<< Chapter < Page Chapter >> Page >

Let G ( x , y ) = x 2 y i ( x + y ) j be a vector field in 2 . What vector is associated with the point ( −2 , 3 ) ?

12 i j

Got questions? Get instant answers now!

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 2 , as is the range. Therefore the “graph” of a vector field in 2 lives in four-dimensional space. Since we cannot represent four-dimensional space visually, we instead draw vector fields in 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 ( 4 , −1 ) is 3 , 1 . Then, we would draw vector 3 , 1 at point ( 4 , −1 ) .

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 2 .

There are two types of vector fields in 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 ( x , y ) is perpendicular to the circle centered at the origin that contains point ( x , y ) , and all other vectors on this circle have the same magnitude.

Drawing a radial vector field

Sketch the vector field F ( x , y ) = x 2 i + y 2 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.

( x , y ) F ( x , y ) ( x , y ) F ( x , y ) ( x , y ) F ( x , y )
( 1 , 0 ) 1 2 , 0 ( 2 , 0 ) 1 , 0 ( 1 , 1 ) 1 2 , 1 2
( 0 , 1 ) 0 , 1 2 ( 0 , 2 ) 0 , 1 ( −1 , 1 ) 1 2 , 1 2
( −1 , 0 ) 1 2 , 0 ( −2 , 0 ) −1 , 0 ( −1 , −1 ) 1 2 , 1 2
( 0 , −1 ) 0 , 1 2 ( 0 , −2 ) 0 , −1 ( 1 , −1 ) 1 2 , 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.

Visual representations of a radial vector field on a coordinate field. The arrows are stretching away from the origin in a radial pattern. The magnitudes increase the further the arrows are from the origin, so the lines are longer. The second version shows concentric circles around the origin to highlight the radial pattern.
(a) A visual representation of the radial vector field F ( x , y ) = x 2 i + y 2 j . (b) The radial vector field F ( x , y ) = x 2 i + y 2 j with overlaid circles. Notice that each vector is perpendicular to the circle on which it is located.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Draw the radial field F ( x , y ) = x 3 i y 3 j .

A visual representation of the given radial field in a coordinate plane. The magnitudes increase further from the origin. The arrow seem to be stretching away from the origin in a rectangular shape.

Got questions? Get instant answers now!

In contrast to radial fields, in a rotational field    , the vector at point ( x , y ) is tangent (not perpendicular) to a circle with radius r = 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....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
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?
Radek Reply
Practice Key Terms 7

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?