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A simple diagram of an increasing concave down curve C in vector field F, with no coordinate plane. Towards the top of the curve, the normal n is drawn perpendicular to the curve C. Another arrow F is drawn sharing n’s endpoint. This flux points up and to the right at about a 90-degree angle to n. The arrows in the vector field to the left of n are drawn pointing straight up. The arrows after n point in the same direction as the flux.
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 r ( t ) = x ( t ) , y ( t ) , a t b . Let n ( t ) = y ( t ) , x ( t ) . The flux of F across C is

C F · N d s = a b F ( r ( t ) ) · n ( t ) d t


The proof of [link] is similar to the proof of [link] . Before deriving the formula, note that n ( t ) = y ( t ) , x ( t ) = ( y ( t ) ) 2 + ( x ( t ) ) 2 = r ( t ) . Therefore,

C F · N d s = C F · n ( t ) n ( t ) d s = a b F · n ( t ) n ( t ) r ( t ) d t = a b F ( r ( t ) ) · n ( t ) d t .

Flux across a curve

Calculate the flux of F = 2 x , 2 y across a unit circle oriented counterclockwise ( [link] ).

A unit circle in a vector field in two dimensions. The arrows point away from the origin in a radial pattern. Shorter vectors are near the origin, and longer ones are further away. A unit circle is drawn around the origin to fit the pattern, and arrowheads are drawn on the circle in a counterclockwise manner.
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 r ( t ) = cos t , sin t , 0 t 2 π . The normal vector to a unit circle is cos t , sin t . Therefore, the flux is

C F · N d s = 0 2 π 2 cos t , 2 sin t · cos t , sin t d t = 0 2 π ( 2 cos 2 t + 2 sin 2 t ) d t = 2 0 2 π ( cos 2 t + sin 2 t ) d t = 2 0 2 π d t = 4 π .
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Calculate the flux of F = x + y , 2 y across the line segment from ( 0 , 0 ) to ( 2 , 3 ) , where the curve is oriented from left to right.


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Let F ( x , y ) = P ( x , y ) , Q ( x , y ) be a two-dimensional vector field. Recall that integral C F · T d s is sometimes written as C P d x + Q d y . Analogously, flux C F · N d s is sometimes written in the notation C Q d x + P d y , because the unit normal vector N is perpendicular to the unit tangent T . Rotating the vector d r = d x , d y by 90° results in vector d y , d x . Therefore, the line integral in [link] can be written as C −2 y d x + 2 x d y .

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: C F · T d s . 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 v ( P ) · T ( P ) . The maximum value of v ( P ) · T ( P ) occurs when the two vectors are pointing in the exact same direction; the minimum value of v ( P ) · T ( P ) occurs when the two vectors are pointing in opposite directions. Thus, the value of the circulation C v · T d s measures the tendency of the fluid to move in the direction of C .

Calculating circulation

Let 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: r ( t ) = cos t , sin t , 0 t 2 π . Then, F ( r ( t ) ) = sin t , cos t and r ( t ) = sin t , cos t . Therefore, the circulation of F along C is

C F · T d s = 0 2 π sin t , cos t · sin t , cos t d t = 0 2 π ( sin 2 t + cos 2 t ) d t = 0 2 π d t = 2 π .

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.

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Introduction about quantum dots in nanotechnology
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s. Reply
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are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
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s. Reply
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or in general
in general
Graphene has a hexagonal structure
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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