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.
A vector field is source-free if there is zero divergence.
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 calculated using the Laplace operator.
For a vector field given as F = <P(x,y), Q(x,y)>, applying the Laplace operator looks like:
where fxx = ∂2f/∂x2 and fyy = ∂2f/∂y2.
For a vector field given as F=<P,Q,R>, applying the Laplace operator looks like:
where fzz = ∂2f/∂z2.