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Terminology nitpick (was: visualizing a non-potential)



In matters of weighty terminological,
with respect to objects geometrical
I often look to folks, authoritical
Misner, Thorne and Wheeler-cal
rather than a modern major-general


It depends.... Here are some true facts that
may or may not answer the question:

1) The gradient of the potential is a vector.

2) The exterior derivative of the potential is a
one-form.

3) If you have a metric, the distinction between
the two is minor. One is the dual of the other.


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.

Item (3) above is why one can represent in first grade the gradient with a
vector field picture. Particularly in a Euclidean space using Cartesian
coordinates the distinction is moot.
_____________________________________

The mu(th) component of the gradient of a function

partial/partial X^mu of F

transforms the way components of one-forms transform, not the way components
of vectors transform.

Joel Rauber