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 x 2 a2 + y2 b2 + z 2 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
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