Talk:Tangent space

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)