Talk:Closed and exact differential forms

Better wording

I would find it clearer to say:

Since d^2 = 0, to be closed is a necessary condition to be exact. Asking whether this is also a sufficient condition is a way of detecting topological information, by differential conditions.

The argument is that it is usually easy to test da=0 and hard to find b with db=a.

Be bold! If you feel it's best to change, change it, by all means. Dysprosia 10:25, 5 Feb 2004 (UTC)

Disagree with merging of this into differential form

This is a nice short article focused on explaining one concept, so it should stay this way. I don't see how this would be made better if part of a bigger article.

What this article needs is an expansion. It neeeds to be made more clear what exactly closedness means, some examples, and a counterexample of a closed form which is not exact.

MarSch, please read Wikipedia_talk:WikiProject_Mathematics#Math_in_the_dock. The biggest problem of math on Wikipedia is that it is too hard to understand for people who know less than the author of an article. So let us try to make things more accessible, not less. Oleg Alexandrov 16:01, 14 September 2005 (UTC)[reply]

Agree with Oleg. Also, this article can be expanded, as it is the doorway to homology and cohomology and harmonic forms and all that. linas 01:05, 15 September 2005 (UTC)[reply]

Error of sign?

Isn't there an error of sign in the expression for $d\alpha$ where $\alpha$ is a one form on the plane? (anonymous) 11:26, 6 October 2005 (UTC)

Different authors have different sign conventions; but you are right, it it disagrees with the convention used in exterior derivative. linas 23:52, 7 October 2005 (UTC)[reply]

Gauge transformation

The article of gauge transformation says:

Mathematically, a gauge is just a choice of a (local) section of some principal bundle. A gauge transformation is just a transformation between two such sections.

The introduction part of the present article says:

The form β is called a "potential form" or "primitive" for α. Since d² = 0, β is not unique, but can be modified by the addition of the differential of a two-step-lower-order form. This is called gauge transformation.

Is it correct? Is it true that this is generally some gauge transformation? Because if yes, then there sholud be somwhere a brincipal bundle and a section. Where are they?
89.135.29.163 (talk) 06:16, 4 March 2011 (UTC)[reply]

Poincare Lemma, is 'smooth' redundant?

The definition given of the Poincare lemma states in part: '...any smooth closed p-form α defined on X is exact...'.

Is the smooth property redundant? Aren't all differential forms smooth? 130.95.51.206 (talk) 04:37, 30 September 2012 (UTC)[reply]

Poincare Lemma: must the homotopy be smooth?

As stated now, it says that Contractibility means that there is a homotopy F ...that continuously deforms X to a point. Since we want to pull back a smooth form along F and obtain a smooth form, don't we need F to be smooth as well?

Kernel of a Form?

It is my understanding that kernels make sense only for linear maps , i.e., for 1-forms. For k-linear maps defined on W:=VxVxVx..xV, the kernel is not a subspace of W. Should one then talk about the kernel of a k-form in general, or just about the kernel of a 1-form? — Preceding unsigned comment added by 150.210.231.36 (talk) 21:42, 31 July 2014 (UTC)[reply]

New material on homotopy operator and Poincare lemma

I suggest that the new material on Homotopy operator should be added or explicitly stated in the current version of the article. The proof of the Poincare lemma can be translated to to the Homotopy Invariance Formula and this is in the classical books on Differential geometry.

Citation of the nLab:Poincare lemma webpage should be added, where similar (I assume the standard) proof is provided, as well as the reference to the source of original idea of the proof.

There should be also note on the complex version of homotopy operator and its connection to the Maurer-Cartan equation to provide full exposition of the material.

Is someone consider moving the material on Poincare lemma to the separate page? — Preceding unsigned comment added by UsernameAlreadyChoosen (talk • contribs) 19:41, 17 July 2020 (UTC)[reply]

I'm not sure it was wise for you to have removed the copy-vio tags, which state that the history page for the article might need a revdel. No administrator was involved, so some check might be needed.

I reverted the page to Your last edit just in case and to not disturb users of Wikipedia, although still I do not see any problem with my edit - all sources were properly cited when the material from them was cited or even followed loosely. It was not a copy-paste - some notation was adjusted to the article and Wikipedia other articles convention - although mathematical formulas are obviously wrong when they are not 'isomorphic' to those from the cited text and reasoning is illogical when not followed as in original source. This was a new material that significantly improves the article and provides a different perspective on the subject. Citing scheme was similar to what was done in the current version of the article, e.g., 'Napier & Ramachandran 2011, pp. 443-444' - properly cited example from the 2011 book was reused and no problem with copyrights. In my opinion it was overreaction. Anyway, until we all resolve what should be added/improved (if yes) in the current version of the article I gave up since archiving day's work is a waste of time. I proposed some directions of development which are in the archive version of my edit (and above in my questions) and can be used in improvement if all agree that it is needed. We should make some consensus.UsernameAlreadyChoosen (talk) 21:06, 17 July 2020 (UTC)[reply]

The problem is with the source that you have used, which was made available this week with a 2020 copyright label. The material in the wikipedia article dates back to the 19th century and is available in many standard text books. The new material might possibly be called original research. Mathsci (talk) 20:12, 17 July 2020 (UTC)[reply]

The proposed material on Homotopy Invariance Formula (I agree that it a bit resemble the current proof of the Poincare lemma, however it extracts this HIF as independent entity and describes its interesting ans useful properties) is typical for this subject and can be found in the today-standard Differential geometry books cited in my edit of this article. The only 'research' material (classified by novelty/date of publication) which was added bases on the recent article that contains homotopy operator in complex manifolds, with proper citation, and summary of the Poincare lemma history up to date. It can be thrown away if it is not clear about how close we should follow derivation of mathematical concepts from properly cited article. We can also ask Author of the article to donate the content if there is need to include this material.UsernameAlreadyChoosen (talk) 21:06, 17 July 2020 (UTC)[reply]

The approach to homotopy invariance is standard; for reference it is now in the article. As with Cartan's formula (connecting the Lie derivative, the interior multiplication and the exterior dervative), the proof of homotopy invariance follows or generalizes the proof of the Poincaré lemma in the article. Homotopy invariance is set as an exercise in Warner's book.

The article on closed and exact diffential forms is fairly rudimentary. Different sections rely on pure and applied mathematics and theoretical physics. It is aimed to be a brief account for advanced mathematics and physics undergraduates or beginning graduates. Other wikipedia articles cover de Rham cohomology, Čech cohomology, simplicial homology, complex manifolds, the Dolbeault cohomology, etc, in more detail. At this stage, the July material does not seem appropriate for an encyclopedia. That is not a judgement on the material in that article. Mathsci (talk) 23:57, 17 July 2020 (UTC)[reply]

I see your point to make it rely on Warner's book. I am glad you provided at least some basic info on homotopy. It is the main tool in many books, e.g. in the book of L. Tu, and R. Bott, L. Tu. I managed to recover revdel info and hopefully it cleans the revision history. UsernameAlreadyChoosen (talk) 07:42, 18 July 2020 (UTC)[reply]

No. This is a rudimentary article on basic topics, and such content is inappropriate. There are existing articles on the Homotopy invariance and Homotopy operator and Poincare lemma but I don't suggest editing those until you get more familiar with how Wikipedia works. Seeing wild copyvio and grand claims about the latest results is not the way to go about things. 67.198.37.16 (talk) 06:18, 17 November 2023 (UTC)[reply]