Recall from a basic course in vector calculus that the divergence of a vector
field represents the local rate of expansion or contraction per unit volume
[5]. So, to find the local expansion or contraction of a flow we must
calculate the divergence of a vector field.
The divergence of a three-dimensional vector field
is
Let V(0) be the measure of an infinitesimal volume centered at 
.
Figure 4.7: Evolution of an infinitesimal volume along a flow line.
Figure 4.7 shows how this volume evolves under the flow; the divergence of the vector field measures the rate at which this initial volume changes,
For instance, the divergence of the vector field for the Lorenz system is
we find that the flow is globally contracting at a
constant rate whenever the sum of 
and 
is positive.
