Re: gradients and exterior derivatives

["RAUBER, JOEL" <JOEL_RAUBER@SDSTATE.EDU>]

My rather rusty understanding matches with everything John Denker wrote
below.

Regarding the "fishscale" visualizations:

If I understand you diagrams correctly, they are more about how to
pictorially represent (visualize) one-forms, rather than how to pictorially
represent non-conservative fields.

The biggest advantage I see to the "fishscale" diagrams is the introduction
of the notion of "up/down" OR "+ or -" with respect to the directionality.
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.

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.

One other slight disadvantage is that the fish scale picture is inherently
2-dimensional, I think.  A family of surfaces can sometimes give you a
3-dimensional perspective impression of the one-form.

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.

Comments?

Joel Rauber

> On 04/28/2003 02:22 PM, RAUBER, JOEL wrote:
> > MTW (Misner, Thorne and Wheeler) give some discussion to
> the above.  They
> > state that most properly one should view the gradient of
> the potential as a
> > one-form.  I.e. the gradient of a function is a one-form,
> i.e. the exterior
> > derivative of the function.
>
> Thanks for pointing this out.
> I am happy to adopt the MTW terminology.
>
> I think we all agree about the physics.
> Let me see if I can re-state the physics
> in the MTW terminology:
>
> 1) The gradient of a scalar potential is most properly
> considered a one-form.
>
> 2) The exterior derivative of a scalar potential
> is a one-form.  It is 100% identical to the gradient,
> because
>     del /\ A = del A
> automatically for any scalar A.  Indeed
>     X /\ A = X A
> for any X whatsoever, provided A is a scalar.
>
> 3) For a non-scalar B, the generic derivative
> del B is not the same as the exterior derivative
> del /\ B, but we don't need to worry about that
> right now.
>
> 4a) The gradient VECTOR that we learned in infancy
> is the dual of the gradient ONE-FORM mentioned in
> item 1.
>
> 4b) You can't form the dual without a metric.
>
> 4c) If you have a nice Cartesian metric, forming
> the dual is trivial:  the components of the vector
> are numerically equal to the components of the
> one-form.
>
> 5) The things you really need, such as Taylor
> expansions, work just fine without a metric.
> Just write the derivative as a one-form and
> turn the crank.
>
> =======================================
>
> On a related topic, I revised and extended the
> writeup on visualizing non-conservative fields.
>    http://www.monmouth.com/~jsd/physics/non-conservative.htm
>
> Among other things, I added a new figure
>
> http://www.monmouth.com/~jsd/physics/non-conservative.htm#fig-betatron
>
> which
>    a) Properly portrays the electric field inside a
>   betatron.
>    b) Has constant curl everywhere.  (Verify this
>   yourself.  It's fun.)
>    c) Demonstrates the power of the fish-scale technique
>   to represent one-forms where the magnitude and direction
>   are both changing from point to point.  (Previous
>   examples had uniform magnitude.)