# 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 $s.$

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
what is biological synthesis of nanoparticles
Sanket Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

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?  By Mariah Hauptman       By Anindyo Mukhopadhyay 