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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 non-magnetic 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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