Re: gradients and exterior derivatives

["John S. Denker" <jsd@MONMOUTH.COM>]

On 04/28/2003 05:35 PM, RAUBER, JOEL wrote:

> The "standard" method (ones I've seen in books) of representing one-forms is
> with a family of surfaces and that method loses the "up/down" sense of
> directionality.

You can color-code the front versus back of
the surface to portray directionality.  Alas
this is a rather dry, mathematical encoding,
not very visually intuitive.

> The family of surfaces method does get the notion of magnitude (The closer
> the surfaces are to each other the larger the magnitude.  It also, when
> combined with arrow pictures of a vector-field allows one to visualize the
> inner-product of a vector with the dual of some one-form.
>
> i.e. <V,F> where V is a vector field and where F is the dual to some
> one-form field.  The magnitude of the inner product is proportional to the
> number of "piercings" of arrows through surfaces.

-- That's true as to surfaces.
-- Fish-scales also portray magnitude in the same way.
   Example:
http://www.monmouth.com/~jsd/physics/non-conservative.htm#fig-betatron

The boundary of each scale is a surface that gets
pierced by an arrow when forming the contraction
between a one-form and a vector.

> One other slight disadvantage is that the fish scale picture is inherently
> 2-dimensional, I think.

In principle, fish-scales generalize to higher dimensions.
What is a contour-line in D=2 becomes a surface in D=3,
also called a shell.

In the D=2 fish-scale diagrams, you see lots of 3/4-complete
circles.  In D=3, the corresponding elements will be
3/4-complete spheres.  The techniques of shading to
portray the orientation of the contour also generalize.

In practice, I'm not volunteering to draw such
a thing.

> A family of surfaces can sometimes give you a
> 3-dimensional perspective impression of the one-form.

Yes, especially if it is an exact one-form.
But if it is a non-exact one-form, you will
need something more, perhaps generalized
fish-scales.  (shells?  shellfish??)

> This visualization is very tricky as you point out and nothing is entirely
> satisfactory it seems.  This may boil down to the need to have several ways
> to visualize of which the "fishscales" adds an important element.

:-)