6.1 Smooth curves in the plane  (Page 2/4)

Recall that a function $f$ that is continuous on a closed interval $\left[a,b\right]$ and continuously differentiable on the open interval $\left(a,b\right)$ is called a smooth function on $\left[a,b\right].$ And, if there exists a partition $\left\{{t}_{0}<{t}_{1}<...<{t}_{n}\right\}$ of $\left[a,b\right]$ such that $f$ is smooth on each subinterval $\left[{t}_{i-1},{t}_{i}\right],$ then $f$ is called piecewise smooth on $\left[a,b\right].$ Although the derivative of a smooth function is only defined and continuous on the open interval $\left(a,b\right),$ and hence possibly is unbounded, it follows from part (d) of [link] that this derivative is improperly-integrable on that open interval.We recall also that just because a function is improperly-integrable on an open interval, its absolute value may not be improperly-integrable.Before giving the formal definition of a smooth curve, which apparently will be related to smooth or piecewise smooth functions,it is prudent to present an approximation theorem about smooth functions. [link] asserts that every continuous function on a closed bounded interval is the uniform limit of a sequence of step functions.We give next a similar, but stronger, result about smooth functions. It asserts that a smooth function can be approximated “almost uniformly” by piecewise linear functions.

Let $f$ be a smooth function on a closed and bounded interval $\left[a,b\right],$ and assume that $|{f}^{\text{'}}|$ is improperly-integrable on the open interval $\left(a,b\right).$ Given an $ϵ>0,$ there exists a piecewise linear function $p$ for which

1.   $|f\left(x\right)-p\left(x\right)|<ϵ$ for all $x\in \left[a,b\right].$
2.   ${\int }_{a}^{b}|{f}^{\text{'}}\left(x\right)-{p}^{\text{'}}\left(x\right)|\phantom{\rule{0.166667em}{0ex}}dx<ϵ.$

That is, the functions $f$ and $p$ are close everywhere, and their derivatives are close on average in the sense that theintegral of the absolute value of the difference of the derivatives is small.

Because $f$ is continuous on the compact set $\left[a,b\right],$ it is uniformly continuous. Hence, let $\delta >0$ be such that if $x,y\in \left[a,b\right],$ and $|x-y|<\delta ,$ then $|f\left(x\right)-f\left(y\right)|<ϵ/2.$

Because $|{f}^{\text{'}}|$ is improperly-integrable on the open interval $\left(a,b\right),$ we may use part (b) of [link] to find a ${\delta }^{\text{'}}>0,$ which may also be chosen to be $<\delta ,$ such that ${\int }_{a}^{a+{\delta }^{\text{'}}}|{f}^{\text{'}}|+{\int }_{b-{\delta }^{\text{'}}}^{b}|{f}^{\text{'}}|<ϵ/2,$ and we fix such a ${\delta }^{\text{'}}.$

Now, because ${f}^{\text{'}}$ is uniformly continuous on the compact set $\left[a+{\delta }^{\text{'}},b-{\delta }^{\text{'}}\right],$ there exists an $\alpha >0$ such that $|{f}^{\text{'}}\left(x\right)-{f}^{\text{'}}\left(y\right)|<ϵ/4\left(b-a\right)$ if $x$ and $y$ belong to $\left[a+{\delta }^{\text{'}},b-{\delta }^{\text{'}}\right]$ and $|x-y|<\alpha .$ Choose a partition $\left\{{x}_{0}<{x}_{1}<...<{x}_{n}\right\}$ of $\left[a,b\right]$ such that ${x}_{0}=a,{x}_{1}=a+{\delta }^{\text{'}},{x}_{n-1}=b-{\delta }^{\text{'}},{x}_{n}=b,$ and ${x}_{i}-{x}_{i-1} for $2\le i\le n-1.$ Define $p$ to be the piecewise linear function on $\left[a,b\right]$ whose graph is the polygonal line joining the $n+1$ points $\left(a,f\left({x}_{1}\right)\right),$ $\left\{\left({x}_{i},f\left({x}_{i}\right)\right)\right\}$ for $1\le i\le n-1,$ and $\left(b,f\left({x}_{n-1}\right)\right).$ That is, $p$ is constant on the outer subintervals $\left[a,{x}_{1}\right]$ and $\left[{x}_{n-1},b\right]$ determined by the partition, and its graph between ${x}_{1}$ and ${x}_{n-1}$ is the polygonal line joining the points $\left\{\left({x}_{1},f\left({x}_{1}\right)\right),...,\left({x}_{n-1},f\left({x}_{n-1}\right)\right)\right\}.$ For example, for $2\le i\le n-1,$ the function $p$ has the form

$p\left(x\right)=f\left({x}_{i-1}\right)+\frac{f\left({x}_{i}\right)-f\left({x}_{i-1}\right)}{{x}_{i}-{x}_{i-1}}\left(x-{x}_{i-1}\right)$

on the interval $\left[{x}_{i-1},{x}_{i}\right].$ So, $p\left(x\right)$ lies between the numbers $f\left({x}_{i-1}\right)$ and $f\left({x}_{i}\right)$ for all $i.$ Therefore,

$|f\left(x\right)-p\left(x\right)|\le |f\left(x\right)-f\left({x}_{i}\right)|+|f\left({x}_{i}\right)-l\left(x\right)|\le |f\left(x\right)-f\left({x}_{i}\right)|+|f\left({x}_{i}\right)-f\left({x}_{i-1}\right)|<ϵ.$

Since this inequality holds for all $i,$ part (1) is proved.

Next, for $2\le i\le n-1,$ and for each $x\in \left({x}_{i-1},{x}_{i}\right),$ we have ${p}^{\text{'}}\left(x\right)=\left(f\left({x}_{i}\right)-f\left({x}_{i-1}\right)\right)/\left({x}_{i}-{x}_{i-1}\right),$ which, by the Mean Value Theorem, is equal to ${f}^{\text{'}}\left({y}_{i}\right)$ for some ${y}_{i}\in \left({x}_{i-1},{x}_{i}\right).$ So, for each such $x\in \left({x}_{i-1},{x}_{i}\right),$ we have $|{f}^{\text{'}}\left(x\right)-{p}^{\text{'}}\left(x\right)|=|{f}^{\text{'}}\left(x\right)-{f}^{\text{'}}\left({y}_{i}\right)|,$ and this is less than $ϵ/4\left(b-a\right),$ because $|x-{y}_{i}|<\alpha .$ On the two outer intervals, $p\left(x\right)$ is a constant, so that ${p}^{\text{'}}\left(x\right)=0.$ Hence,

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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