<< Chapter < Page Chapter >> Page >
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.

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 ( t ) = x ( t ) + i y ( t ) ( x ( t ) , y ( t ) ) , 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 ( a ) and z 2 = f ( b ) 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 ] × [ 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 ' of f . We might also assume that the particle is never at rest as it traces out the curve, i.e., the derivative f ' ( t ) 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.

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Analysis of functions of a single variable' conversation and receive update notifications?

Ask