"Subtracting the drift"

By Matthias Kawski. All rights reserved.
This page illustrates key steps are in a new approach to calculus of vector fields. The central
theme is that any kind of derivative is intimately linked to approximability by linear objects.
This approach allows one to "see" the Jacobian linearization, the curl, the divergence etc.
For more details, and the interactive JAVA-scope visit the author's WWW-site.
 
 

Start with any vector field F on a reasonably small scale.
 

The focus is on how the field changes from point to point.

Fix a point p0
here chosen at the center.

Consider a constant vector field F0
that agrees with F at the point p0
i.e., F0(x)=F(p0) at all points x

Compare the original vector field F
with the "underlying" drift F0.
Subtract the background F0
from the original vector field F.
If the difference (after infinite zooming) is a linear field, it is defined to be the derivative of F at p (modulo a few technicalities). 
In this case the difference appears 
already very close to linear. The scalar
curl is clearly positive, the divergence very clos or equal to zero.