Another quantity that can be calculated directly from the vector
field is the equation of first variation, which provides an approximation
for the evolution of a region about an initial condition 
. The flow 
is a function of both the initial condition
and time. We will often be concerned with the stability of an
initial point in phase space, and thus we are led to consider the
variation about a point while holding the time
t fixed.
Let 
denote differentiation with respect to the phase variables
while holding t fixed. Then from the differential equation for
a flow (eq. (4.3)) we find
which, on applying the chain rule on the right-hand side, yields the equation of first variation ,
This is a linear differential equation for the operator
. 
is the derivative of
at . If the vector field
is n-dimensional, then both 
and
are 
matrices.
Turning once again to the vector field
we find that the equation of first variation for this system is
The superscript i to 
indicates the ith component of flow, and
the subscript j to 
denotes that we are taking the derivative
with respect to the jth phase variable. For example, 
.
The components of 
are the coordinate
positions, so they could be rewritten as
.
The dot, as always, denotes differentiation with respect to time.
