# 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.

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
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do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
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it is a goid question and i want to know the answer as well
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Abigail
for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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NANO
what is fullerene does it is used to make bukky balls
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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?
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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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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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?
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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
how did you get the value of 2000N.What calculations are needed to arrive at it
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