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A simple exposition of Gauss's theorem or the divergence theorem.

Gauss' theorem

Consider the following volume enclosed by a surface we will call S .

Now we will embed S in a vector field:

We will cut the the object into two volumes that are enclosed by surfaces we will call S 1 and S 2 .

Again we embed it in the same vectorfield.
It is clear that flux through S 1 + S 2 is equal to flux through S . This is because the flux through one side of the plane is exactly opposite to theflux through the other side of the plane:
So we see that S F d a = S 1 F d a 1 + S 2 F d a 2 . We could subdivide the surface as much as we want and so for n subdivisions the integral becomes:

S F d a = i = 1 n S i F d a i . What is S i F d a i .? We can subdivide the volume into a bunch of littlecubes:

To first order (which is all that matters since we will take the limit of a smallvolume) the field at a point at the bottom of the box is F z + Δ x 2 F z x + Δ y 2 F z y where we have assumed the middle of the bottom of the box is the point ( x + Δ x 2 , y + Δ y 2 , z ) . Through the top of the box ( x + Δ x 2 , y + Δ y 2 , z + Δ z ) you get F z + Δ x 2 F z x + Δ y 2 F z y + Δ z F z z Through the top and bottom surfaces you get Flux Top - Flux bottom

Which is Δ x Δ y Δ z F z z = Δ V F z z

Likewise you get the same result in the other dimensionsHence S i F d a i = Δ V i [ F x x + F y y + F z z ]

or S i F d a i = F Δ V i S F d a = i = 1 n S i F d a i = i = 1 n F Δ V i

So in the limit that Δ V i 0 and n S F d a = V F V

This result is intimately connected to the fundamental definition of the divergence which is F lim V 0 1 V S F d a where the integral is taken over the surface enclosing the volume V . The divergence is the flux out of a volume, per unit volume, in the limit ofan infinitely small volume. By our proof of Gauss' theorem, we have shown that the del operator acting on a vector field captures this definition.

Questions & Answers

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
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?
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?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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I'm interested in nanotube
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Ramkumar Reply
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Sravani Reply
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Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Waves and optics. OpenStax CNX. Nov 17, 2005 Download for free at http://cnx.org/content/col10279/1.33
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