Esercizio 1 (dal foglio di esercizi)

f((Subscript[x, 1]),(Subscript[x, 2])) che descrive la distanza tra le due curve (Subscript[y, 1])=((Subscript[x, 1])^2) e (Subscript[y, 2])=(Subscript[x, 2])-2.

f[x1_,x2_]:=(((x1-x2))^2)+((((x1^2)-x2+2))^2);

Plot3D[{f[x1,x2]},{x1,1/2+0.5,1/2-0.5},{x2,11/8-0.5,11/8+0.5},Mesh->None]

Esercizio 6.20

Determiniamo la regione D={ (x,y) : 3(x^2)+4(y^2)≤4 , y≥(1/2)}

p1=RegionPlot[{3 (x^2)+4 (y^2)-4≤0&&y≥1/2},{x,-2/Sqrt[3],2/Sqrt[3]},{y,-1,1}];

p2=Plot[y=1/2,{x,-2/Sqrt[3],2/Sqrt[3]}];

p3=ContourPlot[3 (x^2)+4 (y^2)-4==0,{x,-2/Sqrt[3],2/Sqrt[3]},{y,-1,1}];

Show[{p1,p2,p3}]

Comportamento di f sulla curva y=(1/2)

q0=ParametricPlot3D[{x,1/2,(E^((x^2)+(1/4)))-(1/2) (x^2)-(1/4)},{x,-1,1}]

Comportamento di f sull’ellisse

q1=ParametricPlot3D[{x,Sqrt[(1-(3/4) (x^2))],(E^((x^2)+(1-(3/4) (x^2))))-(1/2) (x^2)-(1-(3/4) (x^2))},{x,-2/Sqrt[3],2/Sqrt[3]},PlotRange->{{-2/Sqrt[3],2/Sqrt[3]},{-1,1}}];

q2=ParametricPlot3D[{x,-Sqrt[(1-(3/4) (x^2))],(E^((x^2)+(1-(3/4) (x^2))))-(1/2) (x^2)-(1-(3/4) (x^2))},{x,-2/Sqrt[3],2/Sqrt[3]},PlotRange->{{-2/Sqrt[3],2/Sqrt[3]},{-1,1}}];

Show[{q1,q2}]

q3=ParametricPlot3D[{x,Sqrt[(1-(3/4) (x^2))],(E^((x^2)+(1-(3/4) (x^2))))-(1/2) (x^2)-(1-(3/4) (x^2))},{x,-1,1}]

Comportamento sul bordo di D

Show[{q0,q3}]

Comportamento su D

Plot3D[(E^((x^2)+(y^2)))-(1/2) (x^2)-(y^2),{x,-1,1},{y,1/2,1},RegionFunction->Function[{x,y,z},3 (x^2)+4(y^2)≤ 4&&y≥1/2],Mesh->None]

Comportamento su tutto lo spazio

Plot3D[(E^((x^2)+(y^2)))-(1/2) (x^2)-(y^2),{x,-2,2},{y,-2,2},Mesh->None]

Esercizio 6.21

ContourPlot[{9 (x^2)+(y^2)-1==0},{x,-1/3,1/3},{y,-1,1}]

s1=ParametricPlot3D[{x,Sqrt[(1-9  (x^2))],Log[3 (x^2)-1+9 (x^2)+3]},{x,-1/3,1/3},PlotRange->{{-1/3,1/3},{-1,1}}];

s2=ParametricPlot3D[{x,-Sqrt[(1-9  (x^2))],Log[3 (x^2)-1+9 (x^2)+3]},{x,-1/3,1/3},PlotRange->{{-1/3,1/3},{-1,1}}];

Show[s1,s2]

Esercizio 6.28

ContourPlot3D[(z^2)-x y -1==0,{x,-2,2},{y,-2,2},{z,-2,2},ColorFunction->Function[{x,y,z},Hue[ (1/50) ((x^2)+(y^2)+(z^2))]],Mesh->None,ColorFunctionScaling->False]

Clear[z]

Esercizio 6.37

Disegnamo la retta Γ in (ℝ^3) e la funzione f sulla retta. I suoi valori saranno dati dalla gradazione di colore

pp1=ParametricPlot3D[{x,(4/5) x+(12/5),3},{x,-2,2.3},ColorFunction->Function[{x,y,z},Hue[ ((x^3)+((((4/5) x+(12/5)))^3)-9 x ((4/5) x+(12/5)))/200]],Mesh->None,ColorFunctionScaling->False,PlotStyle->{Tube[.03]}];

pp2=ParametricPlot3D[{2,4,z},{z,-3,5}];

pp3=ParametricPlot3D[{-6/7,12/7,z},{z,-3,5}];

Show[pp1,pp2,pp3]

Esercizio 6.38

Determiniamo la regione D

RegionPlot3D[(x^2) (y^2)+1-(z^2)>=0,{x,-2,2},{y,-2,2},{z,-2,2},Mesh->None];

RegionPlot3D[(x^2)+(y^2)-1≤0,{x,-2,2},{y,-2,2},{z,-2,2},Mesh->None];

RegionPlot3D[(x^2) (y^2)+1-(z^2)>=0&&(x^2)+(y^2)-1≤0,{x,-2,2},{y,-2,2},{z,-2,2},Mesh->None]

Determiniamo anche l’ interezione Γ tra (Subscript[Σ, 1])e (Subscript[Σ, 2])

ContourPlot3D[{(x^2)(y^2)+1-(z^2)==0,(x^2)+(y^2)-1==0},{x,-2,2},{y,-2,2},{z,-2,2},Contours->{0},ContourStyle->Opacity[0],Mesh->None,BoundaryStyle->{1->None,2->None,{1,2}->{{Red,Tube[.03]}}},Boxed->False]

Punti critici su (Subscript[Σ, 1])

ContourPlot3D[-(z^2)+(x^2) (y^2) +1==0,{x,-2,2},{y,-2,2},{z,-2,2},ColorFunction->Function[{x,y,z},Hue[ (1/50) ((x^2)-(y^2)+(z^2))]],Mesh->None,ColorFunctionScaling->False,AxesLabel->{x,y,z}]

Punti critici su (Subscript[Σ, 2])

ContourPlot3D[(x^2) +(y^2) -1==0,{x,-2,2},{y,-2,2},{z,-2,2},ColorFunction->Function[{x,y,z},Hue[ (1/50) ((x^2)-(y^2)+(z^2))]],Mesh->None,ColorFunctionScaling->False,AxesLabel->{x,y,z}]

Punti Critici su Γ

ContourPlot3D[{(x^2)(y^2)+1-(z^2)==0,(x^2)+(y^2)-1==0},{x,-2,2},{y,-2,2},{z,-2,2},Contours->{0},ContourStyle->Opacity[0],Mesh->None,BoundaryStyle->{1->None,2->None,{1,2}->{{Tube[.03]}}},Boxed->False,ColorFunction->Function[{x,y,z},Hue[ (1/30) ((x^2)-(y^2)+(z^2))]],ColorFunctionScaling->False,AxesLabel->{x,y,z}]

Spazio tangente a Γ in ((1/2),((Sqrt[3])/2),((Sqrt[19])/4))

t1=ContourPlot3D[{(x^2)(y^2)+1-(z^2)==0,(x^2)+(y^2)-1==0,(3/4) x+(Sqrt[3]/4) y-(Sqrt[19]/2) z+(13/8)==0,x+Sqrt[3] y-2==0},{x,-1.5,1.5},{y,-1.5,1.5},{z,-2,2},Contours->{0},ContourStyle->Opacity[0],Mesh->None,BoundaryStyle->{1->None,2->None,3->None,4->None,{1,2}->{{Red,Tube[.01]}},{3,4}->{{Purple,Tube[.01]}}},Boxed->False];

t2=ListPointPlot3D[{{1/2,Sqrt[3]/2,Sqrt[19]/4}},PlotStyle->PointSize[.02]];

Show[t1,t2]

Esercizio 6.39

Si deve inserire il valore di c

c=1;

w1=ContourPlot3D[{(x^2)+(y^2)+((z^2)/(c^2))-1==0,x+y+z==0},{x,-2,2},{y,-2,2},{z,-2,2},Contours->{0},ContourStyle->Opacity[0],Mesh->None,BoundaryStyle->{1->None,2->None,{1,2}->{{Tube[.03]}}},Boxed->False,ColorFunction->Function[{x,y,z},Hue[ (1/20) (z)]],ColorFunctionScaling->False,AxesLabel->{x,y,z}];

w2=ListPointPlot3D[{{(Sqrt[((c^2)/(4+2 (c^2)))]),(Sqrt[((c^2)/(4+2 (c^2)))]),-2 (Sqrt[((c^2)/(4+2 (c^2)))])},{-(Sqrt[((c^2)/(4+2 (c^2)))]),-(Sqrt[((c^2)/(4+2 (c^2)))]),2 (Sqrt[((c^2)/(4+2 (c^2)))])}},PlotStyle->PointSize[.02]];

Show[w1,w2]