Divergence

Divergence measures the distortion of a vector field.

Description

The divergence of a vector field given as F = <P(x,y), Q(x,y)> is calculated as:

The divergence of F given as <P(x,y,z), Q(x,y,z), R(x,y,z)> is calculated as:

Note that divergence returns a scalar value.

Properties

The divergence of the curl of a vector field is always 0.

Source-Free Fields

A vector field is source-free if there is zero divergence. This means that there are no sources (points where the field originates) or sinks (points where the field terminates). A consequence is that, for any closed circular path C, movement in is equal to movement out (flux is zero).

The last point leads to one definition of a source-free vector field:

This can be equivalently expressed as:

where P_x = ∂P/∂x and Q_y = ∂Q/∂y. Note furthermore that this equation can only be true if P_x + Q_y = 0, giving an even more succinct formulation.

If a vector field is source-free, there is at least one stream function g. If the field is given as F = <P(x,y), Q(x,y)>, then g must satisfy:

Conservative Fields

A vector field is conservative if it is path independent. There is necessarily at least one function f satisfying F = ∇f. In this case, divergence can be evaluated using the Laplace operator.

where f_xx = ∂²f/∂x² and f_yy = ∂²f/∂y², and

where f_zz = ∂²f/∂z².