# 3.4 Gauss' theorem

 Page 1 / 1
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.

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
Got questions? Join the online conversation and get instant answers!  By   By Mistry Bhavesh   By  