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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+4 x2y2−y4)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)
(x4−4 x2y2−y4)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+10 x2y2+y4)(x+y)(x−y)(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+10 x2y2+y4)(x+y)(x−y)(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, 3 t, 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)
(3 t2, 3, 4 t3)\displaystyle \left(3 \, t^{2},\,3,\,4 \, t^{3}\right)
v(t).cross_product(n)
(−24 t3−24, 24 t3+24 t2, 18 t2−18)\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)
(3 t216 t6+9 t4+9, 316 t6+9 t4+9, 4 t316 t6+9 t4+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 (8 t5+3 t3)t2(16 t6+9 t4+9)32−t16 t6+9 t4+9(4 t6+36 t2+9)t2(16 t6+9 t4+9)2, −3 (8 t5+3 t3)(16 t6+9 t4+9)32(4 t6+36 t2+9)t2(16 t6+9 t4+9)2, −2 (2 (8 t5+3 t3)t3(16 t6+9 t4+9)32−t216 t6+9 t4+9)(4 t6+36 t2+9)t2(16 t6+9 t4+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 (8 t5+3 t3)t3(16 t6+9 t4+9)2(4 t6+36 t2+9)t2(16 t6+9 t4+9)2+6 (2 (8 t5+3 t3)t3(16 t6+9 t4+9)32−t216 t6+9 t4+9)16 t6+9 t4+9(4 t6+36 t2+9)t2(16 t6+9 t4+9)2, 4 (3 (8 t5+3 t3)t2(16 t6+9 t4+9)32−t16 t6+9 t4+9)t316 t6+9 t4+9(4 t6+36 t2+9)t2(16 t6+9 t4+9)2−6 (2 (8 t5+3 t3)t3(16 t6+9 t4+9)32−t216 t6+9 t4+9)t216 t6+9 t4+9(4 t6+36 t2+9)t2(16 t6+9 t4+9)2, 9 (8 t5+3 t3)t2(16 t6+9 t4+9)2(4 t6+36 t2+9)t2(16 t6+9 t4+9)2−3 (3 (8 t5+3 t3)t2(16 t6+9 t4+9)32−t16 t6+9 t4+9)16 t6+9 t4+9(4 t6+36 t2+9)t2(16 t6+9 t4+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 (8 t5+3 t3)t2(16 t6+9 t4+9)32+6 t16 t6+9 t4+9, −18 (8 t5+3 t3)(16 t6+9 t4+9)32, −24 (8 t5+3 t3)t3(16 t6+9 t4+9)32+12 t216 t6+9 t4+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)

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