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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

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Maira Reply
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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
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Crow Reply
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
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Brian Reply
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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
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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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
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Practice Key Terms 2

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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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