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Verify the divergence theorem for vector field F ( x , y , z ) = x + y + z , y , 2 x y and surface S given by the cylinder x 2 + y 2 = 1 , 0 z 3 plus the circular top and bottom of the cylinder. Assume that S is positively oriented.

Both integrals equal 6 π .

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Recall that the divergence of continuous field F at point P is a measure of the “outflowing-ness” of the field at P . If F represents the velocity field of a fluid, then the divergence can be thought of as the rate per unit volume of the fluid flowing out less the rate per unit volume flowing in. The divergence theorem confirms this interpretation. To see this, let P be a point and let B r be a ball of small radius r centered at P ( [link] ). Let S r be the boundary sphere of B r . Since the radius is small and F is continuous, div F ( Q ) div F ( P ) for all other points Q in the ball. Therefore, the flux across S r can be approximated using the divergence theorem:

S r F · d S = B r div F d V B r div F ( P ) d V .

Since div F ( P ) is a constant,

B r div F ( P ) d V = div F ( P ) V ( B r ) .

Therefore, flux S r F · d S can be approximated by div F ( P ) V ( B r ) . This approximation gets better as the radius shrinks to zero, and therefore

div F ( P ) = lim r 0 1 V ( B r ) S r F · d S .

This equation says that the divergence at P is the net rate of outward flux of the fluid per unit volume.

This figure is a diagram of ball B_r, with small radius r centered at P. Arrows are drawn pointing up and to the right across the ball.
Ball B r of small radius r centered at P.

Using the divergence theorem

The divergence theorem translates between the flux integral of closed surface S and a triple integral over the solid enclosed by S . Therefore, the theorem allows us to compute flux integrals or triple integrals that would ordinarily be difficult to compute by translating the flux integral into a triple integral and vice versa.

Applying the divergence theorem

Calculate the surface integral S F · d S , where S is cylinder x 2 + y 2 = 1 , 0 z 2 , including the circular top and bottom, and F = x 3 3 + y z , y 3 3 sin ( x z ) , z x y .

We could calculate this integral without the divergence theorem, but the calculation is not straightforward because we would have to break the flux integral into three separate integrals: one for the top of the cylinder, one for the bottom, and one for the side. Furthermore, each integral would require parameterizing the corresponding surface, calculating tangent vectors and their cross product, and using [link] .

By contrast, the divergence theorem allows us to calculate the single triple integral E div F d V , where E is the solid enclosed by the cylinder. Using the divergence theorem and converting to cylindrical coordinates, we have

s F · d S = E div F d V = E ( x 2 + y 2 + 1 ) d V = 0 2 π 0 1 0 2 ( r 2 + 1 ) r d z d r d θ = 3 2 0 2 π d θ = 3 π .
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Use the divergence theorem to calculate flux integral S F · d S , where S is the boundary of the box given by 0 x 2 , 1 y 4 , 0 z 1 , and F = x 2 + y z , y - z , 2 x + 2 y + 2 z (see the following figure).

This figure is a vector diagram in three dimensions. The box of the figure spans x from 0 to 2; y from 0 to 4; and z from 0 to 1. The vectors point up increasingly with distance from the origin; toward larger x with increasing distance from the origin; and toward smaller y values with increasing height.


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Applying the divergence theorem

Let v = y z , x z , 0 be the velocity field of a fluid. Let C be the solid cube given by 1 x 4 , 2 y 5 , 1 z 4 , and let S be the boundary of this cube (see the following figure). Find the flow rate of the fluid across S .

This is a figure of a diagram of the given vector field in three dimensions. The x components are –y/z, the y components are x/z, and the z components are 0.
Vector field v = y z , x z , 0 .

The flow rate of the fluid across S is S v · d S . Before calculating this flux integral, let’s discuss what the value of the integral should be. Based on [link] , we see that if we place this cube in the fluid (as long as the cube doesn’t encompass the origin), then the rate of fluid entering the cube is the same as the rate of fluid exiting the cube. The field is rotational in nature and, for a given circle parallel to the xy -plane that has a center on the z -axis, the vectors along that circle are all the same magnitude. That is how we can see that the flow rate is the same entering and exiting the cube. The flow into the cube cancels with the flow out of the cube, and therefore the flow rate of the fluid across the cube should be zero.

To verify this intuition, we need to calculate the flux integral. Calculating the flux integral directly requires breaking the flux integral into six separate flux integrals, one for each face of the cube. We also need to find tangent vectors, compute their cross product, and use [link] . However, using the divergence theorem makes this calculation go much more quickly:

S v · d S = C div ( v ) d V = C 0 d V = 0.

Therefore the flux is zero, as expected.

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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
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Crow Reply
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RAW Reply
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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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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?
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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
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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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Adin Reply
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biomolecules are e building blocks of every organics and inorganic materials.
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 3

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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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