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We motivate our third definition of an integral over a curve by returning to physics.This definition is very much a real variable one, so that we think of the plane as R 2 instead of C . A connection between this real variable definition and the complex variable definition of a contour integral will emerge later.

We motivate our third definition of an integral over a curve by returning to physics.This definition is very much a real variable one, so that we think of the plane as R 2 instead of C . A connection between this real variable definition and the complex variable definition of a contour integral will emerge later.

By a vector field on an open subset U of R 2 , we mean nothing more than a continuous function V ( x , y ) ( P ( x , y ) , Q ( x , y ) ) from U into R 2 . The functions P and Q are called the components of the vector field V .

We will also speak of smooth vector fields, by which we will mean vector fields V both of whose component functions P and Q have continuous partial derivatives

t i a l P t i a l x , t i a l P t i a l y , t i a l Q t i a l x a n d t i a l Q t i a l y

on U .

The idea from physics is to think of a vector field as a force field, i.e., something thatexerts a force at the point ( x , y ) with magnitude | V ( x , y ) | and acting in the direction of the vector V ( x , y ) . For a particle to move within a force field, “work” must be done, that is energy must be provided to move the particle against the force,or energy is given to the particle as it moves under the influence of the force field. In either case, the basicdefinition of work is the product of force and distance traveled. More precisely, if a particle is moving in a direction u within a force field, then the work done on the particle is the product of the component of the force field in the direction of u and the distance traveled by the particle in that direction. That is, we must compute dot products of the vectors V ( x , y ) and u ( x , y ) . Therefore, if a particle is moving along a curve C , parameterized with respect to arc length by γ : [ 0 , L ] C , and we write γ ( t ) = ( x ( t ) , y ( t ) ) , then the work W ( z 1 , z 2 ) done on the particle as it moves from z 1 = γ ( 0 ) to z 2 = γ ( L ) within the force field V , should intuitively be given by the formula

W ( z 1 , z 2 ) = 0 L V ( γ ( t ) ) γ ' ( t ) d t = 0 L P ( x ( t ) , y ( t ) ) x ' ( t ) + Q ( x ( t ) , y ( t ) ) y ' ( t ) d t C P d x + Q d y ,

where the last expression is explicitly defining the shorthand notation we will be using.

The preceding discussion leads us to a new notion of what kind of object should be “integrated” over a curve.

A differential form on a subset U of R 2 is denoted by ω = P d x + Q d y , and is determined by two continuous real-valued functions P and Q on U . We say that ω is bounded or uniformly continuous if the functions P and Q are bounded or uniformly continuous functions on U . We say that the differential form ω is smooth of order k if the set U is open, and the functions P and Q have continuous mixed partial derivatives of order k .

If ω = P d x + Q d y is a differential form on a set U , and if C is any piecewise smooth curve of finite length contained in U , then we define the line integral C ω of ω over C by

C ω = C P d x + Q d y = 0 L P ( γ ( t ) ) x ' ( t ) + Q ( γ ( t ) ) y ' ( t ) d t ,

where γ ( t ) = ( x ( t ) , y ( t ) ) is a parameterization of C by arc length.

Questions & Answers

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s. Reply
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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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Graphene has a hexagonal structure
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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