Talk:Cotangent space

vanishing

"Given a function f ∈ I_p (a smooth function vanishing at p) we can form the linear functional df_p as above."

Why is it necessary to restrict to functions that vanish at p? Doesn't df_p exist if f(p) isn't 0?

It does - the reference is to the definition of I_p. By the way, I don't really agree with the way this page does things - I'd prefer to say everything with jets (also). Unfortunately someone still needs to write about jet bundle, so I'll have to be patient. Charles Matthews 10:13, 22 Jun 2004 (UTC)

evaluation

The current exposition lost me when $X_{p}$ (a tangent vector) was "evaluated" at $f$ to produce $X_{p}(f)$. What in the world does this mean? The tangent plane of $f$ at $X_{p}$?- Gauge 02:14, 3 Aug 2004 (UTC)

One way to define tangent vectors is as derivations of functions (see the Tangent space article). This means a tangent vector $X_{p}$ can operate on functions. Loosing speaking, $X_{p}(f)$ is the directional derivative of f in the "direction" $X_{p}$. -- Fropuff 04:34, 2004 Aug 3 (UTC)

Merging

How about merging this article into Cotangent bundle? --MarSch 15:43, 12 Jun 2005 (UTC)

Please don't. Although obviously related, these are very different objects. The cotangent bundle article has a lot of room for expansion. Merging them would eventually lead to an overly long article with no clear emphasis. -- Fropuff 16:28, 2005 Jun 12 (UTC)

Once that happens we could split them and decide what should go where, but at the moment I'm not at all sure what is supposed to go where, although I seem to be one of the few. --MarSch 17:04, 12 Jun 2005 (UTC)

The length really isn't the issue. A cotangent space and a cotangent bundle are very different objects (one is a vector space, the other a manifold) and deserve different articles. If you aren't sure where something should go, then just ask. Myself and others are happy to assist. -- Fropuff 18:10, 2005 Jun 12 (UTC)

How about making one a subarticle of the other? See {{subarticleof}}?--MarSch 14:15, 13 Jun 2005 (UTC)

That doesn't seem appropriate either. Neither article is properly a subarticle of the other. One should be able to learn about cotangent spaces without knowing anything about vector bundles. I think having a link from one article to the other is more than sufficient. -- Fropuff 15:44, 2005 Jun 13 (UTC)

Hmm, I still think they should be merged. This article begins with stating that the cotangent space is something to do with a manifold, which basically means that it is the same thing as the cotangent bundle (yeah ok, not the same, but you know what I mean). I don't see why that would imply that the article would become more difficult. I don't know what good a cotangent space is in isolation, that is: truly without the bundle. In my view its just one of the ways to the cotangent bundle. Basically everything in this article would need to be repeated and expanded upon in the article on the bundle. If you can give me an example of a cotangent space which has nothing to do with cotangent bundle than you can convince me that this article should be more than a redirect. Otherwise I think they should be merged, because this separation is artificial and needlessly complicates the writing and reading of the articles. --MarSch 13:10, 14 Jun 2005 (UTC)

I guess we disagree then. The reader of this article is assumed to know about manifolds, of course, but not necessarily vector bundles. I think having separate articles helps emphasize the distinction between a cotangent space and a cotangent bundle, which might otherwise be easily confused. I believe this separation is natural and not artificial. -- Fropuff 05:06, 15 Jun 2005 (UTC)

So basically you say, if I take the cotangent bundle article and cut all references to anything to do with vector bundles, then I get this article. I don't think crippling articles in that way in any way increases their usefulness. What good are manifolds if you're just gonna pretend they're sets anyway? Maybe, if I can't convince you, I should improve cotangent bundle to what I think it should be and just leave this alone. --MarSch 15:01, 15 Jun 2005 (UTC)

I'm saying it is useful to discuss things at no higher a level then necessary. Of course, the cotangent bundle article should talk about vector bundles. Why don't you edit the cotangent bundle as you see fit, and we can then discuss furthur. I don't think repetition of facts is such a bad thing. -- Fropuff 15:43, 15 Jun 2005 (UTC)

Intuition

The article on Tangent space gives the following 'informal description': one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible "directions" in which one can pass through p and then uses the description to give an intuitive motivation for the definition as directions of curves. Is there a similar intuition/visualisation behind the definition of the Cotangent space using smooth functions? AdamSmithee 10:43, 27 April 2006 (UTC)

I noticed that an alternative definition along the lines you requested has been given, but your request for intuition on the definition has been (as all too often) neglected by the mathematician who wrote it. I've added an informal description of what the definition does (as far as I understand it). Now what we need is something explaining the relationship between the two definitions.--69.212.224.128 03:16, 21 June 2007 (UTC)

Notation

In the section The differential of a function, what does f o y mean? Maybe it is obvious to someone in the field, but it was not obvious to me. --agr 10:06, 18 December 2006 (UTC)

I think what you notate as "y" is supposed to be a "gamma." It would appear the "o" is a substitute for the small circle that denotes function composition, so f o γ takes t to f(γ(t)). — vivacissamamente 14:07, 18 December 2006 (UTC)

tangent covectors vs cotangent vectors

Are the two the same thing? If so, the article should make it clear. -- Taku (talk) 13:46, 13 July 2008 (UTC)

Possible error/ambiguity

In the section "The differential of a function" for the definition of a differential in terms of velocities of curves, it refers to df_p as being on T_p M. But traditionally, the notation is T_(f(p)), as discussed in the wikipedia page on pushforwards [i.e. df:T_p N -> T_(f(p))].

So either the section is wrong, or it departs from conventional differential notation enough to warrant a definition of df_p being given.

I'm merely an undergraduate, and not an expert on differential geometry, so I would prefer if someone else edits the page.