Re: gradients and exterior derivatives

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

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.)