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