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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
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
ya I also want to know the raman spectra
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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.
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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