## Divergence

The divergence of a vector at a point is defined to be the total flux leaving a small volume surrounding the point, divided by the volume.
 Figure 1   The volume element is shown.
 (1)
Since, by Gauss' law,
 (2)

and since,

 (3)

 (4)

This is the differential form of Gauss' law.

Consider the small volume shown in Figure 1. The total flux due to the x component of the vector E is,

 (5)
This flux divided by the volume is,
 (6)
 (7)
Notice that for small D x , equation 7 is the definition of the partial derivative of Ex with respect to x . If you add the flux produced by the y and z components of E.
 (8)
The del operator,
 (9)
can be used to express the divergence of E.
 (10)
Gauss' law in differential form is usually written as,
 (11)
The Divergence Theorem

If the divergence from Equation 1 is multiplied by the volume element ,

 (12)
The right hand side of Equation 12 is the flux out of the volume element . When Equation 12 is integrated over a volume, the total flux is the sum of the flux out of each volume element . Flux between adjacent elements of volume cancel. For two adjacent volume elements, the flux through their common surface is positive for one volume element and negative for the other. Only the flux through the surface of the total volume contributes to the integral.
 (13)

where S is the surface of the volume V.
The integral of the divergence of a vector over a volume is equal to the integral of the vector over the surface of the volume. This is the divergence theorem.

More Electromagnetics

Other formulas from electromagnetic theory can be efficiently written using the del operator. Since,

 (14)
Gauss' law (Equation 11) becomes,
 (15)
The equation in the box is Poisson's equation where,
 (16)
is the Laplacian operator and r is the charge density.

In a region where there is no charge, r is zero and Poisson's equation becomes Laplace's equation.

 (17)
Laplace's equation is a second order partial differential equation that can be solved to find the potential field when the values of the potential are known at the boundaries.