Talk:Tangent space

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This article is within the scope of WikiProject Mathematics.
Mathematics grading:	B Class	High Importance	Field: Geometry and topology
References needed. Several articles here need to be developed together. Geometry guy 22:23, 29 March 2007 (UTC)

[edit] Meaning of derivative/differential

The linear map (df)_p is to be interpreted as the derivative of f at p.

df_p is usually referred to as the differential of f at p, hence the notation. Isn't that a different interpretation than the derivative?

The derivative at a point in calculus is just a number. In multivariable calculus, the derivative at a point is not a number but a matrix or linear map. When dealing with manifolds, it's actually a linear map between the respective tangent spaces. I don't think there's any other notion of derivative at a point than (df)_p. If the map f is differentiable at every point of M, then its derivative is a map df : TM → TN between the tangent bundles which comes from glueing together all the maps (df)_p. AxelBoldt 17:56 Dec 4, 2002 (UTC)

Ok, let me be more particular. I was under the impression that the space of maps of said form was considered to be the dual of the space of derivatives, so that a derivative would be a map back the other way between the cotangent spaces. I forget the reason for this, but it had something to do with there being a natural identification with the standard notion of derivatives in the case that N is the real numbers. I will double check when I get the time.

I think you're right. I replaced derivative with differential. AxelBoldt 22:48 Dec 4, 2002 (UTC)

[edit] discussion at Wikipedia talk:WikiProject Mathematics/related articles

This article is part of a series of closely related articles for which I would like to clarify the interrelations. Please contribute your ideas at Wikipedia talk:WikiProject Mathematics/related articles. --MarSch 14:08, 12 Jun 2005 (UTC)

[edit] p or x?

Do we prefer p or x for a point in M? I don't mind, but it would be nice to be consistent between this article and the tangent bundle article. Geometry guy 23:28, 10 February 2007 (UTC)

In most of the other related articles (admittedly, partly because of my edits), x is used, so I have changed p to x here. Geometry guy 23:32, 11 February 2007 (UTC)