Talk:Pushforward (differential)

message: I wish I remembered more, but isn't this also a term in category theory, where stuff is being pushed along functions? As I said, I really wish I remembered more of those days. --SPUI (talk) 18:39, 2 Mar 2005 (UTC)
replies: text: It could be. My background is mostly differential geometry, and I never met a categorial push forward at graduate school. I wonder what the requester intended. I suspect it probably wasn't what this article covers. mat_x 22:06, 2 Mar 2005 (UTC)

text: Ah, yes, there is pushout. Maybe that was intended. Never heard it called pushforward though. mat_x 22:10, 2 Mar 2005 (UTC)
replies: text: Yes, that was it. --SPUI (talk) 03:36, 3 Mar 2005 (UTC)

text: I've taught this stuff that way, and at least with pictures it's an OK thing to do. Charles Matthews 19:10, 4 Mar 2005 (UTC)
replies: text: I'm sorry if you misunderstood me, there was certainly no questioning your experience. I was just wondering (and not in a passive-aggressive sense) whether an encyclopaedia article should begin with a jargon-blast. mat_x 21:31, 4 Mar 2005 (UTC)
thread_title: Pushforward verses pushout

message: The push forward is a categorical notion.  It has nothing more to do with differential geometry then it does with anything other subject in mathematics.
: Categorical pushforwards are usually called pushouts. I believe that this page should be devoted to pushforward of vector fields, the term pushforward should be mentioned briefly on the pushout page, there should be a separate page on pushforward measures, and a disambiguation page for pushforward. Do others agree? Geometry guy 16:44, 8 February 2007 (UTC)
thread_title: This page needs to be completely rewritten

message: One of the reasons the notion of a pushforward of a vector field is confusing, is that it is actually a section of a pullback bundle. If V->N is a vector bundle over N and F:M->N is a smooth map, then the pullback bundle F*V -> M is the bundle whose fibre at x in M is VF(x). (This is an example of a categorical pullback.) The differential of F, dF (or DF if you prefer), is then a bundle homomorphism from TM to F*TN, or equivalently a section of the bundle Hom(TM,F*TN): these are bundles over M, not N, so I think the notation F* for the differential can be misleading (as is the term pushforward). If X is a vector field on M, then dF(X) (aka F*X) is a section of F*TN. This leads to the notion of a vector field along a map: a section of F*TN is called a vector field along F.
The cleanest way to understand the relation between vector fields on M and N is via the notion of F-related vector fields. If s is a section of a vector bundle V->N, then F*s, the composite of s with F, is a section of F*V over M: thus (F*s)(x)=s(F(x)) in (F*V)x = VF(x). In particular, if Y is a vector field on N, then F*Y is a vector field along F. Vector fields X on TM and Y on TN are said to be F-related if dF(X) = F*Y (both of which are vector fields along F).
Under certain conditions - for example if F is a diffeomorphism - there is a unique Y which is F-related to X, and this is the pushforward "which does not always exist" in the above talk. I think this article could be cleaned up by clarifying all this. Maybe I will find time to do it, if others agree. Geometry guy 17:15, 8 February 2007 (UTC)
replies: text: hello. those seem to be reasonable comments and incorporating them into the article would be a good idea. Mct mht 00:46, 10 February 2007 (UTC)
thread_title: Pushforwards and pullbacks

message: Fropuff: thanks for moving this page (I've only just got move rights), but how about we move it to Pushforward (differential), i.e., pushforward by the differential of a smooth map, and replace this by a disambiguation page which can link to pushforward measure and direct image sheaf as well? Geometry guy 17:49, 11 February 2007 (UTC)
thread_title:  Title and clean-up 

message: I feel it's too easy to chalk the squishing and change of angle in the elipsoid point's tangent space up to roughness or perspective. Should the image be redone so that it's more dramatic, with a lower and nearer-facing point? ᛭ LokiClock (talk) 22:36, 20 July 2011 (UTC)
thread_title:  Image too subtle? 

message: The image is really cool.  Do you have the source code for that?  — Preceding unsigned comment added by 68.147.37.61 (talk) 10:09, 29 October 2012 (UTC)
replies: text: The image is an svg, so the source can be obtained like you would for a normal HTML page. ᛭ LokiClock (talk) 23:04, 25 November 2013 (UTC)
thread_title:  Nice Image 

message: The red and yellow vectors form a basis for the tangent space in the image, and dφ(x) is supposed to be the best linear transform between the tangent spaces. Shouldn't that map red coordinate a in one tangent space to the red coordinate a in the another, and same for yellow coordinates? Yet, the yellow coordinate for the example point being transformed by dφ(x) is a large positive number in the left image and a large negative number in the right image. How does that work? -- 212.149.196.26 (talk) 23:23, 23 November 2013 (UTC)
replies: text: The black arrow is not transforming a specific point, but showing the spaces the function is going from and to. The yellow arrows and red arrows are the only specific vectors whose images are depicted. ᛭ LokiClock (talk) 17:16, 24 November 2013 (UTC)
thread_title:  dφ(x) arrow in the image 

message: Substituted at 02:31, 5 May 2016 (UTC)
thread_title: Assessment comment

message: The notation \mathrm{d} for the pushforward seems to me to be used in only a small minority of differential-geometry texts, and it clashes with the universal notation for the exterior derivative. I propose to change it to ' or the tangent-map notations {}_* or \mathrm{T}.
Example of notation clash:
Consider a function f from a manifold M to the reals, f\colon M \to \mathbf{R}.
The exterior derivative of f, universally denoted \mathrm{d}f, is a map from the tangent space at each point p of the manifold to the reals: \mathrm{d}f(p)\colon \mathrm{T}_pM \to \mathbf{R}.
The pushforward of f, let us denote it f', is instead a map from the tangent space at p to the tangent space at f(p): f'(p) \colon \mathrm{T}_pM \to \mathrm{T}_{f(p)}\mathbf{R}.
They're completely different objects. For the first it also makes sense to consider \mathrm{d}^2, whereas for the second this doesn't exist.  — Preceding unsigned comment added by Pmpgl (talk • contribs) 22:29, 13 September 2017 (UTC)
replies: text: I fully support this motion! Ehaarer (talk) 17:09, 6 July 2023 (UTC)
thread_title:  Should we abandon the <math>\mathrm{d}</math> notation in favour of <math>\mathrm{T}</math> or <math>'</math>? 

message: The text, very bottom of the first/top pic, is innacurate: The image of the pushforward will not not necessarily hit every point p in T_p M, unless \phi is onto. 4.2.109.2 (talk) 20:09, 17 April 2023 (UTC)
thread_title:  First Pic/Drawing 

message: In the intro it says “…the best linear approximation of φ
near x.” Shouldn’t that be best linear approximation of the image of x under φ on N?
Ehaarer (talk) 17:13, 6 July 2023 (UTC)
thread_title:  Intro