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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

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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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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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