# 6.2 Line integrals  (Page 9/20)

 Page 9 / 20 The flux of vector field F across curve C is computed by an integral similar to a vector line integral.

We now give a formula for calculating the flux across a curve. This formula is analogous to the formula used to calculate a vector line integral (see [link] ).

## Calculating flux across a curve

Let F be a vector field and let C be a smooth curve with parameterization $\text{r}\left(t\right)=⟨x\left(t\right),y\left(t\right)⟩,a\le t\le b.$ Let $\text{n}\left(t\right)=⟨{y}^{\prime }\left(t\right),\text{−}{x}^{\prime }\left(t\right)⟩.$ The flux of F across C is

${\int }_{C}\text{F}·\text{N}ds={\int }_{a}^{b}\text{F}\left(\text{r}\left(t\right)\right)·\text{n}\left(t\right)dt$

## Proof

The proof of [link] is similar to the proof of [link] . Before deriving the formula, note that $‖\text{n}\left(t\right)‖=‖⟨y\prime \left(t\right),\text{−}x\prime \left(t\right)⟩‖=\sqrt{{\left(y\prime \left(t\right)\right)}^{2}+{\left(x\prime \left(t\right)\right)}^{2}}=‖{r}^{\prime }\left(t\right)‖.$ Therefore,

$\begin{array}{cc}\hfill {\int }_{C}\text{F}·\text{N}ds& ={\int }_{C}\text{F}·\frac{\text{n}\left(t\right)}{‖\text{n}\left(t\right)‖}ds\hfill \\ & ={\int }_{a}^{b}\text{F}·\frac{\text{n}\left(t\right)}{‖\text{n}\left(t\right)‖}‖{r}^{\prime }\left(t\right)‖dt\hfill \\ & ={\int }_{a}^{b}\text{F}\left(\text{r}\left(t\right)\right)·\text{n}\left(t\right)dt.\hfill \end{array}$

## Flux across a curve

Calculate the flux of $\text{F}=⟨2x,2y⟩$ across a unit circle oriented counterclockwise ( [link] ). A unit circle in vector field F = ⟨ 2 x , 2 y ⟩ .

To compute the flux, we first need a parameterization of the unit circle. We can use the standard parameterization $\text{r}\left(t\right)=⟨\text{cos}\phantom{\rule{0.2em}{0ex}}t,\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩,$ $0\le t\le 2\pi .$ The normal vector to a unit circle is $⟨\text{cos}\phantom{\rule{0.2em}{0ex}}t,\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩.$ Therefore, the flux is

$\begin{array}{cc}\hfill {\int }_{C}\text{F}·\text{N}ds& ={\int }_{0}^{2\pi }⟨2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t,2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩·⟨\text{cos}\phantom{\rule{0.2em}{0ex}}t,\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩\phantom{\rule{0.2em}{0ex}}dt\hfill \\ & ={\int }_{0}^{2\pi }\left(2\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{2}t+2\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}t\right)\phantom{\rule{0.2em}{0ex}}dt=2{\int }_{0}^{2\pi }\left({\text{cos}}^{2}t+{\text{sin}}^{2}t\right)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ & =2{\int }_{0}^{2\pi }dt=4\pi .\hfill \end{array}$

Calculate the flux of $\text{F}=⟨x+y,2y⟩$ across the line segment from $\left(0,0\right)$ to $\left(2,3\right),$ where the curve is oriented from left to right.

3/2

Let $\text{F}\left(x,y\right)=⟨P\left(x,y\right),Q\left(x,y\right)⟩$ be a two-dimensional vector field. Recall that integral ${\int }_{C}\text{F}·\text{T}ds$ is sometimes written as ${\int }_{C}Pdx+Qdy.$ Analogously, flux ${\int }_{C}\text{F}·\text{N}ds$ is sometimes written in the notation ${\int }_{C}\text{−}Qdx+Pdy,$ because the unit normal vector N is perpendicular to the unit tangent T . Rotating the vector $d\text{r}=⟨dx,dy⟩$ by 90° results in vector $⟨dy,\text{−}dx⟩.$ Therefore, the line integral in [link] can be written as ${\int }_{C}-2ydx+2xdy.$

Now that we have defined flux, we can turn our attention to circulation. The line integral of vector field F along an oriented closed curve is called the circulation    of F along C . Circulation line integrals have their own notation: ${\oint }_{C}\text{F}·\text{T}ds.$ The circle on the integral symbol denotes that C is “circular” in that it has no endpoints. [link] shows a calculation of circulation.

To see where the term circulation comes from and what it measures, let v represent the velocity field of a fluid and let C be an oriented closed curve. At a particular point P , the closer the direction of v ( P ) is to the direction of T ( P ), the larger the value of the dot product $\text{v}\left(P\right)·\text{T}\left(P\right).$ The maximum value of $\text{v}\left(P\right)·\text{T}\left(P\right)$ occurs when the two vectors are pointing in the exact same direction; the minimum value of $\text{v}\left(P\right)·\text{T}\left(P\right)$ occurs when the two vectors are pointing in opposite directions. Thus, the value of the circulation ${\oint }_{C}\text{v}·\text{T}ds$ measures the tendency of the fluid to move in the direction of C .

## Calculating circulation

Let $\text{F}=⟨-y,x⟩$ be the vector field from [link] and let C represent the unit circle oriented counterclockwise. Calculate the circulation of F along C .

We use the standard parameterization of the unit circle: $\text{r}\left(t\right)=⟨\text{cos}\phantom{\rule{0.2em}{0ex}}t,\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩,0\le t\le 2\pi .$ Then, $\text{F}\left(\text{r}\left(t\right)\right)=⟨\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}t,\text{cos}\phantom{\rule{0.2em}{0ex}}t⟩$ and ${r}^{\prime }\left(t\right)=⟨\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}t,\text{cos}\phantom{\rule{0.2em}{0ex}}t⟩.$ Therefore, the circulation of F along C is

$\begin{array}{cc}\hfill {\oint }_{C}\text{F}·\text{T}ds& ={\int }_{0}^{2\pi }⟨\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}t,\text{cos}\phantom{\rule{0.2em}{0ex}}t⟩·⟨\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}t,\text{cos}\phantom{\rule{0.2em}{0ex}}t⟩dt\hfill \\ & ={\int }_{0}^{2\pi }\left({\text{sin}}^{2}t+{\text{cos}}^{2}t\right)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ & ={\int }_{0}^{2\pi }dt=2\pi .\hfill \end{array}$

Notice that the circulation is positive. The reason for this is that the orientation of C “flows” with the direction of F . At any point along the circle, the tangent vector and the vector from F form an angle of less than 90°, and therefore the corresponding dot product is positive.

#### Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
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
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?
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
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
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?
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
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!    By By     By David Martin