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created gradient
created stubs for curl and divergence
Thanks!
I have added more links and a TOC.
I added redirect rotation of a vector field to the more cryptic anglosaxon alternative curl (and which I could prefer to be a title in fact, wikipedia givee 2.1 mil hits, for curl of a vactor field 320 thousand hits).
added a link to and created a stub for symplectic gradient. By trhe way, I notice now that moment map is still very stubby; i hope to be able to expand that soon.
People are forgetting that you need an orientation, not just a metric, to do Hodge duals (and thus, for example, to define the curl of a vector field as a vector field on a Riemannian $3$-fold).
go on, jump in, I seem not to have time these days to reconcentrate on this topic, which I started
Anyway, I did expand moment map following Joel Robbin’s notes (well, I think I have learned the first steps into the subject from my advisor, so I follow his notes). Still not much there but at least the basic reasoning leading to the definition is produced (and the literature).
Unhappy that we had the curl only in the classical $3$ dimensions, I discussed its relation to the cross product and described nonclassical cross products. The latter page is now massively general!
Well, a further justification should go from the Stokes theorem. I mean, there is a limit of an integral formula for the rotation of the vector field, similar to the one for the divergence.
I mean
$rot \vec\mathcal{A} = lim_{vol D\to 0} \frac{1}{vol D} \oint_{\partial D} \vec{n}\times \vec\mathcal{A} d S$ $div \vec\mathcal{A} = lim_{vol D\to 0} \frac{1}{vol D} \oint_{\partial D} \vec{n}\cdot \vec\mathcal{A} d S$where $D$ runs over the domains with smooth boundary containing point $x$ at which the rotation or the divergence of the vector field is calculated. These formulas are invariant.
I added the above remarks in the case of divergence (in which case this works in any dimension).
New entry nabla and additions to rotation of a vector field.
Right, I have left it in a rudimentary form, planning something similar and I am happy with your way of finishing the job!
There is something funny going on with the rendering: at least on my system that
\nabla \odot T
in the entry sometimes (such as in the big displayed formula) comes out as “$\nabla box T$”.
You probably caught this while I was in the midst of editing for itex’s capabilities. Reload the page and check again.
Ah, right, now it dsiplays correctly.
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