Talk:Jet (mathematics)

Jet bundles and jets

The definition in Jet bundles does not agree with the definition I am familiar with. There are various definitions in the literature, depending on the context. If anyone happens to be familiar with the original Ehresmann paper, please check it out:

• Ehresmann, C., "Introduction a la théorie des structures infinitésimales et des pseudo-groupes de Lie." Geometrie Differentielle, Colloq. Inter. du Centre Nat. de la Recherche Scientifique, Strasbourg, 1953, 97-127.

In short, iterated jet spaces are not the same as jet spaces. There are various types of holonomic constraints that people impose on iterated jets; for instance holonomic (pure) jets, semi-holonomic jets, and non-holonomic jets. In any case, (pure) jets are fundamental for distinguishing between these cases.151.204.6.171

The higher order tangent bundles T^kM with k > 1 are not vector bundles. The transition functions are certainly non-singular, but for a vector bundle you also need them to be linear in the fibre cdoordinates.83.104.131.53 12:44, 5 February 2006 (UTC)

Mistake in subsection "Taylor's theorem"

It says that there's an isomorphism between $J_{p}^{k}({\mathbb {R} }^{n},{\mathbb {R} }^{m})$ and ${\mathbb {R} }^{m}[z]/(z^{k+1})$. $n$ does not appear on the right hand side. I think it should read something like ${\mathbb {R} }^{m}[z_{1},\dots ,z_{n}]/$(all products of $k+1$ variables $z_{i}$)

129.199.98.79 (talk) 13:01, 21 April 2009 (UTC)

algebro-geometric definition is unclear and possibly wrong

What is the ring C_p(R^m,R^n? what is the product in this ring?

Amitushtush (talk) 11:17, 9 October 2009 (UTC)

It is definitely wrong.

Comments in response to the above

The ring is equipped with conponentwise multiplication. There was indeed a mistake in the discussion, namely the claim that the germs vanishing at p form the maximal ideal. This is true for m = 1, but false in general (the germs whose first coordinate function vanishes at p form a strictly larger ideal that is not the unit ideal). — Preceding unsigned comment added by 81.107.34.217 (talk) 01:45, 10 June 2012 (UTC)

wow

This article is really good. Props to everyone who wrote it! 70.112.187.225 (talk) 00:27, 26 January 2010 (UTC)

Introduction is wrong

I find the first senctences "In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions." to be slightly wrong. Here are my reasons:

1. a function in mathematics is not something living on $\mathbb{R}^n$ but more generally on some manifolds, varieties etc

2. Once you are in this more general setting saying that you associate a polynomial to a function is simply wrong. A polynomial on which space? Even the second part of the senctence claiming that it is an abstract polynomial is wrong.

The author is right that jets are relatedt to taylor expansion, but in a sense jets are about making taylor expansion coordinate-free, and once you do this jets are not polynomials. Otherwise this whole article could just be renamed to taylor expansions.

Merge with Jet Bundle?

I've also written there that I don't see a reason to have two pages on Jets. I find the page Jet bundle to be better written, so I propose to replace the current Jet (mathematics) with that one.

—Preceding unsigned comment added by 134.157.61.172 (talk) 16:40, 27 May 2010 (UTC)

Merger

I think that the article Jet bundle should be merged into Jet (mathematics). Clearly these two articles are intimately linked and so should be merged. As to where they should go, well. The Jet (mathematics) article is where most people would start out; it's more basic in nature. The Jet Bundle article is very generalised and abstract in nature, and so should be come under a generalisation subsection of the Jet (mathematics) article. (For the record, I am not, and do not know who is, 134.157.61.172 from the preceding merge request.) — Fly by Night (talk) 16:02, 11 August 2010 (UTC)

I think there is sufficient scope for separate articles. This article appears to be more about jets themselves (as opposed to the bundle structure) and the jet bundle is about the bundle structure. While these topics don't necessarily have to be treated separately, the two articles are already a respectable length. Also I think that achieving harmony in the notation will be a major headache. The notation in the jet bundle article is also not good for someone just learning, I think, but is obviously desirable from the perspective of studying the bundle. Sławomir Biały (talk) 14:54, 5 March 2011 (UTC)

"This forms a real vector space"

I could be mistaken, but I don't think $J_{0}^{k}({\mathbb {R} },M)_{p}$ is a vector space in any canonical way. If it really is, then when you put that back in you should carefully explain why.

If we put coordinates (x, y) (x and y are functions $M\to \mathbb {R}$), then we give $J_{0}^{k}({\mathbb {R} },M)_{p}$ the structure of a vector space. Say M is 2-dimensional, and we're looking at jets up to order 2. We can assume the coordinates vanish at p, so the curve $\gamma \colon \mathbb {R} \to M$ looks like (at + bt^2, ct + dt^2) (up to second order). This is a 4-dimensional vector space (a, b, c, and d can be any real number).

But suppose we change to coordinates (x', y'). Say x' looks like x + x^2 up to second order. A curve $[\gamma ]$ giving (t, 0) in the old coordinates looks like (t + t^2, 0) in the new coordinates. I claim that the new coordinates give a different vector space structure. Write $\cdot _{1}$ and $\cdot _{2}$ for scalar multiplication in old and new coordinates, respectively.

Then $2\cdot _{1}[\gamma ]$ gives (2t, 0) in old coordinates, so it gives (2t + 4t^2, 0) in new coordinates. But $2\cdot _{2}[\gamma ]$ gives (2(t + t^2), 0) in new coordinates. So $\cdot _{1}\neq \cdot _{2}$.

Kier07 (talk) 15:11, 9 May 2012 (UTC)

Thanks for the explanation. Of course you're right that it isn't a real vector space. I misunderstood the reason for removal, thinking it was a misreading of the paragraph rather than a correction of a mathematical error. Best, Sławomir Biały (talk) 15:21, 9 May 2012 (UTC)

Ahh, cool, thanks. So my confusion was justified :) Kier07 (talk) 15:26, 9 May 2012 (UTC)

Clarify notation in Mappings between Euclidean spaces

Can you clarify what is meant by the notation $z^{\otimes k}$ ?

Also is $z\in \mathbb {R} ^{n}$ ?