Talk:Skew gradient

Cleanup needed

This article is currently so difficult to interpret that it does not actually communicate what it is in tended to. My impression is of several omissions and probable errors. Though I am not familiar with the skew gradient, the following observations seem to apply:

• The function f is presumably intended to be restricted to being holomorphic but this is not stated. Otherwise the rest simply has no possibility of making sense. In fact, without this its mention seems utterly irrelevant.
• Orthogonality and the dot/inner product have not been defined. These are not, in general, defined on complex numbers as a field, but since the algebra has not been specified, what is meant cannot be inferred. Whether (x,y) and (u,v) represent vectors in a real vector space, numbers in the complex field, or something else, this is not made clear.
• The operators ∇ and ∇^⊥ are not adequately defined.
• The reference is a bit long to go through to find its definition of skew gradient. It would be good if a page number was given.

— Quondum^☏✎ 08:46, 25 January 2012 (UTC)

Wonderful remarks! I defined f to be an analytic function, and rewrote the definition in a way that makes more sense. I also defined the algebras where every definition is taking place. This is my first article, so, by all means, please let me know whether it is still suffering!

— Meldraft 01:27, 09 February 2012 (UTC)

Much improved, making it much more accessible.

I've rewritten the lead with attention to links. I may have got some details wrong so please check; I've introduced the requirement that the function be harmonic (which seems to ensure that the skew gradient exists and is assumed by the source, and is probably necessary for it to exist as defined but I'm not sure about this). I added a few more requirements that were previously implicit.

• You might consider a start such as In vector calculus, ..., depending on what field of study you identify it as belonging to.
• You may want to slip in that the function is a scalar field, e.g. A skew gradient of a scalar field that is a harmonic function..., though this may be clumsy.
• The article still needs something to motivate its existence: applicability, examples of use or somesuchlike. The reference seems to go some way towards that. The scalar product used to define orthogonality and magnitude should probably still be defined, though most would assume it as the standard inner product.
• Further properties would be useful

Some observations on the subject matter, not on the article:

• It does not appear to generalize directly to any number of dimensions other than 2, unlike the gradient.
• It probably does generalize to "inner products" that are not positive-definite, but my guess is that the reference does not deal with this.
• There seem to be two vector fields that will satisfy the definition: a skew gradient and its negative. One has a choice of rotating the gradient either "clockwise" or "anticlockwise". It would be good to note this and see what the reference has to say on it.
• It would appear to be equivalent to the Hodge dual of the gradient in geometric algebra, which suffers none of the limitations above. This could be noted as a generalization if you wanted. It is similar to the shortcomings of the cross product versus the exterior product of geometric algebra.

Happy writing. — Quondum^☏✎ 07:33, 9 February 2012 (UTC)