文档介绍：Chapter Fifiteen
Surfaces Revisited
Vector Description of Surfaces
We look now at the very special case of functions r: D ® R 3 , where D Ì R 2 is a
nice subset of the plane. We suppose r is a nice function. As the point (s,t) Î D moves
around in D, if we place the tail of the vector r(s, t) at the origin, the nose of this vector
will trace out a surface in three-space. Look, for example at the function r:D ® R 3 ,
where r(s, t) = si + tj + (s2 + t 2 )k , and D = {(s,t) Î R 2:- 1 £ s,t £ 1}. It shouldn't be
difficult to convince yourself that if the tail of r(s, t) is at the origin, then the nose will be
on the paraboloid z = x2 + y 2 , and for all (s,t) Î D, we get the part of the paraboloid
above the square - 1 £ x, y £ 1. It is sometimes helpful to think of the function r as
providing a map from the region D to the surface.
The vector function r is called a vector description of the surface. This is, of course,
exactly the two dimensional analogue of the vector description of a curve.
For a curve, r is a function from a nice piece of the real line into three space; and for a
surface, r is a function from a nice piece of the plane into three space.
Let's look at another example. Here, let
r(s, t) = coss sint i + sin s sin t j + cost k ,
for 0 £ t £ p and 0 £ s £ 2p . What have we here? First, notice that
| r(s,t)|2 = (coss sin t) 2 + (sin s sin t) 2 + (cost ) 2
= sin 2 t (cos2 s + sin 2 s) + cos2 t
= sin 2 t + cos2 t = 1
Thus the nose of r is always on the sphere of radius one and centered at the origin.
Notice next, that the variable, or parameter, s is the longitude of r(s, t) ; and the variable t
is the latitude of r(s, t) . (More precisely, t is co-latitude.) A moment's reflection on this
will convince you that as r is a description of the entire sphere. We have a map of the
sphere on the rectangle
Observe that the entire lower edge of the rectangle (the line from (0,0) to (2p ,0) ) is
mapped by r onto th