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Why is it necessary to restrict to functions that vanish at p? Doesn't dfp exist if f(p) isn't 0?
The current exposition lost me when (a tangent vector) was "evaluated" at to produce . What in the world does this mean? The tangent plane of at ?- Gauge 02:14, 3 Aug 2004 (UTC)
How about merging this article into Cotangent bundle? -- MarSch 15:43, 12 Jun 2005 (UTC)
How about making one a subarticle of the other? See {{ subarticleof}}?-- MarSch 14:15, 13 Jun 2005 (UTC)
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)
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)
Are the two the same thing? If so, the article should make it clear. -- Taku ( talk) 13:46, 13 July 2008 (UTC)
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.
This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
Why is it necessary to restrict to functions that vanish at p? Doesn't dfp exist if f(p) isn't 0?
The current exposition lost me when (a tangent vector) was "evaluated" at to produce . What in the world does this mean? The tangent plane of at ?- Gauge 02:14, 3 Aug 2004 (UTC)
How about merging this article into Cotangent bundle? -- MarSch 15:43, 12 Jun 2005 (UTC)
How about making one a subarticle of the other? See {{ subarticleof}}?-- MarSch 14:15, 13 Jun 2005 (UTC)
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)
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)
Are the two the same thing? If so, the article should make it clear. -- Taku ( talk) 13:46, 13 July 2008 (UTC)
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.