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Gauss and Stokes Theorem
Lachlan
July 2024

1

Gauss Theorem

Imagine you have a blob somewhere in space (A hollow shell, a surface), and there are some
vectors (F) inside the blob
...

y
∇ · F dV
(2)
V

This integral adds together all the vectors flowing out of all the tiny volumes of dV
...
We finally have
Z
∇ · F dV
(3)
V

Another way to find at the total divergence is to notice that as a result from these little
divergences from every tiny volume in space, there will be a total divergence passing through
the surface of the blob
...
The reason for dotting is because the ”effective”
component of the vector must be normal to the blob, so dotting F with the normal vector of
the blob ”filters” out everything but the ”effective” component
...

Z
ˆ dS
F·n
(5)
S

This is equal to the previous method in (3), so we can say
Z
Z
ˆ dS
∇ · F dV =
F·n
V

(6)

S

Which tells us that in a given region (enclosed by a mathematical blob), tiny divergences at
every small volume in space within that region will result in a net divergence at the surface of
the region
...
A
question you might ask is how much the vectors are swirling around a given point on the surface
on the plate? We can answer that by the curl
∇×F

(7)

To find how much swirling is going on in total, we can just need to add up the tiny bits of curl
over the entire surface of the plate
...
As a result of these little curls, there will be
a total curl around the whole plate
...
Now we can just sum over the entire
closed loop by a line integral to get the total curl
I
F · dl
(10)
Relating the above to (8), we will have
Z
I
∇ × F dS = F · dl

(11)

S

Which simply states that a bunch of small curls on a given surface (a kind of mathematical
plate that needn’t be flat) will make the whole plate rotate with a net curl found by the line
integral on the right
...
1

Applications
Application of Gauss Theorem - Coloumb from Maxwell

An application of Gauss theorem can be found in electrodynamics
...
We also know the charge density integrates to a charge of Q in
total
...
Now we can modify (12) to say that for
a point charge,
∇ · E = Qδ 3 (x)
2

(14)

We seek to find the electric field evaluated at a distance r away from the particle
...
Mathematically, we can say E · n
the delta function produces a field that points radially outwards, so it will always be normal to
ˆ )
...
This gives
Z
∇ · E dV = E4πr2
(18)
V

Now let’s deal with the right hand side of (15)
...
Here we have a constant function
Q, so naturally, it just pulls out the constant itself
...

E=

Q
4πr2

(21)

Which is precisely Coloumb’s law in scalar form and natural units
...


3

(22)

3
...
This time we seek to
find the magnetic field around a straight current carrying wire at distance r where the electric
field does not change with time
...
This means the current density j can be found by
j =< jδy (y)δz (z), 0, 0 >

(23)

The fourth Maxwell equation states that
∇×B=

1
j
c2

(24)

We can take an infinitesimally thin slice of the wire and construct a mathematical ”plate” of
radius r around it
...

Z
Z
1
j dS
(25)
∇ × B dS = 2
c S
S
Dealing with the left hand side, we can use the Stokes theorem, which gives us
Z
I
∇ × B dS = B · dl

(26)

S

Again, we can pull out the magnetic field because it is the same if evaluated at a distance r
and does not depend on the angle of displacement
...
Using this fact, we now have
Z
∇ × B dS = 2πrB
(28)
S

Now we can deal with the left hand side of (25)
...
A slice of the surface is given by dS = dxdydz
...
Now we can combine the two
sides of the equation to get
2πrB =

j
c2

(30)

After some rearranging, we are left with an expression for B
B=

j
2πc2 r

Which is exactly Orsted’s law in natural units and in scalar form

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