 
Summary: The flow of a vector field.
Suppose F = Pi + Qj is a vector field in the plane1
Associated to F is its flow which, for each time t is
a transformation
ft(x, y) = (ut(x, y), vt(x, y))
and which is characterized by the requirements that
(1) f0(x, y) = (x, y)
and
(2)
d
dt
ft(x, y) = F(ft(x, y)).
That is, for each (x, y), t ft(x, y) is a path whose velocity at time t is the vector that F assigns to ft(x, y).
Example. Let F = yi + xj. Draw a picture of F. Note that
ft(x, y) = (x cos t  y sin t, x sin t + y cos t).
That is, ft is counterclockwise rotation of R2
through an angle of t radians.
Example. Let F = xi + yj. Draw a picture of F. Note that
ft(x, y) = et
(x, y).
