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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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