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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

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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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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