Definitions and Plots

Curves are defined in terms of functions of a single variable or parametrized by one parameter [ y=f(x) or R(t)=(x(t), y(t), z(t)) ]. They can also be implicitly defined an expression containing two variables, for example, xy=1. Surfaces can be defined in terms of functions of two variables,

z=S(x,y)

parametrized by two parameters,

R(u,v)=(x(u,v), y(u,v), z(u,v)

or implicitly defined by an expression containing three variables.

x^2+y^2+z^3=1.

Here is the sphere x^2+y^2+z^1=4 plotted in all three ways.

In[88]:=

  upper=Plot3D[Sqrt[4-x^2-y^2],{x,-2,2},{y,-2,2},AspectRatio->Automatic];
  lower=Plot3D[-Sqrt[4-x^2-y^2],{x,-2,2},{y,-2,2},AspectRatio->Automatic];
  Show[upper,lower];

In[89]:=

  Show[upper,lower,ViewPoint->{0,1,0}];

In[90]:=

  ParametricPlot3D[{2 Cos[u] Sin[v],2 Sin[u] Sin[v], 2 Cos[v]},{u,0,2 Pi},{v,0,Pi}];

In[91]:=

  ContourPlot3D[x^2+y^2+z^2,{x,-2,2},{y,-2,2},{z,-2,2},Contours->{4.}];

Think intuitively of a surface for the moment.

A smooth curve has a continuously turning tangent. A smooth surface has a continuously varying normal. A surface is piecewise smooth if it consists of a finite number of smooth parts joined together. (The surface of a sphere is smooth, the surface of a cube is piecewise smooth.)

A closed curve encloses a region (area). A closed surface encloses a volume.

A simple curve does not intersect itself. A simple surface does not intersect itself.

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