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Independence of parameterization

Find the value of integral C ( x 2 + y 2 + z ) d s , where C is part of the helix parameterized by r ( t ) = cos ( 2 t ) , sin ( 2 t ) , 2 t , 0 t π . Notice that this function and curve are the same as in the previous example; the only difference is that the curve has been reparameterized so that time runs twice as fast.

As with the previous example, we use [link] to compute the integral with respect to t . Note that f ( r ( t ) ) = cos 2 ( 2 t ) + sin 2 ( 2 t ) + 2 t = 2 t + 1 and

( x ( t ) ) 2 + ( y ( t ) ) 2 + ( z ( t ) ) 2 = ( sin t + cos t + 4 ) = 2 2

so we have

C ( x 2 + y 2 + z ) d s = 2 2 0 π ( 1 + 2 t ) d t = 2 2 [ t + t 2 ] 0 π = 2 2 ( π + π 2 ) .

Notice that this agrees with the answer in the previous example. Changing the parameterization did not change the value of the line integral. Scalar line integrals are independent of parameterization, as long as the curve is traversed exactly once by the parameterization.

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Evaluate line integral C ( x 2 + y z ) d s , where C is the line with parameterization r ( t ) = 2 t , 5 t , t , 0 t 10 . Reparameterize C with parameterization s ( t ) = 4 t , 10 t , −2 t , 0 t 5 , recalculate line integral C ( x 2 + y z ) d s , and notice that the change of parameterization had no effect on the value of the integral.

Both line integrals equal 1000 30 3 .

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Now that we can evaluate line integrals, we can use them to calculate arc length. If f ( x , y , z ) = 1 , then

C f ( x , y , z ) d s = lim n i = 1 n f ( t i * ) Δ s i = lim n i = 1 n Δ s i = lim n length ( C ) = length ( C ) .

Therefore, C 1 d s is the arc length of C .

Calculating arc length

A wire has a shape that can be modeled with the parameterization r ( t ) = cos t , sin t , t , 0 t 4 π . Find the length of the wire.

The length of the wire is given by C 1 d s , where C is the curve with parameterization r . Therefore,

The length of the wire = C 1 d s = 0 4 π r ( t ) d t = 0 4 π ( sin t ) 2 + cos 2 t + t d t = 0 4 π 1 + t d t = [ 2 ( 1 + t ) 3 / 2 3 ] 0 4 π = 2 3 ( ( 1 + 4 π ) 3 / 2 1 ) .
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Find the length of a wire with parameterization r ( t ) = 3 t + 1 , 4 2 t , 5 + 2 t , 0 t 4 .

4 17

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Vector line integrals

The second type of line integrals are vector line integrals, in which we integrate along a curve through a vector field. For example, let

F ( x , y , z ) = P ( x , y , z ) i + Q ( x , y , z ) j + R ( x , y , z ) k

be a continuous vector field in 3 that represents a force on a particle, and let C be a smooth curve in 3 contained in the domain of F . How would we compute the work done by F in moving a particle along C ?

To answer this question, first note that a particle could travel in two directions along a curve: a forward direction and a backward direction. The work done by the vector field depends on the direction in which the particle is moving. Therefore, we must specify a direction along curve C ; such a specified direction is called an orientation of a curve    . The specified direction is the positive direction along C ; the opposite direction is the negative direction along C . When C has been given an orientation, C is called an oriented curve ( [link] ). The work done on the particle depends on the direction along the curve in which the particle is moving.

A closed curve    is one for which there exists a parameterization r ( t ) , a t b , such that r ( a ) = r ( b ) , and the curve is traversed exactly once. In other words, the parameterization is one-to-one on the domain ( a , b ) .

Two images, labeled A and B. Image A shows a curve C that is an oriented curve. It is a curve that connects two points; it is a line segment with curves. Image B, on the other hand, is a closed curve. It has no endpoints and completely encloses an area.
(a) An oriented curve between two points. (b) A closed oriented curve.

Questions & Answers

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?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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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
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
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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
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
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.
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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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