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In [link] , what if we had oriented the unit circle clockwise? We denote the unit circle oriented clockwise by C . Then

C F · T d s = C F · T d s = −2 π .

Notice that the circulation is negative in this case. The reason for this is that the orientation of the curve flows against the direction of F .

Calculate the circulation of F ( x , y ) = y x 2 + y 2 , x x 2 + y 2 along a unit circle oriented counterclockwise.

2 π

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Calculating work

Calculate the work done on a particle that traverses circle C of radius 2 centered at the origin, oriented counterclockwise, by field F ( x , y ) = −2 , y . Assume the particle starts its movement at ( 1 , 0 ) .

The work done by F on the particle is the circulation of F along C : C F · T d s . We use the parameterization r ( t ) = 2 cos t , 2 sin t , 0 t 2 π for C . Then, r ( t ) = −2 sin t , 2 cos t and F ( r ( t ) ) = −2 , 2 sin t . Therefore, the circulation of F along C is

C F · T d s = 0 2 π −2 , 2 sin t · −2 sin t , 2 cos t d t = 0 2 π ( 4 sin t + 4 sin t cos t ) d t = [ −4 cos t + 4 sin 2 t ] 0 2 π = ( −4 cos ( 2 π ) + 2 sin 2 ( 2 π ) ) ( −4 cos ( 0 ) + 4 sin 2 ( 0 ) ) = −4 + 4 = 0.

The force field does zero work on the particle.

Notice that the circulation of F along C is zero. Furthermore, notice that since F is the gradient of f ( x , y ) = −2 x + y 2 2 , F is conservative. We prove in a later section that under certain broad conditions, the circulation of a conservative vector field along a closed curve is zero.

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Calculate the work done by field F ( x , y ) = 2 x , 3 y on a particle that traverses the unit circle. Assume the particle begins its movement at ( −1 , 0 ) .

0

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Key concepts

  • Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along the x -axis, but the domain of integration in a line integral is a curve in a plane or in space.
  • If C is a curve, then the length of C is C d s .
  • There are two kinds of line integral: scalar line integrals and vector line integrals. Scalar line integrals can be used to calculate the mass of a wire; vector line integrals can be used to calculate the work done on a particle traveling through a field.
  • Scalar line integrals can be calculated using [link] ; vector line integrals can be calculated using [link] .
  • Two key concepts expressed in terms of line integrals are flux and circulation. Flux measures the rate that a field crosses a given line; circulation measures the tendency of a field to move in the same direction as a given closed curve.

Key equations

  • Calculating a scalar line integral
    C f ( x , y , z ) d s = a b f ( r ( t ) ) ( x ( t ) ) 2 + ( y ( t ) ) 2 + ( z ( t ) ) 2 d t
  • Calculating a vector line integral
    C F · d s = C F · T d s = a b F ( r ( t ) ) · r ( t ) d t
    or
    C P d x + Q d y + R d z = a b ( P ( r ( t ) ) d x d t + Q ( r ( t ) ) d y d t + R ( r ( t ) ) d z d t ) d t
  • Calculating flux
    C F · n ( t ) n ( t ) d s = a b F ( r ( t ) ) · n ( t ) d t

True or False? Line integral C f ( x , y ) d s is equal to a definite integral if C is a smooth curve defined on [ a , b ] and if function f is continuous on some region that contains curve C .

True

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True or False? Vector functions r 1 = t i + t 2 j , 0 t 1 , and r 2 = ( 1 t ) i + ( 1 t ) 2 j , 0 t 1 , define the same oriented curve.

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True or False? C ( P d x + Q d y ) = C ( P d x Q d y )

False

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True or False? A piecewise smooth curve C consists of a finite number of smooth curves that are joined together end to end.

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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?
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research.net
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sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
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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.
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Do you know which machine is used to that process?
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
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or in general
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in general
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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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