# 6.5 Divergence and curl  (Page 3/9)

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## Determining whether a field is source free

Is field $\text{F}\left(x,y\right)=⟨{x}^{2}y,5-x{y}^{2}⟩$ source free?

Note the domain of F is ${ℝ}^{2},$ which is simply connected. Furthermore, F is continuous with differentiable component functions. Therefore, we can use [link] to analyze F . The divergence of F is

$\frac{\partial }{\partial x}\left({x}^{2}y\right)+\frac{\partial }{\partial y}\left(5-x{y}^{2}\right)=2xy-2xy=0.$

Therefore, F is source free by [link] .

Let $\text{F}\left(x,y\right)=⟨\text{−}ay,bx⟩$ be a rotational field where a and b are positive constants. Is F source free?

Yes

Recall that the flux form of Green’s theorem says that

${\oint }_{C}\text{F}·\text{N}ds={\iint }_{D}{P}_{x}+{Q}_{y}dA,$

where C is a simple closed curve and D is the region enclosed by C . Since ${P}_{x}+{Q}_{y}=\text{div}\phantom{\rule{0.2em}{0ex}}\text{F},$ Green’s theorem is sometimes written as

${\oint }_{C}\text{F}·\text{N}ds={\iint }_{D}\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}dA.$

Therefore, Green’s theorem can be written in terms of divergence. If we think of divergence as a derivative of sorts, then Green’s theorem says the “derivative” of F on a region can be translated into a line integral of F along the boundary of the region. This is analogous to the Fundamental Theorem of Calculus, in which the derivative of a function $f$ on a line segment $\left[a,b\right]$ can be translated into a statement about $f$ on the boundary of $\left[a,b\right].$ Using divergence, we can see that Green’s theorem is a higher-dimensional analog of the Fundamental Theorem of Calculus.

We can use all of what we have learned in the application of divergence. Let v be a vector field modeling the velocity of a fluid. Since the divergence of v at point P measures the “outflowing-ness” of the fluid at P , $\text{div}\phantom{\rule{0.2em}{0ex}}\text{v}\left(P\right)>0$ implies that more fluid is flowing out of P than flowing in. Similarly, $\text{div}\phantom{\rule{0.2em}{0ex}}\text{v}\left(P\right)<0$ implies the more fluid is flowing in to P than is flowing out, and $\text{div}\phantom{\rule{0.2em}{0ex}}\text{v}\left(P\right)=0$ implies the same amount of fluid is flowing in as flowing out.

## Determining flow of a fluid

Suppose $\text{v}\left(x,y\right)=⟨\text{−}xy,y⟩,y>0$ models the flow of a fluid. Is more fluid flowing into point $\left(1,4\right)$ than flowing out?

To determine whether more fluid is flowing into $\left(1,4\right)$ than is flowing out, we calculate the divergence of v at $\left(1,4\right)\text{:}$

$\text{div}\left(\text{v}\right)=\frac{\partial }{\partial x}\left(\text{−}xy\right)+\frac{\partial }{\partial y}\left(y\right)=\text{−}y+1.$

To find the divergence at $\left(1,4\right),$ substitute the point into the divergence: $-4+1=-3.$ Since the divergence of v at $\left(1,4\right)$ is negative, more fluid is flowing in than flowing out ( [link] ).

For vector field $\text{v}\left(x,y\right)=⟨\text{−}xy,y⟩,y>0,$ find all points P such that the amount of fluid flowing in to P equals the amount of fluid flowing out of P .

All points on line $y=1.$

## Curl

The second operation on a vector field that we examine is the curl, which measures the extent of rotation of the field about a point. Suppose that F represents the velocity field of a fluid. Then, the curl of F at point P is a vector that measures the tendency of particles near P to rotate about the axis that points in the direction of this vector. The magnitude of the curl vector at P measures how quickly the particles rotate around this axis. In other words, the curl at a point is a measure of the vector field’s “spin” at that point. Visually, imagine placing a paddlewheel into a fluid at P , with the axis of the paddlewheel aligned with the curl vector ( [link] ). The curl measures the tendency of the paddlewheel to rotate.

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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
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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?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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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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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?