Calculus III:
Tangency and Normals

10/03/2013

Watershed Map

This plot shows a function of two variables mimicking some aspects of the map on your handout.

Graph

Plot3D[-x^4/4 + 4x^2+ 2y ,{x,-4,4},{y,0,5}]

Watershed Map

Contour Plot

Here are the contours for the function -- observe the similar shape

ContourPlot[-x^4/4 + 4x^2 +2y,{x,-4,4},{y,0,5}]

Ellipse

p1=ParametricPlot[{Sqrt[3] *Cos [t] , Sqrt[6]* Sin[t]},{t, -Pi, Pi}];

p2=ParametricPlot[{1+4t,2+4t},{t, -0.25, 0.25}];

p3 = Plot[{-x+3},{x, 0, 2}];

Show[p1,p2,p3,PlotRange->All]

Elliptic Paraboloid

p4=ContourPlot3D[2x^2+5y^2 == z, {x,-3,4},{y,-3,3},{z,0,15}];

p5=ContourPlot3D[4x+10y-z==7,{x,-3,4},{y,-3,3},{z,0,15}];

p6=ParametricPlot3D[{1+4t,1+10t,7-t},{t, -0.5,0.5}];

Show[p4,p5,p6,PlotRange->All]

Degree 6 surface

p7=ContourPlot3D[x * y^3 * z^2 ==2,{x,-4,4},{y,-4,4},{z,-4,4}];

p8=ContourPlot3D[x+6y-4z==12,{x,-4,4},{y,-4,4},{z,-4,4}];

p9=ParametricPlot3D[{2+t,1+6t,-1-4t},{t,-1,1}];

Show[p7,p8,p9]

Intersecting cylinder and cone

h=x^2 + 3y^2 - 2z^2;

g=x^2+y^2-1;

p10=ContourPlot3D[{h==0,g==0},{x,-2,2},{y,-2,2},{z,-2,2},MeshFunctions->{Function[{x,y,z,f},h-g]},MeshStyle->{{Thick,Blue}},Mesh->{{0}},ContourStyle->Directive[Orange,Opacity[0.5],Specularity[White,30]]]

Intersecting cylinder and cone

p11=ParametricPlot3D[{t,1,Sqrt[1.5]},{t,-2,2},PlotStyle->Thickness[0.01]];

Show[p10,p11]

Intersecting plane and cylinder

h=y+z-3;

g=x^2+y^2-5;

p12=ContourPlot3D[{h==0,g==0},{x,-2.5,2.5},{y,-2.5,2.5},{z,0,7},MeshFunctions->{Function[{x,y,z,f},h-g]},MeshStyle->{{Thick,Blue}},Mesh->{{0}},ContourStyle->Directive[Orange,Opacity[0.5],Specularity[White,30]]]

Intersecting plane and cylinder

p13=ParametricPlot3D[{1-4t,2+2t,1-2t},{t,-2,2},PlotStyle->Thickness[0.01]];

Show[p12,p13]

Normal to a sphere

p14=ContourPlot3D[x^2+y^2+z^2==1,{x,-1.5,1.5},{y,-1.5,1.5},{z,-1.5,1.5},ContourStyle->Directive[Opacity[0.5]],Mesh->None];

p15=ParametricPlot3D[{t,0,0},{t,-1.5,1.5},PlotStyle->Thickness[0.01]];

Show[p14,p15]

Tangential Surfaces

h=3x^2+2y^2+z^2-9;

g=x^2+y^2+z^2-8x-6y-8z+24;

p16=ContourPlot3D[{h==0,g==0},{x,0,2},{y,0,2},{z,1,3}]