Divergence - Properties

Properties

The following properties can all be derived from the ordinary differentiation rules of calculus. Most importantly, the divergence is a linear operator, i.e.

for all vector fields F and G and all real numbers a and b.

There is a product rule of the following type: if is a scalar valued function and F is a vector field, then

or in more suggestive notation

Another product rule for the cross product of two vector fields F and G in three dimensions involves the curl and reads as follows:

or

The Laplacian of a scalar field is the divergence of the field's gradient:

The divergence of the curl of any vector field (in three dimensions) is equal to zero:

If a vector field F with zero divergence is defined on a ball in R3, then there exists some vector field G on the ball with F = curl(G). For regions in R3 more complicated than this, the latter statement might be false (see Poincaré lemma). The degree of failure of the truth of the statement, measured by the homology of the chain complex

(where the first map is the gradient, the second is the curl, the third is the divergence) serves as a nice quantification of the complicatedness of the underlying region U. These are the beginnings and main motivations of de Rham cohomology.