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 JAVAscope visit
the author's WWWsite.
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 p_{0}
here chosen at the center. Consider a constant vector field F_{0}


Compare the original vector field F
with the "underlying" drift F_{0}. 

Subtract the background F_{0}
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. 