# Talk:Tangent space

## Contents

- 1 Meaning of derivative/differential
- 2 discussion at Wikipedia talk:WikiProject Mathematics/related articles
- 3 p or x?
- 4 Jacobian?
- 5 Is this definition right?
- 6 Formulas
- 7 "Tangent map"
- 8 Where is this formula?
- 9 Most...Rubbish...Intro...Ever
- 10 Applications to graphics programming?
- 11 Relation to exponential map for Lie groups?
- 12 Twice the dimension? Is that related to Jolt cola having twice the caffiene?
- 13 The tangent space at x is not the vector space of derivations at x of smooth functions

## Meaning of derivative/differential

The linear map (d*f*)_{p} is to be interpreted as the derivative of *f* at *p*.

d*f*_{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 (d
*f*)_{p}. If the map*f*is differentiable at every point of*M*, then its derivative is a map d*f*: T*M*→ T*N*between the tangent bundles which comes from glueing together all the maps (d*f*)_{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)

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)

## 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)

## Jacobian?

As I understand it, if I have a function, *f*(*x*) defined on a manifold, the gradient of *f* at a point, *x* is in the tangent space of *x*. But what about the Jacobian? Would one say it's in the tangent space of *x*? (I assume not because it's a matrix not a vector.) What space is the Jacobian in? —Ben FrantzDale 21:57, 4 May 2007 (UTC)

- as you say, the Jacobian at a given point is a linear map between tangent spaces. see Pushforward (differential). Mct mht 02:11, 5 May 2007 (UTC)

- That isn't quite what I meant. I was curious what space you'd say it is in. That is, if grad
*f*is in the tangent space (and so the field grad*f*is a section of the tangent bundle), then the Jacobian should be in a somehow related space. I suppose it should be in the square of the tangent space, that is . However that space is constructed, the connection for the tangent space should define a connection for this new space of rank-two symmetric tensors. The word I'm looking for may be associated bundle. —Ben FrantzDale 12:50, 6 May 2007 (UTC)

- That isn't quite what I meant. I was curious what space you'd say it is in. That is, if grad

- This may become clear if you are more specific about the nature of f. If f is a map from M to the real numbers R, then df is a map from TM to TR, namely the push forward. It is also a form on M (as opposed to a vector) which takes in a section v of TM by df(v(p))=v(f)(p) (here v is a directional derivative.) In turn (f)(p) is a real number which may be thought of as an element of the (one dimensional) tangent space to f(p) in R.
- If f is a map from M to another manifold N then it is nice to work with local coordinates on N; you can write the k-th component of f (giving the k-th coordinate of the image of a point) as a function from M to R and consider its differential. The collection of all these forms (one for each value of k) is is a representation of df specified by the choice of local coordinates on N. If local coordinates are used on M as well then df has a (coordinate system dependent) matrix representation (acquired using the coordinate bases on both M and N) called the Jacobian of f.
- I hope this clears a few things up, but I'm not sure I know the answer to your question. I feel similarly to you about the possibility of considering it an element of a bundle associated to T*M, but I'm not sure how to reconcile this with the dependance on local coordinates on N. While I'm curious, I don't think that the addition of a discussion of such an associated bundle would add much to this page, or the related pages.Dewa (talk) —Preceding comment was added at 22:37, 31 May 2008 (UTC)

## Is this definition right?

The subsection "Definition as directions of curves" defines a map (dφ)_{x} : T_{x}*M* → **R**^{n} by

- (dφ)
_{x}(γ'(0)) = (φ o γ)'(0).

Now, given the given type of (dφ)_{x}, I'd expect its argument to be an element of T_{x}*M*, which is is defined as the set of all tangent vectors, where a tangent vector is an equivalence class of curves. So the argument should be a class of curves [γ] and I'd be willing to accept the following definition:

- (dφ)
_{x}([γ]) = (φ o γ)'(0),

whose required independence of the choice of representative γ follows immediately from the equivalence relation. I'm not sufficiently familiar with the topic to be fully confident that this is the correct improvement and apply the change myself. --Lambiam 07:47, 4 September 2007 (UTC)

- The article says a couple of lines before: "The equivalence class of the curve γ is written as γ'(0)". So γ'(0) is just a different notation (and the usual one) for [γ]. -- Jitse Niesen (talk) 07:57, 4 September 2007 (UTC)

- Thanks, but what a weird convention, seeing how γ'(0) in the l.h.s. has a completely other meaning then than (φ o γ)'(0) in the r.h.s. --Lambiam 09:26, 4 September 2007 (UTC)

- I see what you mean: if the prime means what Nissen says then on the LHS the prime denotes an equivalence class of curves on
*M*with equivalence determined by behavior at*x*while on the RHS it denotes an equivalence class of curves on**R**^{n}with equivalence (implicitly) determined by behavior at φ(*x*). But this does not give the desired construction; it does not map an equivalence class of curves to an n-touple. So Lambiam believes that the prime on the RHS is meant to be a derivative WRT*t*, which would mean this article made double notational use of the prime. I will change it.Dewa (talk) 23:01, 31 May 2008 (UTC)

- I see what you mean: if the prime means what Nissen says then on the LHS the prime denotes an equivalence class of curves on

- I'm not sure the change is correct, as (φ o γ)(0) is not a function of a variable
*t*. The following would be correct (I think):- (dφ)
_{x}(γ'(0)) =*D*(0), where*D*(*t*) = (*d*/*dt*)(φ o γ)(*t*).

- (dφ)
- I don't see how to say that in an essentially simpler way without overloading the prime in one formula. --Lambiam 07:33, 1 June 2008 (UTC)

- I'm not sure the change is correct, as (φ o γ)(0) is not a function of a variable

## Formulas

Some formulas in the latter part of the article appear to be written in LaTEX, but they don't parse! —Preceding unsigned comment added by 217.127.9.138 (talk) 22:32, 4 September 2008 (UTC)

- Everything seems to render correctly for me. Sometimes LaTeX formulas, which render as PNG graphics, don't get cached right away if the server load is high, and that might be the problem. At any rate, give it a few hours, and it should be ok. siℓℓy rabbit (talk) 22:46, 4 September 2008 (UTC)

## "Tangent map"

The term "tangent map" appears in the Wiki list of missing math topics. Could somebody familiar with this article "Tangent space" please work it in? TIA --LDH (talk) 14:18, 28 November 2008 (UTC)

## Where is this formula?

I didn't really read the whole article, but I noticed that it doesn't include the formula fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0)-(z-z0) = 0. Is there a reason for this? --BiT (talk) 15:32, 28 April 2009 (UTC)

- this ought to be included. I was just about to say the same thing. BriEnBest (talk) 00:23, 16 November 2010 (UTC) ... well, at least something similar:
**n**dot (position - point) = zero -> normal vector dot ((x,y,z) - (x_{0},y_{0},z_{0})) = 0 BriEnBest (talk) 00:27, 16 November 2010 (UTC)

## Most...Rubbish...Intro...Ever

Please! We *can* do better than this atrocious opening paragraph. Blitterbug (talk) 16:03, 5 December 2010 (UTC)

## Applications to graphics programming?

I found this term in an article about "how to be a graphics programmer". Should there be some mention of applications in this article? —Preceding unsigned comment added by 24.99.60.219 (talk) 18:48, 12 May 2011 (UTC)

- I found this article which provides an onward reference but I do not see much on Wikipedia about the subject - I guess I'm looking in the wrong place! 188.220.56.222 (talk) 17:01, 19 August 2011 (UTC)

## Relation to exponential map for Lie groups?

There is no mention of how this concept relates to the exponential map for Lie groups. My understanding is that it is closely related, but must admit, I'm confused exactly how it works. My sense is that the Lie algebra is the tangent space at the identity, , but I don't know how that relates to the tangent space at other group elements. I think it's basically that you can talk about the tangent space to another group element, by thinking of a tangent vector as though it is in the tangent space of the identity and then defining the exponential map from the tangent space of *g* as

That is, following the geodesic defined by is the same as using the exponential map at the origin and then "transporting" that transform over to *g*. Is that right? Which version is right? I think this also has to do with using a group automorphism to "use" the tangent space at the origin everywhere. Apologies for the vagueness. —Ben FrantzDale (talk) 15:52, 9 June 2011 (UTC)

The article says "all the tangent spaces of a manifold form another manifold of twice the dimension"

Twice the dimension of WHAT? The original manifold? If so, please add that.

**Question that doesn't belong on the talk page but I'm asking anyway because I don't give a rats ass in hell about any rule that limits my understanding:**

Okay, a sphere has dimension two and is embedded in 3-space. Every tangent space of that sphere is a plane. Combining them all gives you the universe, minus the inside of the sphere, and is called the tangent bundle. It has dimension 3. **In what sense does the tangent bundle have "twice the dimension" of something?**
Helvitica**Bold** 07:42, 27 February 2012 (UTC)

- You're right, there is room for improvement in this article. Something that isn't made perfectly clear is that these tangent spaces, as described in this article, are treated as abstract manifolds. For your sphere embedded in R^3, the tangent planes are not considered as subsets of the same R^3, but as vector spaces that have an independent existence. So the tangent bundle of a two-dimensional sphere doesn't live inside 3-space; it really is a four dimensional manifold (a two-dimensional family of two-dimensional planes; two plus two makes four). Jowa fan (talk) 08:27, 27 February 2012 (UTC)

## The tangent space at x is not the vector space of derivations at x of smooth functions

It is the vector space of derivations of the algebra of germs of smooth functions at x. — Preceding unsigned comment added by 68.193.47.165 (talk) 19:09, 23 September 2012 (UTC)