# 6.5 Vector fields, differential forms, and line integrals  (Page 2/3)

REMARK There is no doubt that the integral in this definition exists, because $P$ and $Q$ are continuous functions on the compact set $C,$ hence bounded, and ${\gamma }^{\text{'}}$ is integrable, implying that both ${x}^{\text{'}}$ and ${y}^{\text{'}}$ are integrable. Therefore $P\left(\gamma \left(t\right)\right){x}^{\text{'}}\left(t\right)+Q\left(\gamma \left(t\right)\right){y}^{\text{'}}\left(t\right)$ is integrable on $\left(0,L\right).$

These differential forms $\omega$ really should be called “differential 1-forms.”For instance, an example of a differential 2-form would look like $R\phantom{\rule{0.166667em}{0ex}}dxdy,$ and in higher dimensions, we could introduce notions of differential forms of higher and higher orders, e.g., in 3 dimension things like $P\phantom{\rule{0.166667em}{0ex}}dxdy+Q\phantom{\rule{0.166667em}{0ex}}dzdy+R\phantom{\rule{0.166667em}{0ex}}dxdz.$ Because we will always be dealing with ${R}^{2},$ we will have no need for higher order differential forms, but the study of such things is wonderful.Take a course in Differential Geometry!

Again, we must see how this quantity ${\int }_{C}\omega$ depends, if it does,on different parameterizations. As usual, it does not.

Suppose $\omega =Pdx+Qdy$ is a differential form on a subset $U$ of ${R}^{2}.$

1. Let $C$ be a piecewise smooth curve of finite length contained in $U$ that joins ${z}_{1}$ to ${z}_{2}.$ Prove that
${\int }_{C}\omega ={\int }_{C}P\phantom{\rule{0.166667em}{0ex}}dx+Q\phantom{\rule{0.166667em}{0ex}}dy={\int }_{a}^{b}P\left(\phi \left(t\right)\right){x}^{\text{'}}\left(t\right)+Q\left(\phi \left(t\right)\right){y}^{\text{'}}\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt$
for any parameterization $\phi :\left[a,b\right]\to C$ having components $x\left(t\right)$ and $y\left(t\right).$
2. Let $C$ be as in part (a), and let $\stackrel{^}{C}$ denote the reverse of $C,$ i.e., the same set $C$ but thought of as a curve joining ${z}_{2}$ to ${z}_{1}.$ Show that ${\int }_{\stackrel{^}{c}}\omega =-{\int }_{C}\omega .$
3. Let $C$ be as in part (a). Prove that
$|{\int }_{C}P\phantom{\rule{0.166667em}{0ex}}dx+Q\phantom{\rule{0.166667em}{0ex}}dy|\le \left({M}_{P}+{M}_{Q}\right)L,$
where ${M}_{P}$ and ${M}_{Q}$ are bounds for the continuous functions $|P|$ and $|Q|$ on the compact set $C,$ and where $L$ is the length of $C.$

The simplest interesting example of a differential form is constructed as follows. Suppose $U$ is an open subset of ${R}^{2},$ and let $f:U\to R$ be a differentiable real-valued function of two real variables; i.e., both of its partial derivatives exist at every point $\left(x,y\right)\in U.$ (See the last section of Chapter IV.) Define a differential form $\omega =df,$ called the differential of $f,$ by

$df=\frac{tialf}{tialx}\phantom{\rule{0.166667em}{0ex}}dx+\frac{tialf}{tialy}\phantom{\rule{0.166667em}{0ex}}dy,$

i.e., $P=tialf/tialx$ and $Q=tialf/tialy.$ These differential forms $df$ are called exact differential forms.

REMARK Not every differential form $\omega$ is exact, i.e., of the form $df.$ Indeed, determining which $\omega$ 's are $df$ 's boils down to what may be the simplest possible partial differential equation problem.If $\omega$ is given by two functions $P$ and $Q,$ then saying that $\omega =df$ amounts to saying that $f$ is a solution of the pair of simultaneous partial differential equations

$\frac{tialf}{tialx}=P\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\frac{tialf}{tialy}=Q.$

See part (b) of the exercise below for an example of a nonexact differential form.

Of course if a real-valued function $f$ has continuous partial derivatives of the second order, then [link] tells us that the mixed partials ${f}_{xy}$ and ${f}_{yx}$ must be equal. So, if $\omega =Pdx+Qdy=df$ for some such $f,$ Then $P$ and $Q$ would have to satisfy $tialP/tialy=tialQ/tialx.$ Certainly not every $P$ and $Q$ would satisfy this equation, so it is in fact trivial to find examples of differential forms that are not differentials of functions.A good bit more subtle is the question of whether every differential form $Pdx+Qdy,$ for which $tialP/tialy=tialQ/tialx,$ is equal to some $df.$ Even this is not true in general, as part (c) of the exercise below shows. The open subset $U$ on which the differential form is defined plays a significant role, and, in fact, differential forms provide a way of studyingtopologically different kinds of open sets.

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
Do somebody tell me a best nano engineering book for beginners?
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
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
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!