# 6.1 Vector fields  (Page 2/15)

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Let $\text{G}\left(x,y\right)={x}^{2}y\text{i}-\left(x+y\right)\text{j}$ be a vector field in ${ℝ}^{2}.$ What vector is associated with the point $\left(-2,3\right)?$

$12\text{i}-\text{j}$

## Drawing a vector field

We can now represent a vector field in terms of its components of functions or unit vectors, but representing it visually by sketching it is more complex because the domain of a vector field is in ${ℝ}^{2},$ as is the range. Therefore the “graph” of a vector field in ${ℝ}^{2}$ lives in four-dimensional space. Since we cannot represent four-dimensional space visually, we instead draw vector fields in ${ℝ}^{2}$ in a plane itself. To do this, draw the vector associated with a given point at the point in a plane. For example, suppose the vector associated with point $\left(4,-1\right)$ is $⟨3,1⟩.$ Then, we would draw vector $⟨3,1⟩$ at point $\left(4,-1\right).$

We should plot enough vectors to see the general shape, but not so many that the sketch becomes a jumbled mess. If we were to plot the image vector at each point in the region, it would fill the region completely and is useless. Instead, we can choose points at the intersections of grid lines and plot a sample of several vectors from each quadrant of a rectangular coordinate system in ${ℝ}^{2}.$

There are two types of vector fields in ${ℝ}^{2}$ on which this chapter focuses: radial fields and rotational fields. Radial fields model certain gravitational fields and energy source fields, and rotational fields model the movement of a fluid in a vortex. In a radial field    , all vectors either point directly toward or directly away from the origin. Furthermore, the magnitude of any vector depends only on its distance from the origin. In a radial field, the vector located at point $\left(x,y\right)$ is perpendicular to the circle centered at the origin that contains point $\left(x,y\right),$ and all other vectors on this circle have the same magnitude.

## Drawing a radial vector field

Sketch the vector field $\text{F}\left(x,y\right)=\frac{x}{2}\phantom{\rule{0.1em}{0ex}}\text{i}+\frac{y}{2}\phantom{\rule{0.1em}{0ex}}\text{j}.$

To sketch this vector field, choose a sample of points from each quadrant and compute the corresponding vector. The following table gives a representative sample of points in a plane and the corresponding vectors.

 $\left(x,y\right)$ $\text{F}\left(x,y\right)$ $\left(x,y\right)$ $\text{F}\left(x,y\right)$ $\left(x,y\right)$ $\text{F}\left(x,y\right)$ $\left(1,0\right)$ $⟨\frac{1}{2},0⟩$ $\left(2,0\right)$ $⟨1,0⟩$ $\left(1,1\right)$ $⟨\frac{1}{2},\frac{1}{2}⟩$ $\left(0,1\right)$ $⟨0,\frac{1}{2}⟩$ $\left(0,2\right)$ $⟨0,1⟩$ $\left(-1,1\right)$ $⟨-\frac{1}{2},\frac{1}{2}⟩$ $\left(-1,0\right)$ $⟨-\frac{1}{2},0⟩$ $\left(-2,0\right)$ $⟨-1,0⟩$ $\left(-1,-1\right)$ $⟨-\frac{1}{2},-\frac{1}{2}⟩$ $\left(0,-1\right)$ $⟨0,-\frac{1}{2}⟩$ $\left(0,-2\right)$ $⟨0,-1⟩$ $\left(1,-1\right)$ $⟨\frac{1}{2},-\frac{1}{2}⟩$

[link] (a) shows the vector field. To see that each vector is perpendicular to the corresponding circle, [link] (b) shows circles overlain on the vector field.

Draw the radial field $\text{F}\left(x,y\right)=-\frac{x}{3}\phantom{\rule{0.1em}{0ex}}\text{i}-\frac{y}{3}\phantom{\rule{0.1em}{0ex}}\text{j}.$

In contrast to radial fields, in a rotational field    , the vector at point $\left(x,y\right)$ is tangent (not perpendicular) to a circle with radius $r=\sqrt{{x}^{2}+{y}^{2}}.$ In a standard rotational field, all vectors point either in a clockwise direction or in a counterclockwise direction, and the magnitude of a vector depends only on its distance from the origin. Both of the following examples are clockwise rotational fields, and we see from their visual representations that the vectors appear to rotate around the origin.

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 to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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