2D

Suppose that  F  is a constant vector field (defined at every point) and  dX  is a fixed vector.

If we magnify a small increment of the parametric curve (and scale  F[X]  to the same magnitude) we see approximately a constant flow along a straight line segment.

A small increment  X[t+δt]-X[t]  is approximately  X'[t]δt  on a scale of    (assuming   ).  The flow along this segment is thus approximately  F[X[t]]•X'[t]δt  and if we add all the segments along the curve we obtain the flow along the curve,

Rate of Flow Across a Straight Segment

Suppose that  F  is a constant vector field (defined at every point) and  dX  is a fixed vector.

In one second the fluid particles that have crossed the  dX  segment form a parallelogram,

Flow of  F  across dX to the right =

Microscopic Rate of Flow Across:

Created by Mathematica  (July 2, 2004)