# 6.5 Vector fields, differential forms, and line integrals

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 $\stackrel{\to }{V}\left(x,y\right)\equiv \left(P\left(x,y\right),Q\left(x,y\right)\right)$ from $U$ into ${R}^{2}.$ The functions $P$ and $Q$ are called the components of the vector field $\stackrel{\to }{V}.$

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

$\frac{tialP}{tialx},\frac{tialP}{tialy},\frac{tialQ}{tialx}and\frac{tialQ}{tialy}$

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 $\left(x,y\right)$ with magnitude $|\stackrel{\to }{V}\left(x,y\right)|$ and acting in the direction of the vector $\stackrel{\to }{V}\left(x,y\right).$ 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 $\stackrel{\to }{u}$ 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 $\stackrel{\to }{u}$ and the distance traveled by the particle in that direction. That is, we must compute dot products of the vectors $\stackrel{\to }{V}\left(x,y\right)$ and $\stackrel{\to }{u}\left(x,y\right).$ Therefore, if a particle is moving along a curve $C,$ parameterized with respect to arc length by $\gamma :\left[0,L\right]\to C,$ and we write $\gamma \left(t\right)=\left(x\left(t\right),y\left(t\right)\right),$ then the work $W\left({z}_{1},{z}_{2}\right)$ done on the particle as it moves from ${z}_{1}=\gamma \left(0\right)$ to ${z}_{2}=\gamma \left(L\right)$ within the force field $\stackrel{\to }{V},$ should intuitively be given by the formula

$\begin{array}{ccc}\hfill W\left({z}_{1},{z}_{2}\right)& =& {\int }_{0}^{L}⟨\stackrel{\to }{V}\left(\gamma \left(t\right)\right)\mid {\gamma }^{\text{'}}\left(t\right)⟩\phantom{\rule{0.166667em}{0ex}}dt\hfill \\ & =& {\int }_{0}^{L}P\left(x\left(t\right),y\left(t\right)\right){x}^{\text{'}}\left(t\right)+Q\left(x\left(t\right),y\left(t\right)\right){y}^{\text{'}}\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt\hfill \\ & \equiv & {\int }_{C}P\phantom{\rule{0.166667em}{0ex}}dx+Q\phantom{\rule{0.166667em}{0ex}}dy,\hfill \end{array}$

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 $\omega =Pdx+Qdy,$ and is determined by two continuous real-valued functions $P$ and $Q$ on $U.$ We say that $\omega$ 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 $\omega$ 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 $\omega =Pdx+Qdy$ 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 ${\int }_{C}\omega$ of $\omega$ over $C$ by

${\int }_{C}\omega ={\int }_{C}P\phantom{\rule{0.166667em}{0ex}}dx+Q\phantom{\rule{0.166667em}{0ex}}dy={\int }_{0}^{L}P\left(\gamma \left(t\right)\right){x}^{\text{'}}\left(t\right)+Q\left(\gamma \left(t\right)\right){y}^{\text{'}}\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt,$

where $\gamma \left(t\right)=\left(x\left(t\right),y\left(t\right)\right)$ is a parameterization of $C$ by arc length.

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!