Re: do I understand co-variant vs. contra-variant?
From: J Jensen (jjensen14_at_hotmail.com)
Date: 10/30/04
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Date: 30 Oct 2004 13:55:22 -0700
"Van Jacques" <vanjac12@yahoo.com> wrote in message news:<1099130160.830473.193140@z14g2000cwz.googlegroups.com>...
> If you are from physics, take a look at Misner, Thorne, and Wheeler
> "Gravitation". They have vectors as arrows (tangent vectors to curves),
> and 1-forms as level surfaces of functions f, so df is rep by a series
> of surfaces ortho to the vector grad(f). 2df puts the surfaces twice as
> close
> together. <v,df> = df(v) = # of times the vector v pierces the
> level surfaces of the function f.
>
> Van
Yes, evidently then this is the same representation as I mentioned reading about
in the American Journal of Physics earlier in this thread. For me though,
I am not so interested in *visualizing* vectors or co-vectors; I merely
need to confirm that I have distilled the correct information about
what defines "Co-variant" and "Contra-variant", as I mentioned in my
original posting. In other words, I am forced to cope with this
terminology while I am trying to research something else, and I am trying
to double check that I haven't missed some crucial detail.
--Jeff
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