Re: visualizing a non-potential

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

On 04/27/2003 01:44 AM, Bob Sciamanda wrote:
> It looks like there is a height change if one takes a step radially.  In the
> betatron E field, moving at right angles to the circular field lines involves no
> work.

Yes, this property of the diagram is an unwanted artifact.

But it is very minor, because if you go a long distance
in the radial direction, there is no cumulative up/down
motion.  Also, this artifact goes away entirely in the
continuum limit.

I added a discussion of this point to
http://www.monmouth.com/~jsd/physics/non-potential.htm#sec-fish-scale

>  Perhaps you need steps in the form of radial strips, rather than small
> circles.

That would work fine in this particular example, but
wouldn't handle the general case.  We want to be able
to represent non-exact one-forms such as PdV, where
no easy simplifications are available.

On 04/27/2003 02:24 AM, Bob Sciamanda wrote:

> A non-conservative E field is a vector function of position (and
> time, in the general case). It is representable by continuous field
> lines, tangent to the direction of the local E vector.

Yes, that is one possible valid representation.
It is not the only possible valid representation.

In particular, depending on what one is trying to
represent, sometimes the one-form representation
is more apt than the vector representation.

> Be satisfied with the field line representation of the E field - that
>  IS a genuine (vector) function of position.

That is optionally but not exclusively satisfactory.  I can
also be satisfied with the fish-scale representation - that
IS a genuine (one-form) function of position.

If (!) we have a metric (which gives us a notion of length
and a notion of angles) then every one-form has a dual
representation in terms of vectors.  But there are cases of
interest, notably thermodynamics, where we have no metric.
Vector representations are not satisfactory for representing
things like PdV.

I also added a discussion of this point to the web page.

> ... slope ...

I extirpated all mention of "slope".

For a one-form, we can define proper notions of upward
and downward steps, but the usual notion of "slope" does
not carry over particularly well from vector-fields to
one-forms.