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Determining whether a field is source free

Is field F ( x , y ) = x 2 y , 5 x y 2 source free?

Note the domain of F is 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

x ( x 2 y ) + y ( 5 x y 2 ) = 2 x y 2 x y = 0 .

Therefore, F is source free by [link] .

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Let F ( x , y ) = a y , b x be a rotational field where a and b are positive constants. Is F source free?


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

C F · N d s = D P x + Q y d A ,

where C is a simple closed curve and D is the region enclosed by C . Since P x + Q y = div F , Green’s theorem is sometimes written as

C F · N d s = D div F d A .

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 [ a , b ] can be translated into a statement about f on the boundary of [ a , b ] . 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 , div v ( P ) > 0 implies that more fluid is flowing out of P than flowing in. Similarly, div v ( P ) < 0 implies the more fluid is flowing in to P than is flowing out, and div v ( P ) = 0 implies the same amount of fluid is flowing in as flowing out.

Determining flow of a fluid

Suppose v ( x , y ) = x y , y , y > 0 models the flow of a fluid. Is more fluid flowing into point ( 1 , 4 ) than flowing out?

To determine whether more fluid is flowing into ( 1 , 4 ) than is flowing out, we calculate the divergence of v at ( 1 , 4 ) :

div ( v ) = x ( x y ) + y ( y ) = y + 1 .

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

A vector field in two dimensions with negative divergence at (1,4). The arrows are very flat but become more vertical closer to the y axis. Above the x axis, the arrows point up and towards the y axis on either side of it. Below the x axis, the arrows point down and away from the y axis on either side of it.
Vector field v ( x , y ) = x y , y has negative divergence at ( 1 , 4 ) .
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For vector field v ( x , y ) = x y , 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 .

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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.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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what school?
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Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
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Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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