In vector calculus, the divergence of a vector field is defined as:
where is the gradient operator taken over the dot product with the respective , , and components of the field. The divergence is distributive.
Geometrically, it represents how much the field diverges at a given point, i.e., how much of a vector field is “created” or “expelled” at a point.
If the divergence is zero, , we have a solenoidal or divergence-free field.
Properties
Formal statement
For a closed Gaussian surface enclosing a charge, we let shrink arbitrarily small to , and the enclosed volume . We can express in terms of a volume charge density, .
By Gauss’ law for :
Then as we let approach 0:
The formal statement of divergence.
See also
- Curl, the cross product equivalent