6.1 Smooth curves in the plane

Our first project is to make a satisfactory definition of a smooth curve in the plane, for there is a good bit of subtlety to such a definition.In fact, the material in this chapter is all surprisingly tricky, and the proofs are good solid analytical arguments, with lots of $ϵ$ 's and references to earlier theorems.

Whatever definition we adopt for a curve, we certainly want straight lines, circles, and other natural geometric objects to be covered by our definition. Our intuition is that a curve in the plane should be a “1-dimensional” subset, whatever that may mean.At this point, we have no definition of the dimension of a general set, so this is probably not the way to think about curves. On the other hand, from the point of view of a physicist, we might well define a curve as the trajectory followedby a particle moving in the plane, whatever that may be. As it happens, we do have some notion of how to describe mathematically the trajectory of a moving particle.We suppose that a particle moving in the plane proceeds in a continuous manner relative to time. That is, the position of the particle at time $t$ is given by a continuous function $f\left(t\right)=x\left(t\right)+iy\left(t\right)\equiv \left(x\right(t),y(t\left)\right),$ as $t$ ranges from time $a$ to time $b.$ A good first guess at a definition of a curve joining two points $z_1$ and $z_2$ might well be that it is the range $C$ of a continuous function $f$ that is defined on some closed bounded interval $[a,b].$ This would be a curve that joins the two points $z_1=f\left(a\right)$ and $z_2=f\left(b\right)$ in the plane. Unfortunately, this is also not a satisfactory definition of a curve, because of the followingsurprising and bizarre mathematical example, first discovered by Guiseppe Peano in 1890.

THE PEANO CURVE The so-called “Peano curve” is a continuous function $f$ defined on the interval $[0,1],$ whose range is the entire unit square $[0,1]\times [0,1]$ in $R^2.$

Be careful to realize that we're talking about the “range” of $f$ and not its graph. The graph of a real-valued function could never be the entire square.This Peano function is a complex-valued function of a real variable. Anyway, whatever definition we settle on for a curve, we do not want the entire unit square tobe a curve, so this first attempt at a definition is obviously not going to work.

Let's go back to the particle tracing out a trajectory. The physicist would probably agree that the particle should have a continuously varying velocity at all times, or at nearly all times,i.e., the function $f$ should be continuously differentiable. Recall that the velocity of the particle is defined to be the rate of change of the positionof the particle, and that's just the derivative $f^{\text{'}}$ of $f.$ We might also assume that the particle is never at rest as it traces out the curve, i.e., the derivative $f^{\text{'}}\left(t\right)$ is never 0. As a final simplification,we could suppose that the curve never crosses itself, i.e., the particle is never at the same position more than once during the time interval from $t=a$ to $t=b.$ In fact, these considerations inspire the formal definition of a curve that we will adopt below.