{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "# Surfaces, continuité et courbes de niveau\n", "\n", "\n", "Soit $\\mathcal{D} \\subseteq \\mathbb{R}^n$ une région de l'espace (on peut penser que $n=2$), et $f:\\mathcal{D} \\to \\mathbb{R}$ une fonction. Soit en plus $\\bf{a}\\in \\mathcal{D}$. On dit que la fonction $f(\\bf{x})$ tend vers $L$ lorsque $\\bf{x}$ tend vers $\\bf{a}$ si pour tout réel $\\epsilon > 0 $ il existe un réel $\\delta >0$ tel que $|f(\\bf{x}) - L| < \\epsilon$ aussitôt que $|\\bf{x} - \\bf{a}| < \\delta$. Autrement dit, la distance entre $f(\\bf{x})$ et $L$ est aussi petite que l'on veut, pourvu que la distance entre $\\bf{x}$ et $\\bf{a}$ soit suffisament petite. On écrit dans ce cas $\\displaystyle \\lim_{\\bf{x} \\to \\bf{a}} f(\\bf{x} ) = L$\n", "\n", "La fonction $f$ est continue en $\\bf{a}$ si $\\displaystyle \\lim_{ \\bf{x} \\to \\bf{a} } f(x)= f( \\bf{a} )$. Ceci présuppose notamment que $\\bf{a}$ fait partie du domaine de $f$. Ceci signifie que les valeurs de $f$ ne varient pas brusquement lorsque $\\bf{x}$ varie aux alentours de $\\bf{a}$. Dans le cas où $n=2$ (deux variables) l'interprétation de ceci en termes de la surface $z=f(x,y)$ est essentiellement la même que celle de la contuité d'une fonction $y=f(x)$ en termes de sa courbe : **il n'y a pas de déchirures**. Par ailleurs, en plusieurs variables on dispose de l'outil supplémentaire qui sont les courbes de niveau.\n", "\n", "Voyons quelques exemples." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Exemple 0\n", "\n", "On considère le fonction $f(x,y) = \\frac{y}{x^2+y^2}$. Ci après on trouve ce qu'il faut pour tracer la surface, joliment coloriée et le diagramme des courbes de niveau." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": "\n\n" }, "execution_count": 1, "metadata": { }, "output_type": "execute_result" }, { "data": { "image/png": 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" }, "execution_count": 1, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('x,y')\n", "cm = colormaps.Blues\n", "def f(x,y) :\n", "\treturn y/(x^2 + y^2)\n", "\n", "def c(x,y):\n", "\treturn 0.5+0.7*y/(x^2 + y^2+0.01)\n", "S=plot3d(f,(x,-2,2),(y,-2,2),color = (c,cm), opacity=1, mesh=1)\n", "show(S, aspect_ratio=[10,10,1])\n", "C=contour_plot(f, (x,-2, 2), (y,-2, 2), cmap='Blues',linestyles='solid', contours=[-5,-3,-2,-1,0,1,2,3,5], colorbar=True)\n", "show(C,figsize=4)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Exemple 1\n", "On considère la fonction\n", "\n", "$$ f(x,y) = \\left\\{ \\begin{array}{lcl} \\frac{x^2-y^2}{x^2+y^2} & \\text{ si } & (x,y)\\ne (0,0)\\\\ 0& \\text{ si } & (x,y) =(0,0) \\end{array}\\right.$$\n", "On voit bien qu'il y a un problème à l'origine (le 11e commandement).\n", "Voyons la surface et les courbes de niveau. Voyons les courbes de niveau : on voit bien que si l'on se rapproche de l'origine par différentes droites, les valeurs de $f$ sont différentes aussi. La limite de $f$ en $(0,0)$ n'existe simplement pas, cette fonction ne peut pas être continue en ce point." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": "\n\n" }, "execution_count": 2, "metadata": { }, "output_type": "execute_result" }, { "data": { "image/png": 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" }, "execution_count": 2, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('x,y')\n", "f(x,y)=(x^2-y^2)/(x^2+y^2)\n", "cm = colormaps.Blues\n", "def c(x,y):\n", "\treturn 0.5+0.5*(x^2-y^2)/(x^2+y^2+0.005)# Un bricolage pour colorier... la discontinuité pose problème\n", "S=plot3d(f,(x,-1,1),(y,-1,1),color = (c,cm), opacity=1, mesh=1)\n", "show(S)\n", "C=contour_plot(f, (x,-1, 1), (y,-1, 1), cmap='Blues',linestyles='solid', colorbar=True)\n", "show(C,figsize=4)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Exemple 2\n", " considérons la fonction $g(x,y) = \\arctan\\left( \\frac{y}{x}\\right)$ lorsque $x\\ne 0$.\n", "La présence du $x$ au dénominateur indique qu'il y aura des problèmes pour des valeurs de $x$ très petites." ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "var('x,y')\n", "g(x,y)=arctan(y/x)\n", "cm = colormaps.Blues\n", "def c(x,y):\n", "\treturn (1/pi.n())*arctan(y/x) + 1/2\n", "S=plot3d(g,(x,-1,1),(y,-1,1),color=(c,cm), opacity=0.55, mesh=1)\n", "show(S)\n", "C=contour_plot(g, (x,-1, 1), (y,-1, 1),cmap='Blues',linestyles='solid', colorbar=True)\n", "show(C,figsize=4)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Exemple 3\n", "Voyons maintenant à quoi ressemble une surface continue. Soit $h$ la fonction définie par\n", "\n", "$$ h(x,y) = \\left\\{ \\begin{array}{lcl} \\frac{3x^2 y}{x^2+y^2} & \\text{ si } & (x,y)\\ne (0,0)\\\\ 0& \\text{ si } & (x,y) =(0,0) \\end{array}\\right.$$\n", "\n", "Ici on pourrait penser que le même problème que ci-haut va se présenter, mas ce n'est pas le cas, en fait, le numérateur est de degré 3, le dénominateur de degré 2. Le numérateur tend vers 0 plus vite que le dénominateur." ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "var('x,y')\n", " = colormaps.Blues\n", "h(x,y)= 3*x^2*y/(x^2+y^2)\n", "def c(x,y):\n", "\treturn 0.5+1.5*x^2*y/(x^2+y^2+0.005)#\n", "\n", "\n", "S=plot3d(h,(x,-1,1),(y,-1,1), color = (c,cm), opacity=1, mesh=1)\n", "show(S)\n", "C=contour_plot(h, (x,-1, 1), (y,-1, 1),cmap='Blues',linestyles='solid', colorbar=True)\n", "show(C,figsize=4)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Exemple 4\n", "Soit maintenant la fonction définie par $\\displaystyle h(x,y) = \\frac{xy^2}{x^2+y^4}$ pour $(x,y)\\ne (0,0)$ Nous avons vu en classe que si $\\mathcal{C}_m$ est la droite de pente $m$, alors $$\\lim_{\\stackrel{(x,y)\\to (0,0)}{\\mathcal{C}_m}} f(x,y) = 0$$\n", "\n", "mais sur la parabole $x=y^2$, la limite a une autre valeur. Ainsi, la limite de $f(x,y)$ lorsque $(x,y)$ tend vers $0$ n'existe pas. Voyons la surface, mais mieux encore, les courbes de niveau." ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "var('x,y,s,t')\n", "h(x,y)= x*y^2/(x^2+y^4)\n", "cm = colormaps.Blues\n", "def c(x,y):\n", "\treturn 0.6+x*y^2/(x^2+y^4+0.005)# Colorier ceci pose des problèmes, à cause de la singularité.\n", "S=plot3d(h,(x,-1,1),(y,-1,1),color = (c,cm), opacity=1, mesh=1)\n", "show(S)\n", "C=contour_plot(h, (x,-1, 1), (y,-1, 1),cmap='Blues',linestyles='solid', colorbar=True)\n", "show(C,figsize=4)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "display_name": "SageMath (stable)", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.15" } }, "nbformat": 4, "nbformat_minor": 0 }