Lecture 1 5

Figure 1.4. A sphere with two handles.

We begin by asking what sorts of equations are acceptable. By

moving all the terms to the same side, any equation in x, y, and

z can be written in the form F (x, y, z) = 0. If we hope to get a

smooth surface, we must demand that the function F be at least

differentiable—any of the equations (1.1), (1.2), (1.3), and (1.4) can

be written in this form with a quadratic polynomial as the function

F . But why are the sphere, the cylinder, and the ellipsoid all smooth,

while the cone has a special point? The difference is clearly seen in the

geometric description of the surfaces, since the line we use to define

the cone passes through the axis of rotation, but it is not so easy to

see what feature of the equations is responsible. How does this point

of non-smoothness turn up in the equations?

The answer is that the origin is a critical point of the function

x2 + y2 − a2z2 and lies on the surface defined by (1.3), while the other

functions, x2 + y2 + z2 − R2, x2 + y2 − R2, and

x2

a2

+

y2

b2

+

z2

c2

− 1,

have no critical points at the zero level. Thus, if we want to define a

smooth surface in R3 by an equation of the form F (x, y, z) = 0, the

function F should have no critical points at the zero level.

Turning to the other half of the relationship between surfaces

and equations, we find that not every geometric object which com-

mon sense would call a surface can be represented as the solution set

of an equation. One diﬃculty is caused by boundaries—notice that

the cylinder defined in (1.2) is unbounded, and extends infinitely far

in both the positive and negative z-directions. Suppose we want to

consider a finite cylinder, which may be obtained by rotating an inter-

val around a parallel line, or by rolling up a rectangular sheet of paper