Visualisering i (ℝ^2)

Graf av funktion av en variabel: f:ℝ->ℝ

Plot[Sin[x],{x,-Pi,Pi}]

Kurva på parameterform: (OverBar[r]):ℝ->(ℝ^2)

ParametricPlot[{Sin[u],Sin[2u]},{u,0,2Pi}]

Kurva på implicit form: f(x,y)=0, f:(ℝ^2)->ℝ

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

Visualisering i (ℝ^3)

Graf av funktion av två variabler: f:(ℝ^2)->ℝ

Plot3D[Sin[x]+Cos[y],{x,-Pi,Pi},{y,-Pi,Pi}]

Kurva på parameterform: (OverBar[r]):ℝ->(ℝ^3)

ParametricPlot3D[{Sin[u],Cos[u],u/10},{u,0,20}]

Yta på parameterform: (OverBar[r]):(ℝ^2)->(ℝ^3)

ParametricPlot3D[{4+(3+Cos[v]) Sin[u],4+(3+Cos[v]) Cos[u],4+Sin[v]},{u,0,2Pi},{v,0,2Pi}]

Yta på implicit form: f(x,y,z)=0, f:(ℝ^3)->ℝ

ContourPlot3D[(x^2)+(y^2)+(z^2)==1,{x,-1,1},{y,-1,1},{z,-1,1}]

Standardytor (grad två) i (ℝ^3)

Ellipsoid

Manipulate[ContourPlot3D[((x^2)/(a^2))+((y^2)/(b^2))+((z^2)/(c^2))==1,{x,-2,2},{y,-2,2},{z,-2,2}],{a,1,2},{b,1,2},{c,1,2}]

Hyperboloid (enmantlad, tvåmantlad), koner

Manipulate[ContourPlot3D[((x^2)/(a^2))+((y^2)/(b^2))-((z^2)/(c^2))==s,{x,-2,2},{y,-2,2},{z,-2,2}],{a,1,2},{b,1,2},{c,1,2},{s,1,-1,-1}]

Paraboloid (hyperbolisk, elliptisk)

Manipulate[ContourPlot3D[((x^2)/(a^2))+s*((y^2)/(b^2))==z,{x,-2,2},{y,-2,2},{z,-2,2}],{a,1,2},{b,1,2},{s,-1,1,2}]

Cylindrar - elliptiska

Manipulate[ContourPlot3D[((x^2)/(a^2))+((y^2)/(b^2))==1,{x,-2,2},{y,-2,2},{z,-2,2}],{a,1,2},{b,1,2}]

Cylindrar - hyperboliska

Manipulate[ContourPlot3D[((x^2)/(a^2))-((y^2)/(b^2))==s,{x,-2,2},{y,-2,2},{z,-2,2}],{a,1,2},{b,1,2},{s,-1,1,1}]

Cylindrar - paraboliska

Manipulate[ContourPlot3D[s*((y^2)/(b^2))==x,{x,-2,2},{y,-2,2},{z,-2,2}],{b,1,2},{s,-1,1,2}]

1.10. Rita mängderna

(Subscript[M, 1])={(x,y,z); x+y+z≤1, x≥0, y≥0, z≥0}

RegionPlot3D[x+y+z≤1 ,{x,0,1},{y,0,1},{z,0,1}]

Hitta randpunkter genom att byta ut olikheter mot likhet.  Ofta enkelt att betrakta koordinatplanen x=0, y=0, z=0.

RegionPlot3D[x+y+z<=1 &&x==0,{x,0,1},{y,0,1},{z,0,1}]

RegionPlot3D[x+y+z<=1 &&y==0,{x,0,1},{y,0,1},{z,0,1}]

RegionPlot3D[x+y+z<=1 &&z==0,{x,0,1},{y,0,1},{z,0,1}]

ContourPlot3D[x+y+z==1,{x,0,1},{y,0,1},{z,0,1}]

(Subscript[M, 2]),... , (Subscript[M, 5])

RegionPlot3D[(x^2)+(y^2)+(z^2)≤4 &&z≥0,{x,-2,2},{y,-2,2},{z,0,4}]

RegionPlot3D[(x^2)+(y^2)+4(z^2)≤1,{x,-1,1},{y,-1,1},{z,-1,1}]

RegionPlot3D[(x^2)+(y^2)≤z &&z≤1,{x,-1,1},{y,-1,1},{z,0,2}]

RegionPlot3D[(Sqrt[(x^2)+(y^2)])≤z &&z≤1,{x,-1,1},{y,-1,1},{z,0,2}]

Nivåkurvor

Problem 1.14

Rita nivåkurvor: {(x,y); f(x,y)=k}

ContourPlot[(Sqrt[(x^2)+(y^2)]),{x,-2,2},{y,-2,2},ContourShading->None,Contours->{1,2,3}]

ContourPlot[(x^2)+4(y^2),{x,-2,2},{y,-2,2},ContourShading->None,Contours->{1,2,3}]

ContourPlot[(x^2)-(y^2),{x,-2,2},{y,-2,2},ContourShading->None,Contours->{-1,1,2}]

Nivåytor

Problem 1.19

Rita nivåytor: {(x,y,z); f(x,y,z)=k}

ContourPlot3D[z-(Sqrt[1-(x^2)-(y^2)]),{x,-2,2},{y,-2,2},{z,0,2},Contours->{0,1}]

Kurvor

Problem 1.23

Hitta skärningen mellan kurvorna :

Parameterform: r(t)=(x(t),y(t)}, t∈ℝ

(x,y)=((t^2),t+1)

Implicit form: f(x,y)=0

5(x^2)+5x*y+3(y^2)-8x-6y+3=0

Show[ContourPlot[5(x^2)+5x*y+3(y^2)-8x-6y+3==0,{x,-1,2},{y,-1,2}],ParametricPlot[{(t^2),t+1},{t,-2,1}]]

Gränsvärden

Problem 1.24 (c)

Plot3D[((x-y)/(x-1)),{x,0,2},{y,0,2}]

Problem 1.24 (e)

Plot3D[(((x^3)-(x^2)*y)/((x^2)+(y^2)+x*y)),{x,-1,1},{y,-1,1}]

Problem 1.29 (a)

Plot3D[((Sin[(x^2)+(y^2)])/((x^2)+(y^2))),{x,-1,1},{y,-1,1}]