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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 ) .


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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
    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 .


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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 )


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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

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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?
Radek Reply
Practice Key Terms 8

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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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