre ordinateur : -{{ sage }} fonctionne en mode client serveur et peut fonctionner sur un réseau (par -exemple internet).

    Grâce à {{ sage }}, il est possible de partager ou publier -simplement vos feuilles de calcul, figures et graphiques en utilisant les -fonctionnalités du bloc-note.
    - -
    -{{ sage }} est principalement disponible en anglais. Une partie de sa -documentation a cependant été traduite en français: - -L'intégralité de la documentation en anglais est accessible -ici. Elle regroupe les versions originales (parfois plus à jour que -leurs traductions) de ces documents, ainsi que plusieurs manuels qui -n'ont pas encore été traduits: Constructions, -Manuel de référence, Guide du développeur... -
    - - - -
    -Voir aussi le livre électronique en français - Calcul mathématique avec {{ sage }}.
    - - - -
    - - -{% endblock %} diff --git a/src/fr/telecharger.html b/src/fr/telecharger.html deleted file mode 100755 index 439492a5..00000000 --- a/src/fr/telecharger.html +++ /dev/null @@ -1,16 +0,0 @@ -{% extends 'fr/_fr.html' %} -{% set title = 'Télécharger' %} - -{% block content %} -

    Télécharger {{ sage }}

    - - -{% endblock %} \ No newline at end of file diff --git a/src/fr/tour-plotting-fr.html b/src/fr/tour-plotting-fr.html deleted file mode 100755 index 779ffb03..00000000 --- a/src/fr/tour-plotting-fr.html +++ /dev/null @@ -1,645 +0,0 @@ -{% extends 'fr/_fr.html' %} -{% set title = "Désignation d'une fonction" %} - -{% block content %} -

    Désignation d'une fonction

    - -
    - Il existe plusieurs façons de définir une fonction dans {{ sage }}. Nous allons présenter chacune d'elles et exposer les avantages et inconvénients, selon qu'il s'agit d'une fonction d'une ou plusieurs variables, et que la fonction comporte ou non des paramètres. -
    - -

    Le cas particulier des fonctions usuelles

    - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    -

    Fonctions usuelles

    -
    -
    sage: plot(atan,(-2,2))
    -
    -
    - -
    -
    - Les fonctions usuelles existent en tant qu'objet dans {{ sage }} et peuvent être utilisées sans indiquer de variable. -
    - -
    - -
    - - Tracé de la fonction tangente, sans variable -
    - - -
    - - -
    - Les fonctions usuelles se notent : -abs, -acos ou arccos, -acosh ou arccosh, -acot ou arccot, -acoth ou arccoth, -acsc ou arccsc, -acsch ou arccsch, -asec ou arcsec, -asech ou arcsech, -asin ou arcsin, -asinh ou arcsinh, -atan ou arctan, -atanh ou arctanh, -ceil, -cos, -cosh, -cot, -coth, -csc, -csch, -erf, -exp, -floor, -log, -sec, -sech, -sin, -sinh, -sqrt, -tan, -tanh. -
    - - - - - -

    Fonction d'une variable, sans paramètre

    -
    - Remarque : il faut déclarer au préalable la variable utilisée. -
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    -

    Utilisation d'une expression pour manipuler la fonction

    -
    -
    sage: var('t')
    -sage: f = 2^t
    -sage: plot(f)
    -
    -
    - -
    -
    - La fonction peut être utilisée comme une expression. Dans ce cas, la variable ne figure pas dans le nom de la fonction. -
    - -
    - -
    - Fonction d'une  variable définie par une expression -
    - - -
    -

    Définition fonctionnelle abrégée

    -
    -
    sage: var('x')
    -sage: f(x) = x^2 - 0.5
    -sage: plot(f)
    -
    -
    - -
    -
    - La fonction peut être définie de façon implicite comme application en faisant figurer dans son nom la variable choisie, entre parenthèse, comme dans la notation mathématique $$ f(x) $$. -
    - -
    - -
    - Définition fonctionnelle abrégée d'une fonction d'une variable -
    - - -
    -

    Définition fonctionnelle explicite

    -
    -
    sage: g = lambda x: -x^2 + 0.3*cos(20*x)
    -sage: plot(g)
    -
    -
    - -
    -
    - La fonction peut être définie de façon explicite comme un objet dont l'évaluation nécessite un élément en entrée. On utilise pour cela le mot clef lambda de Python. -
    - -
    - -
    - Définition fonctionnelle explicite d'une fonction d'une variable -
    - - -
    - - - -

    Fonction d'une variable avec paramètre

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    -

    Définition fonctionnelle explicite

    -
    -
    sage: var('k, t')
    -sage: f = lambda t: k^t
    -sage: k = 0.5
    -sage: p1 = plot(f(t), (0.2,3), rgbcolor='red')
    -sage: k = 2
    -sage: p2 = plot(f(t), (0.2,3), rgbcolor='purple')
    -sage: show(p1 + p2)
    -
    - -
    -
    - C'est la méthode la moins ambiguë pour distinguer la variable du paramètre. On constate que dans notre exemple, les modifications du paramètre sont bien prises en compte. -
    - -
    - -
    - Définition fonctionnelle explicite d'une fonction d'une variable avec paramètre -
    - - -
    -

    Définition par une expression

    -
    -
    sage: k = 2  # DANGER
    -sage: var('t')
    -sage: f = k^t
    -sage: k = 0.5
    -sage: plot(f,(t,0.2,3))
    -
    - -
    -
    - C'est une définition qui comporte des risques et que nous déconseillons.
    - Premier piège : le paramètre $$k$$ a déjà une valeur lors de la définition de $$f$$. Celle-ci n'est plus modifiable par la suite. -
    - -
    - -
    - -
    - - -
    -
    sage: k = 2  # DANGER
    -sage: var('t, k')
    -sage: f = k^t
    -sage: k = 0.5
    -sage: plot(f(t),(t,0.2,3))
    -
    - -
    -
    - Deuxième piège : bien que le paramètre $$k$$ soit explicitement déclaré comme étant une inconnue, la définition de $$f$$ ne précise pas qui de $$k$$ et de $$t$$ est la variable. {{ sage }} considère alors qu'il s'agit de la première inconnue rencontrée. De ce fait, l'expression $$f(t)$$ dans le plot devient $$t^t$$, ce qui justifie le tracé mais surprendra certainement l'utilisateur. -
    - -
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    Fonction de plusieurs variables

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    -

    Définition fonctionnelle explicite

    -
    -
    sage: var('u, v')
    -sage: h = lambda u,v: u^2 + 2*v^2
    -sage: plot3d(h, (u,-1,1), (v,-1,1))
    -
    - -
    -
    - Cette notation généralise celle rencontrée pour une variable.
    - Remarque : les deux variables ne doivent pas être encadrées par des parenthèses ! -
    - -
    - -
    - -
    - - -
    -
    sage: var('u, v')
    -sage: h = lambda u,v: u^2 + 2*v^2
    -sage: h(2,1)
    -6
    -
    - -
    -
    - La distinction entre les deux variables est parfaite. -
    - -
    -

    Définition fonctionnelle abrégée

    -
    -
    sage: var('u, v')
    -sage: h(u,v) = u^2 - v^2
    -sage: plot3d(h, (u,-1,1), (v,-1,1))
    -
    - -
    -
    - Lorsque la fonction ne comporte pas de paramètres, ce mode de définition ne présente aucun danger. C'est certainement ce qui s'approche le plus des notations mathématiques standards. -
    - -
    - -
    - -
    - - -
    -

    Définition par une expression

    -
    -
    sage: var('u, v')
    -sage: f1=(4+(3+cos(v))*sin(u),
    -     4+(3+cos(v))*cos(u), 4+sin(v))
    -sage: f2=(8+(3+cos(v))*cos(u),
    -     3+sin(v), 4+(3+cos(v))*sin(u))
    -sage: p1=parametric_plot3d(f1,(u,0,2*pi),(v,0,2*pi),
    -     color="red", opacity=0.5, aspect_ratio=[1,1,1])
    -sage: p2=parametric_plot3d(f2,(u,0,2*pi),(v,0,2*pi),
    -     color="blue", opacity=0.5)
    -sage: p1+p2
    -
    - -
    -
    - -
    - Encore une fois, nous déconseillons vivement l'utilisation de ce mode de définition de fonctions.
    - Parfois, l'absence de paramètre permet l'utilisation de cette notation pour les tracés de surfaces, à condition de préciser la variable affectée à chaque intervalle. -
    - -
    - -
    - - -
    -
    sage: var('u, v')
    -sage: f1=(4+(3+cos(v))*sin(u),
    -    4+(3+cos(v))*cos(u),4+sin(v))
    -sage: f1(0,0)
    -Traceback (most recent call last): - File "<stdin>", line 1, in <module> - File "/home/phil/.sage/sage_notebook/worksheets/admin/30/code/36.py", line 8, in <module> - exec compile(ur'f1(Integer(0),Integer(0))' + '\n', '', 'single') - File "/donnees/SAGE-MATH/sage-3.1.1-debian32-intel-i686-Linux/local/lib/python2.5/site-packages/sympy/thirdparty/pyglet/", line 1, in <module> - -TypeError: 'tuple' object is not callable - - -
    -
    - Mais il devient impossible de distinguer une variable d'une autre et, à cause de cette ambiguïté, certains calculs simples sont interdits. -
    - -
    -{% endblock %} \ No newline at end of file diff --git a/src/pt/index.html b/src/pt/index.html deleted file mode 100755 index d951dcb6..00000000 --- a/src/pt/index.html +++ /dev/null @@ -1,57 +0,0 @@ -{% extends 'base.html' %} -{% set title = "Portuguese" %} - -{% block logo_title %} -
    Português -{% endblock %} - -{% block header_text %} -back to {{ sage }} main page -{% endblock %} - -{% block menu %} -
  • Index
  • -{% endblock %} - - -{% block content %} -
    - -

    O que é {{ sage }}?

    - -

    {{ sage }} é um software matemático livre e de código aberto (open-source), -desenvolvido sob a licença GPL por uma comunidade de programadores e matemáticos, que busca -ser uma alternativa para os principais sistemas proprietários de software matemático como o -Magma, Maple, Mathematica e Matlab. Ele engloba e se utiliza de um grande número de pacotes -pré-existentes como Maxima, GAP, Pari/GP, softwares de renderização de imagens e muitos outros, -integrando-os em uma interface única que busca ser amigável e de fácil assimilação. -Todos os principais pacotes são instalados juntamente com o {{ sage }} e muitos outros pacotes existem -para extensões em áreas específicas. Por este motivo {{ sage }} é adequado para uso em ensino e pesquisa. -{{ sage }} pode ser utilizado por meio de comandos de linhas interativos ou de um Notebook, uma interface -acionada de dentro de um browser onde os passos são armazenados em páginas separadas por usuário. -Este Notebook pode estar conectado à instalação local do software ou remotamente por meio de rede -e internet e permite a criação de gráficos e expressões matemáticas bem renderizadas que podem ser -reutilizados, ampliados ou excluídos, e partilhados com outros usuários através da rede.

    - -

    Uma das vantagens evidentes de se utilizar software livre é a garantia de que qualquer outro -pesquisador ou estudante que receba páginas de {{ sage }} será capaz de exibí-las e manipulá-las sem -a necessidade de aquisição de software proprietário. Além disto qualquer pessoa interessada poderá -investigar, e alterar se necessário, o código fonte, situação bem diversa do que ocorre com o software -proprietário onde o usuário desconhece os algoritmos usados pelo programa.

    - -

    Visite a homepage de {{ sage }} ou sua documentação completa, em inglês.

    -

    Visite uma turnê pelo {{ sage }}, em português.

    -

    Um tutorial de {{ sage }}, em português.

    - - - - -
    -{% endblock %} diff --git a/src/ru/index.html b/src/ru/index.html deleted file mode 100755 index 97f957f9..00000000 --- a/src/ru/index.html +++ /dev/null @@ -1,93 +0,0 @@ -{% extends 'base.html' %} -{% set title = "русский" %} - -{% block logo_title %} -русский -{% endblock %} - -{% block header_text %} -back to {{ sage }} main page -{% endblock %} - -{% block menu %} -
  • Index
  • -
  • Documentation
  • -
  • Tutorial
  • -
  • PDF
  • -{% endblock %} - - -{% block content %} - -
    - -

    {{ sage }} (анг. 'Мудрец')

    -
    -{{ sage }} (анг. 'Мудрец') — система компьютерной алгебры покрывающая много областей математики, включая алгебру, комбинаторику, вычислительную математику и матанализ. -Первая версия {{ sage }} была выпущена 24 февраля 2005 года в виде свободного программного обеспечения с лицензией GNU GPL. -Первоначальной целью проекта было "создание открытого программного обеспечения альтернативного системам Magma, Maple, Mathematica, и MATLAB". -Разработчиком {{ sage }} является Уильям Стейн — математик Университета Вашингтона. Официальный сайт: sagemath.org. -
    - -

    Video Tutorials

    -credits: © 2010 Темирлан Кумаргажин, Рахим Давлеткалиев, Carleton University, Оттава, Канада
    - -

    S1

    - -
    - - - - - - - - -
    -

    S2

    - -
    - - - - - - - - -
    -

    S3

    - -
    - - - - - - - - -
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    S4

    - -
    - - - - - - - - -
    - -
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