# 6.1 Smooth curves in the plane  (Page 3/4)

$\begin{array}{ccc}\hfill {\int }_{a}^{b}|{f}^{\text{'}}-{p}^{\text{'}}|& =& \sum _{i=1}^{n}{\int }_{{x}_{i-1}}^{{x}_{i}}|{f}^{\text{'}}-{p}^{\text{'}}|\hfill \\ & =& {\int }_{a}^{{x}_{1}}|{f}^{\text{'}}|+\sum _{i=2}^{n-1}|{f}^{\text{'}}-{p}^{\text{'}}|+{\int }_{{x}_{n-1}}^{b}|{f}^{\text{'}}|\hfill \\ & \le & {\int }_{a}^{a+{\delta }^{\text{'}}}|{f}^{\text{'}}|+{\int }_{b-{\delta }^{\text{'}}}^{b}|{f}^{\text{'}}|+\frac{ϵ}{4\left(b-a\right)}{\int }_{{x}_{1}}^{{x}_{n-1}}1\hfill \\ & <& ϵ.\hfill \end{array}$

The proof is now complete.

REMARK It should be evident that the preceding theorem can easily be generalized to a piecewise smooth function $f,$ i.e., a function that is continuous on $\left[a,b\right],$ continuously differentiable on each subinterval $\left({t}_{i-1},{t}_{i}\right)$ of a partition $\left\{{t}_{0}<{t}_{1}<...<{t}_{n}\right\},$ and whose derivative ${f}^{\text{'}}$ is absolutely integrable on $\left(a,b\right).$ Indeed, just apply the theorem to each of the subintervals $\left({t}_{i-1},{t}_{i}\right),$ and then carefully piece together the piecewise linear functions on those subintervals.

Now we are ready to define what a smooth curve is.

By a smooth curve from a point ${z}_{1}$ to a different point ${z}_{2}$ in the plane, we mean a set $C\subseteq C$ that is the range of a 1-1, smooth, function $\phi :\left[a,b\right]\to C,$ where $\left[a,b\right]$ is a bounded closed interval in $R,$ where ${z}_{1}=\phi \left(a\right)$ and ${z}_{2}=\phi \left(b\right),$ and satisfying ${\phi }^{\text{'}}\left(t\right)\ne 0$ for all $t\in \left(a,b\right).$

More generally, if $\phi :\left[a,b\right]\to {R}^{2}$ is 1-1 and piecewise smooth on $\left[a,b\right],$ and if $\left\{{t}_{0}<{t}_{1}<...<{t}_{n}\right\}$ is a partition of $\left[a,b\right]$ such that ${\phi }^{\text{'}}\left(t\right)\ne 0$ for all $t\in \left({t}_{i-1},{t}_{i}\right),$ then the range $C$ of $\phi$ is called a piecewise smooth curve from ${z}_{1}=\phi \left(a\right)$ to ${z}_{2}=\phi \left(b\right).$

In either of these cases, $\phi$ is called a parameterization of the curve $C.$

Note that we do not assume that $|{\phi }^{\text{'}}|$ is improperly-integrable, though the preceding theorem might have made you think we would.

REMARK Throughout this chapter we will be continually faced with the fact that a given curve can have many different parameterizations.Indeed, if ${\phi }_{1}:\left[a,b\right]\to C$ is a parameterization, and if $g:\left[c,d\right]\to \left[a,b\right]$ is a smooth function having a nonzero derivative, then ${\phi }_{2}\left(s\right)={\phi }_{1}\left(g\left(s\right)\right)$ is another parameterization of $C.$ Since our definitions and proofs about curves often involve a parametrization, we will frequently need to prove that the results we obtain are independent of the parameterization.The next theorem will help; it shows that any two parameterizations of $C$ are connected exactly as above, i.e., there always is such a function $g$ relating ${\phi }_{1}$ and ${\phi }_{2}.$

Let ${\phi }_{1}:\left[a,b\right]\to C$ and ${\phi }_{2}:\left[c,d\right]\to C$ be two parameterizations of a piecewise smooth curve $C$ joining ${z}_{1}$ to ${z}_{2}.$ Then there exists a piecewise smooth function $g:\left[c,d\right]\to \left[a,b\right]$ such that ${\phi }_{2}\left(s\right)={\phi }_{1}\left(g\left(s\right)\right)$ for all $s\in \left[c,d\right].$ Moreover, the derivative ${g}^{\text{'}}$ of $g$ is nonzero for all but a finite number of points in $\left[c,d\right].$

Because both ${\phi }_{1}$ and ${\phi }_{2}$ are continuous and 1-1, it follows from [link] that the function $g={\phi }_{1}^{-1}\circ {\phi }_{2}$ is continuous and 1-1 from $\left[c,d\right]$ onto $\left[a,b\right].$ Moreover, from [link] , it must also be that $g$ is strictly increasing or strictly decreasing. Write ${\phi }_{1}\left(t\right)={u}_{1}\left(t\right)+i{v}_{1}\left(t\right)\equiv \left({u}_{1}\left(t\right),{v}_{1}\left(t\right)\right),$ and ${\phi }_{2}\left(s\right)={u}_{2}\left(s\right)+i{v}_{2}\left(s\right)\equiv \left({u}_{2}\left(s\right),{v}_{2}\left(s\right)\right).$ Let $\left\{{x}_{0}<{x}_{1}<...<{x}_{p}\right\}$ be a partition of $\left[a,b\right]$ for which ${\phi }_{1}^{\text{'}}$ is continuous and nonzeroon the subintervals $\left({x}_{j-1},{x}_{j}\right),$ and let $\left\{{y}_{0}<{y}_{1}<...<{y}_{q}\right\}$ be a partition of $\left[c,d\right]$ for which ${\phi }_{2}^{\text{'}}$ is continuous and nonzero on the subintervals $\left({y}_{k-1},{y}_{k}\right).$ Then let $\left\{{s}_{0}<{s}_{1}<...<{s}_{n}\right\}$ be the partition of $\left[c,d\right]$ determined by the finitely many points $\left\{{y}_{k}\right\}\cup \left\{{g}^{-1}\left({x}_{j}\right)\right\}.$ We will show that $g$ is continuously differentiable at each point $s$ in the subintervals $\left({s}_{i-1},{s}_{i}\right).$

Fix an $s$ in one of the intervals $\left({s}_{i-1},{s}_{i}\right),$ and let $t={\phi }_{1}^{-1}\left({\phi }_{2}\left(s\right)\right)=g\left(s\right).$ Of course this means that ${\phi }_{1}\left(t\right)={\phi }_{2}\left(s\right),$ or ${u}_{1}\left(t\right)={u}_{2}\left(s\right)$ and ${v}_{1}\left(t\right)={v}_{2}\left(s\right).$ Then $t$ is in some one of the intervals $\left({x}_{j-1},{x}_{j}\right),$ so that we know that ${\phi }_{1}^{\text{'}}\left(t\right)\ne 0.$ Therefore, we must have that at least one of ${u}_{1}^{\text{'}}\left(t\right)$ or ${v}_{1}^{\text{'}}\left(t\right)$ is nonzero. Suppose it is ${v}_{1}^{\text{'}}\left(t\right)$ that is nonzero. The argument, in case it is ${u}_{1}^{\text{'}}\left(t\right)$ that is nonzero, is completely analogous. Now, because ${v}_{1}^{\text{'}}$ is continuous at $t$ and ${v}_{1}^{\text{'}}\left(t\right)\ne 0,$ it follows that ${v}_{1}$ is strictly monotonic in some neighborhood $\left(t-\delta ,t+\delta \right)$ of $t$ and therefore is 1-1 on that interval. Then ${v}_{1}^{-1}$ is continuous by [link] , and is differentiable at the point ${v}_{1}\left(t\right)$ by the Inverse Function Theorem. We will show that on this small interval $g={v}_{1}^{-1}\circ {v}_{2},$ and this will prove that $g$ is continuously differentiable at $s.$

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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