# 6.2 Line integrals  (Page 9/20)

 Page 9 / 20

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] ).

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.

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
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
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
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
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?