# 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.

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 $f\left(t\right)=x\left(t\right)+iy\left(t\right)\equiv \left(x\left(t\right),y\left(t\right)\right),$ 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 $\left[a,b\right].$ 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 $\left[0,1\right],$ whose range is the entire unit square $\left[0,1\right]×\left[0,1\right]$ 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.

#### Questions & Answers

what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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