# 3.4 Gauss' theorem

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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 $S\text{.}$ 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 ${\oint }_{S}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{a}={\oint }_{{S}_{1}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{1}}+{\oint }_{{S}_{2}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{2}}\text{.}$ We could subdivide the surface as much as we want and so for $n$ subdivisions the integral becomes:

${\oint }_{S}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{a}=\sum _{i=1}^{n}{\oint }_{{S}_{i}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{i}}\text{.}$ What is ${\oint }_{{S}_{i}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{i}}$ .? 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 ${F}_{z}+\frac{\Delta x}{2}\frac{\partial {F}_{z}}{\partial x}+\frac{\Delta y}{2}\frac{\partial {F}_{z}}{\partial y}$ where we have assumed the middle of the bottom of the box is the point $\left(x+\frac{\Delta x}{2},y+\frac{\Delta y}{2},z\right)$ . Through the top of the box $\left(x+\frac{\Delta x}{2},y+\frac{\Delta y}{2},z+\Delta z\right)$ you get ${F}_{z}+\frac{\Delta x}{2}\frac{\partial {F}_{z}}{\partial x}+\frac{\Delta y}{2}\frac{\partial {F}_{z}}{\partial y}+\Delta z\frac{\partial {F}_{z}}{\partial z}$ Through the top and bottom surfaces you get Flux Top - Flux bottom 

Which is $\Delta x\Delta y\Delta z\frac{\partial {F}_{z}}{\partial z}=\Delta V\frac{\partial {F}_{z}}{\partial z}$

Likewise you get the same result in the other dimensionsHence ${\oint }_{{S}_{i}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{i}}=\Delta {V}_{i}\left[\frac{\partial {F}_{x}}{\partial x}+\frac{\partial {F}_{y}}{\partial y}+\frac{\partial {F}_{z}}{\partial z}\right]$

or ${\oint }_{{S}_{i}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{i}}=\stackrel{⃗}{\nabla }\cdot \stackrel{⃗}{F}\Delta {V}_{i}$ $\begin{array}{c}{\oint }_{S}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{a}=\sum _{i=1}^{n}{\oint }_{{S}_{i}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{i}}\\ =\sum _{i=1}^{n}\stackrel{⃗}{\nabla }\cdot \stackrel{⃗}{F}\Delta {V}_{i}\end{array}$

So in the limit that $\Delta {V}_{i}\to 0$ and $n\to \infty$ ${\oint }_{S}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{a}={\oint }_{V}\stackrel{⃗}{\nabla }\cdot \stackrel{⃗}{F}dV$

This result is intimately connected to the fundamental definition of the divergence which is $\stackrel{⃗}{\nabla }\cdot \stackrel{⃗}{F}\equiv \underset{V\to 0}{{lim}}\frac{1}{V}{\oint }_{S}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{a}$ 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.

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yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
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Rafiq
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Damian
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LITNING
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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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
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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 ?
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?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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