# 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.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
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?