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The Divergence Theorem
David Cheng's Fundamentals of Electromagnetics states that:
...the volume integral of the divergence of a vector field equals the total outward flux
of the vector through the surface that bounds the volume.
So, for a vector field A and volume V (bounded by surface S):
/
/


 (del * A) dv = () A * ds


/v
/s
This allows us to reduce a triple integral over a volume to a double integral over a
surface. We can often utilize this to establish other theorems and relations in
electromagnetics. Knowing the divergence theorem allows you to remember these new
relations better, because you have a general idea of where they cam from.
For example, one of the postulates of electomagnetics in free space is:
(free
charge density)
(del * E) = 
(epsilon
)
0
This, combined with the divergence theorem, gives us Gauss's Law:
/
/

1

 (del * E)dV =   (free
charge density) dV

epsilon

/v
0
/v
Therefore, from the divergence theorem :
/

Q
() E * ds = 
,
Q=total free charge

epsilon
/s
0
Which is simply a statement of Gauss's Law
Stoke's Theorem
Again according to Fundamentals of Engineering Electromagnetics, Stoke's theorem states
that:
... the surface integral of the curl of a vector field over an open surface is equal to
the closed line integral of the vector along the contour bounding the surface.
/
/


 (del ¡¿ A) * ds = () A * dL


/s
/c
This allows us to reduce certain surface integral to line integrals and vice versa. Like
the divergence theorem, it is very useful in analyzing vector fields in electromagnetics,
as it allow us to establish other theorems and relations. Hopefully, by knowing where
these theorems come from, you can better understand them and apply them rather than just
memorizing the theorem or relation itself.
For example, Ampere's circuital law in nonmagnetic material can be derived from Stoke's
theorem and one of the postulates of magnetostatics. This postulate states:
(del ¡¿B) = u J , u = the permeability of free space.
0
0
This , combined with stoke's theorem, gives us Ampere's circuital law:
/
/


 (del ¡¿B) * dS = u  J * dS

0

/s
/s
Therefore, from stoke's theorem:
/

() B * dL = u I

0
/c
Which is simply a statement of Ampere's circuital law.
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