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There are two main parts to the vector field ANALYZER.
The first part helps visualize the derivatives (and their principal
geometric components) by simple zooming -- just as easy as in the
calculus of functions of a single variable -- after sufficient
magnification you obtain linear vector fields, i.e. the Jacobian,
the curl, the divergence etc.
The vector field analyzer is fully interactive -- drag the lens
to any point and change the magnification factor. Several
important fields are predefined, or enter any formula (the
analyzer comes with a powerful parser....)
Challenge: Take a look and SEE that the derivative of the
flow past a cylinder has (OBVIOUS) winding number 3/2!
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The differential part of the analyzer is complemented by
interactive tools to explore and study
the integral aspects of vector fields:
The main innovation allows for interactive explorations of
the nonlinear flow, of the linearized flow, and of the flows
of the principal geometric components of the linearized flow.
To visualize these define (draw!) your own regions
and follow their evolution --
e.g. see that the exponential of a linear map
is linear (preserves polygonal character), see orthogonality,
and area-preservation at work and discover a whole new world
of exciting links between vector calculus, differential equations,
linear algebra and baby-Lie groups.
The screen shot shows part of a live animation of the area-preserving flow associated to the magentic field Im(dz/z). |
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