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Author: Juan Carlos Bustamante
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%typeset_mode True

Exercice n,2 du devoir n.2

On commence par le dessin de surface.

var('x,y') f(x,y) = (x^3*y-x*y^3)/(x^2+ y^2) S=plot3d(f,(x,-2,2),(y,-1,1), color="lightgreen", opacity = 0.45, mesh = 0.2) S.show()
(x, y)
3D rendering not yet implemented

Calcul des dérivées partielles, pour (x,y)(0,0)(x,y) \ne (0,0).

fx(x,y) = diff(f(x,y),x).factor() fx(x,y)
(x4+4x2y2y4)y(x2+y2)2\displaystyle \frac{{\left(x^{4} + 4 \, x^{2} y^{2} - y^{4}\right)} y}{{\left(x^{2} + y^{2}\right)}^{2}}
fy(x,y) = diff(f(x,y),y).factor() fy(x,y)
(x44x2y2y4)x(x2+y2)2\displaystyle \frac{{\left(x^{4} - 4 \, x^{2} y^{2} - y^{4}\right)} x}{{\left(x^{2} + y^{2}\right)}^{2}}

Voyons les défivées secondes, et leur continuité : on voit bien, dans le diagramme des courbes de niveau, qu'il y a un problème de continuité.

fxy(x,y) = diff(f(x,y),x,y).factor() fxy(x,y)
(x4+10x2y2+y4)(x+y)(xy)(x2+y2)3\displaystyle \frac{{\left(x^{4} + 10 \, x^{2} y^{2} + y^{4}\right)} {\left(x + y\right)} {\left(x - y\right)}}{{\left(x^{2} + y^{2}\right)}^{3}}
cmsel = [colormaps['autumn'](i) for i in sxrange(0,1,0.05)] C= contour_plot(fxx, (x,-0.5, 0.5), (y,-0.5, 0.5),cmap='autumn',linestyles='solid', fill=True) show(C,figsize=4)
fyx(x,y) = diff(f(x,y),y,x).factor() fyx(x,y)
(x4+10x2y2+y4)(x+y)(xy)(x2+y2)3\displaystyle \frac{{\left(x^{4} + 10 \, x^{2} y^{2} + y^{4}\right)} {\left(x + y\right)} {\left(x - y\right)}}{{\left(x^{2} + y^{2}\right)}^{3}}
%md ## Question 1 on considère la courbe $\mathbf{r}(t) = (t^3, 3t,t^4)$. On veut savoir s'il existe un point où son plan osculateur est parallè au plan $x+y+z=1$.

Question 1

on considère la courbe r(t)=(t3,3t,t4)\mathbf{r}(t) = (t^3, 3t,t^4). On veut savoir s'il existe un point où son plan osculateur est parallè au plan x+y+z=1x+y+z=1.

var('t') assume(t, "real") def x(t) : return t^3 def y(t): return(3*t) def z(t) : return(t^4)
t\displaystyle t
def r(t) : return(vector([x(t), y(t), z(t)])) r(t)
(t3,3t,t4)\displaystyle \left(t^{3},\,3 \, t,\,t^{4}\right)
n=vector([6,6,-8])
def v(t) : return(diff(r(t),t)) v(t)
(3t2,3,4t3)\displaystyle \left(3 \, t^{2},\,3,\,4 \, t^{3}\right)
v(t).cross_product(n)
(24t324,24t3+24t2,18t218)\displaystyle \left(-24 \, t^{3} - 24,\,24 \, t^{3} + 24 \, t^{2},\,18 \, t^{2} - 18\right)
%md ### Le calcul du vecteur binormal $\mathbf{B}(t)$ demande un peu plus de travail

Le calcul du vevteur binormal B(t)\mathbf{B}(t) demande un peu plus de travail

def T(t) : return v(t)/v(t).norm() T(t)
(3t216t6+9t4+9,316t6+9t4+9,4t316t6+9t4+9)\displaystyle \left(\frac{3 \, t^{2}}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}},\,\frac{3}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}},\,\frac{4 \, t^{3}}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}}\right)
def N(t) : return(diff(T(t),t) / diff(T(t),t).norm().factor()) N(t)
(3(8t5+3t3)t2(16t6+9t4+9)32t16t6+9t4+9(4t6+36t2+9)t2(16t6+9t4+9)2,3(8t5+3t3)(16t6+9t4+9)32(4t6+36t2+9)t2(16t6+9t4+9)2,2(2(8t5+3t3)t3(16t6+9t4+9)32t216t6+9t4+9)(4t6+36t2+9)t2(16t6+9t4+9)2)\displaystyle \left(-\frac{\frac{3 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{\frac{3}{2}}} - \frac{t}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}}}{\sqrt{\frac{{\left(4 \, t^{6} + 36 \, t^{2} + 9\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2}}}},\,-\frac{3 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{\frac{3}{2}} \sqrt{\frac{{\left(4 \, t^{6} + 36 \, t^{2} + 9\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2}}}},\,-\frac{2 \, {\left(\frac{2 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)} t^{3}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{\frac{3}{2}}} - \frac{t^{2}}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}}\right)}}{\sqrt{\frac{{\left(4 \, t^{6} + 36 \, t^{2} + 9\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2}}}}\right)
def B(t): return(N(t).cross_product(T(t))) B(t)
(12(8t5+3t3)t3(16t6+9t4+9)2(4t6+36t2+9)t2(16t6+9t4+9)2+6(2(8t5+3t3)t3(16t6+9t4+9)32t216t6+9t4+9)16t6+9t4+9(4t6+36t2+9)t2(16t6+9t4+9)2,4(3(8t5+3t3)t2(16t6+9t4+9)32t16t6+9t4+9)t316t6+9t4+9(4t6+36t2+9)t2(16t6+9t4+9)26(2(8t5+3t3)t3(16t6+9t4+9)32t216t6+9t4+9)t216t6+9t4+9(4t6+36t2+9)t2(16t6+9t4+9)2,9(8t5+3t3)t2(16t6+9t4+9)2(4t6+36t2+9)t2(16t6+9t4+9)23(3(8t5+3t3)t2(16t6+9t4+9)32t16t6+9t4+9)16t6+9t4+9(4t6+36t2+9)t2(16t6+9t4+9)2)\displaystyle \left(-\frac{12 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)} t^{3}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2} \sqrt{\frac{{\left(4 \, t^{6} + 36 \, t^{2} + 9\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2}}}} + \frac{6 \, {\left(\frac{2 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)} t^{3}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{\frac{3}{2}}} - \frac{t^{2}}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}}\right)}}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9} \sqrt{\frac{{\left(4 \, t^{6} + 36 \, t^{2} + 9\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2}}}},\,\frac{4 \, {\left(\frac{3 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{\frac{3}{2}}} - \frac{t}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}}\right)} t^{3}}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9} \sqrt{\frac{{\left(4 \, t^{6} + 36 \, t^{2} + 9\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2}}}} - \frac{6 \, {\left(\frac{2 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)} t^{3}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{\frac{3}{2}}} - \frac{t^{2}}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}}\right)} t^{2}}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9} \sqrt{\frac{{\left(4 \, t^{6} + 36 \, t^{2} + 9\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2}}}},\,\frac{9 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2} \sqrt{\frac{{\left(4 \, t^{6} + 36 \, t^{2} + 9\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2}}}} - \frac{3 \, {\left(\frac{3 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{\frac{3}{2}}} - \frac{t}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}}\right)}}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9} \sqrt{\frac{{\left(4 \, t^{6} + 36 \, t^{2} + 9\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{2}}}}\right)
diff(T(t),t)
(18(8t5+3t3)t2(16t6+9t4+9)32+6t16t6+9t4+9,18(8t5+3t3)(16t6+9t4+9)32,24(8t5+3t3)t3(16t6+9t4+9)32+12t216t6+9t4+9)\displaystyle \left(-\frac{18 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)} t^{2}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{\frac{3}{2}}} + \frac{6 \, t}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}},\,-\frac{18 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{\frac{3}{2}}},\,-\frac{24 \, {\left(8 \, t^{5} + 3 \, t^{3}\right)} t^{3}}{{\left(16 \, t^{6} + 9 \, t^{4} + 9\right)}^{\frac{3}{2}}} + \frac{12 \, t^{2}}{\sqrt{16 \, t^{6} + 9 \, t^{4} + 9}}\right)
bf45850e-0d9b-47e2-a728-caa66e658541