Talk:Differential form

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Serious Revisions Needed!!![edit]

This page has too many problems. It should begin with a disclaimer. —Preceding unsigned comment added by MephJones (talkcontribs) 20:06, 12 March 2010 (UTC)

Maybe you should be more specific... I for one think the concept section (at least) is terrific! —Preceding unsigned comment added by 66.97.107.188 (talk) 01:04, 3 April 2010 (UTC)

Some of the MANY problems this article presents to me (in the lede):
A. "Multivariable". "Multivariant" is by far the more common term - in my experience (USA, science)
B. Parsing the 1st paragraph,
i) Differential forms "are an approach to..." - no they're not - they are USED in (or for) an approach.
ii) They provide "an approach to defining integrands"... So, what happened to the differential calculus portion of 'multivariable calculus'? I realize that separating integral and differential calculus is only possible in simple, 'pure' cases, but this 'explanation' lacks clarity, imho.
iii) After reading the introductory paragraph I have learned that they involve multivariant calculus, that they are "an approach", and they have many applications (and also the editors can't decide if they are plural or singular - is it 'they are' or is it 'it is'? - make up your mind). No description is offered of what they are. Then several totally confusing examples are given without ANY attempt made to describe the context of the provided equations, or define what the meaning of the equations are.
iv) the first example states that f(x)dx is a 1-form. Well, no, its not. If x is a variable and f() is a function of that variable and if dx exists then the expression f(x)dx has meaning.
v) the first example goes on to claim that this 1-form can be integrated over an interval [a,b]. Well, no it can't. At least not necessarily, and I believe all contributors to this article know why that just isn't true.
vi) The 2nd 'example' is even more horrible: "f(x,y,z) dx∧dy + g(x,y,z) dx∧dz + h(x,y,z) dy∧dz". As a clue, the wedge operator has not yet here been defined, and it does NOT form part of a typical calculus education (hence not explaining it PRIOR to using it is simply poor practice, very poor). Further, why are there 3 functions (f(),g(), and h())??? Is that required? What if there were just two? What if there were four? Are the 3 wedge products necessary for a 2-form? Is their order significant? What about f(x,y,z) dx∧dy + f(x,y,z) dz∧dx or f(x,y,z) dx∧dy + g(r,s,t) dr∧ds or f(x,y,z)dx∧dy + g(r,s) dr∧ds or f(x,y,z)dx∧dy + g(r,s,t,u) dr∧ds. Teaching by example requires presentation of an EXHAUSTIVE series of examples - either that or (prior) explanation of specific and general cases. I could go on and on with these questions. What I'm trying to convey is that the examples are almost totally useless, one should use examples AFTER the definition/explanation, not before.
vii) To continue with the second example: the claim is made that the expression has a surface integral. Why? Why is it necessary for the surface integral to exist? Magic? Definitions not provided to the reader? Requirements for proper structuring of the expression? What?? Also, it should be clear that the expression BELONGS in parentheses if the entire expression is to be integrated over S. ∫f(x,y,z) dx∧dy + g(..) is OBVIOUSLY unclear compared to ∫{f(x,y,z) dx∧dy + g(..) }.... and thats as far as I've gotten... I should note that I am encountering the term n-form more and more often, so its either the flavor of the month or is going 'mainstream' and displacing less precise terminology.173.189.78.173 (talk) 15:03, 1 September 2014 (UTC)
It seems to me that you misunderstand the purpose of the article. Wikipedia is not a textbook. The article is not structured to give a careful, didactic exposition of differential forms and how they are used. It is, like all Wikipedia articles, an attempt to be an encyclopedic account of its subject. I will concede that the article is not perfect, and glancing through it again I see some things I would change, but your primary objection seems to be that the article is not something that it does not wish to be.
In regards to your specific objections, I more often hear the noun phrase "multivariable calculus" that the adjective-plus-noun "multivariate calculus", and I've never seen or heard "multivariant calculus". The first example f(x) dx is indeed a 1-form, and just as the article claims, it can be integrated over a closed integral (or at least attempted to be integrated; you could get ∞, −∞, or an undefined quantity such as ∞−∞); the result is what we expect from basic single-variable calculus. The second example is a 2-form, as claimed, and it can be integrated over a surface. There are many different 2-forms, and you wrote down some other possibilities above. An n-form is a 0-form, 1-form, 2-form, etc. according to the (non-negative integer) value of n.
Differential forms have been around for about 100 years and are a mainstream, indispensable tool. I recommend that you read one of the books listed in the references. Flanders is the most elementary and physically motivated, I think. Ozob (talk) 16:19, 1 September 2014 (UTC)
There is actually a bit of confusion in the lead: it uses the expression \int_a^b f(x)\,dx, which is an abuse of notation if dx is to be treated as a one-form (only curves are valid; curves are not specified by their endpoints on a general manifold). This abuse in this context seems designed to confuse, since it clearly uses a familiar notation without pointing out that the meaning has changed. The IP's objection that the notation for exterior product is used without mention or linking is also valid. —Quondum 17:57, 1 September 2014 (UTC)

Closed form redirect[edit]

I think that closed form should redirect here, rather than to de Rham cohomology as at present; and also should be disambiguated with respect to the 'closed form solution' meaning.

Charles Matthews 14:03, 11 Nov 2003 (UTC)

Disagree with merging closed and exact differential forms into here[edit]

See Talk:Closed and exact differential forms (unsigned comment by Oleg Alexandrov (talk))

symplectic form[edit]

http://en.wikipedia.org/skins-1.5/common/images/button_math.png Mathematical formula (LaTeX) Hi, sorry if this is a mixture or request/comment/question, but i was just wondering if symplectic differential form has any relation with this, and what exactly the correct definition would be. The preceding unsigned comment was added by Evilbu (talk • contribs) .

Either a 2-form, or a closed 2-form - probably the latter, in contemporary literature. Charles Matthews 22:25, 6 February 2006 (UTC)
In addition to being closed, a symplectic form must also be nondegenerate. See symplectic manifold for details. There are words like almost-symplectic, which means not necessarily closed, etc. Orthografer 00:57, 3 November 2006 (UTC)

multilinear map from \wedge^n\ TM?[edit]

The article claims "At any point p on a manifold, a k-form gives a multilinear map from the k-th exterior power of the tangent space at p to R." Wouldn't this be just a linear map from the exterior power (which itself, however, might be thought of as a alternating multilinear map on TM\times\cdots\times TM)? Tesseran 05:59, 24 July 2006 (UTC)

We cannot consider a multilinear map on TM\times\cdots\times TM because this is not a linear space. Commentor (talk) 02:00, 27 March 2008 (UTC)

Why[edit]

you people don't explain why you use indexes below and above without any care? Why don't you explain that there is an advantage by indexing above for coordinated function? --kiddo 01:56, 2 November 2006 (UTC)

Page complexity[edit]

I think this page is too complex for general wikipedia users. Any one expert on this topic please make the page more comphrensive. —Preceding unsigned comment added by 76.184.2.133 (talk) 11:19, 18 October 2007 (UTC)

notation[edit]

Why is a basic 1-form written in two different ways in this article. Sometimes it's dx^i, sometimes it's dx^I? Randomblue (talk) 18:04, 2 February 2008 (UTC)

A form with capital indices is a k-form rather than a 1-form. This is explained in the article. Silly rabbit (talk) 18:12, 2 February 2008 (UTC)

When we integrate a function f over an m-dimensional subspace S of \mathbb{R}^n, we write it as

\int_S f\,{\mathrm d}x^1 \cdots {\mathrm d}x^m.

Consider {\mathrm d}x^1, ...,{\mathrm d}x^n for a moment as formal objects themselves

In the one case we go up to dx^m, and in the other dx^n. Is this an error? Randomblue (talk) 22:25, 2 March 2008 (UTC)

Yes, an error. It's so minor that I am not sure it's worth fixing it --- maybe somebody will be motivated by that to do some more substantial improvements :) 02:04, 27 March 2008 (UTC) —Preceding unsigned comment added by Commentor (talkcontribs)
But we are integrating here over S, not over Rn. S is a submanifold of dimension m. So one integrates m-forms over S, not n-forms. silly rabbit (talk) 02:20, 27 March 2008 (UTC)

Stylistic elements[edit]

Umm... so is a math article supposed to make me laugh out loud multiple times? I think this page should be reworked stylistically.
Example: Consider dx1, ...,dxn for a moment as formal objects themselves, rather than tags appended to make integrals look like Riemann sums.
And worse: where dxI and friends represent basic k-forms
And "Gentle introduction" is a rather interesting section name... —Preceding unsigned comment added by 24.12.151.56 (talkcontribs)

So fix it. siℓℓy rabbit (talk) 13:14, 11 August 2008 (UTC)

error in the lead paragraph[edit]

Not every differential form is a wedge product of exterior derivatives, contrary to the claim in the introduction attributed to Cartan. Since I am not sure what the editor's intention was here, I cannot correct it easily. Certainly Cartan did not say any such thing. Katzmik (talk) 07:33, 14 August 2008 (UTC)

Attributions aside, I'm not sure what the issue is here. On a smooth manifold the differential forms are defined locally as linear combinations of wedge products as in \sum_if_i(\mathbf{x})dx_{i,1}\wedge\ldots dx_{i,k}, where $k$ is some fixed nonnegative integer. Does my edit work for you? Orthografer (talk) 00:06, 17 August 2008 (UTC)

exterior algebra and exterior derivative[edit]

The two exterior items mentioned in the lead are closely related but the use of the word "exterior" for both is actually very misleading. The former is essentially a concept in linear algebra, while the latter requires a differentiable structure. In particular, contrary to the claim in the lead paragraph, differential forms do not form an exterior algebra (the article on exterior algebra is purely linear-algebraic and deals with the finite dimensional case), but rather a differential algebra, or more precisely a differential graded associative algebra. Katzmik (talk) 07:40, 14 August 2008 (UTC)

This situation always arises with spaces of sections, and it is clearly better to avoid abuse of language whereever possible. There is no requirement in particular that exterior algebras must be finite dimensional, although I suppose I must assume some responsibility for the article Exterior algebra treating primarily the finite dimensional case. Nevertheless, a more accurate description would be that differential forms are sections of a sheaf of exterior algebras, or that they are sections of the exterior algebra of the cotangent bundle. siℓℓy rabbit (talk) 13:31, 14 August 2008 (UTC)

Motivation[edit]

Here is a proposal for a different way to motivate the concept. Start with volume computation in \mathbb R^n. Observe that it is invariant under linear transformations with unit determinant. Therefore, the minimal structure needed to define volumes has less structure than the full coordinate system. Argue that volume is naturally a signed quantity. Then generalize to smooth manifolds. This gives the top dimensional differential form and justifies the anti-commutativity. To motivate the intermediate dimensional forms discuss signed lengths of curves invariantly. (Maybe this would be too long?) Opinions? Oded (talk) 16:11, 17 August 2008 (UTC)

This is nice, but somewhat non-standard. Ultimately we have to be able back up what we write by reliable secondary sources, so we have to take a fairly standard approach. Unfortunately most of this article is still rather weak: I expanded the vanilla motivation to draw attention to this weakness. I hope other editors who agree with my view that differential forms are rather important (!) will help to fix the basic shortcomings of the article. The relation to cohomology and homology needs massive expansion. The treatment of operations is utterly inadequate: what is the Lie derivative, the exterior derivative, Cartan's identity, etc. etc.? I'm looking forward to the luxury of fine-tuning the motivation and lead, but right now, this article has other issues. Geometry guy 20:40, 17 August 2008 (UTC)
The motivation in terms of determinants and more generally minors is done very nicely at exterior algebra (work by silly rabbit I believe and others). There is no need to duplicate it here. Katzmik (talk) 09:27, 20 August 2008 (UTC)

From my perspective, the main use of differential forms is in order to integrate them. They are the "right thing" to integrate, so to speak. Therefore, this seems like the most useful motivation. Of course, I realize there could be other perspectives as well. Oded (talk) 17:50, 20 August 2008 (UTC)

I agree. My first comment also was in agreement with your remarks, particularly concerning invariance under SL(n). I was merely pointing out that some of this discussion is available at the other page. You are welcome to copy some of it over to here if you like, or present your own perspective. Katzmik (talk) 08:29, 21 August 2008 (UTC)
Over compact oriented submanifolds. In the (more natural) cooriented situation, you need to use multivector densities. But in any case, it's not our job on Wikipedia to present our own perspective. Geometry guy 14:23, 21 August 2008 (UTC)

error in definition of de Rham cohomology[edit]

In the definition of the exterior differential complex, it seems to me that the first term "R" should be deleted. Otherwise an interval has trivial 0-dimensional cohomology. Katzmik (talk) 09:00, 21 August 2008 (UTC)

Fixed. Geometry guy 14:00, 21 August 2008 (UTC)

Unclearness in expression of 1-form[edit]

The expression \mathrm d f = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \mathrm{d} x^i used to introduce the 1-form is circular because it invokes the 1-form dxi. The circularity could be removed if the value of dxi had been given previously. The preceding paragraph seems to attempt to describe dxi but only gets as far as ∂xi / ∂xj which is not enough. I'm speaking as someone who has learned multivariable calculus but not yet differential forms. Halberdo (talk) 21:48, 12 January 2009 (UTC)

It seems that dxip(a) is the projection of a onto the xi axis. As I am new to this subject, I hesitate to add that explanation to the article, in case it is wrong; but if it is right, would someone please do so?Halberdo (talk) 22:17, 12 January 2009 (UTC)

Right, \scriptstyle{dx^i}_p(a)=a^i, where \scriptstyle a=a^1e_1+\cdots+a^ne_n in case of euclidean forms. These \scriptstyle{dx^i}_p are the generators of the dual space of the tangent space \scriptstyle T_pM, which in turn also is generated by the derivations \scriptstyle\frac{\partial}{\partial x^i}, --kmath (talk) 23:39, 18 January 2009 (UTC)

K-form redirect[edit]

It would be helpful if this article could be found by searching for "k-form". At he moment "k-form" redirects to "linear equation", an article in which the term "k-form" isn't even mentioned.Dependent Variable (talk) 04:03, 2 October 2009 (UTC)

Done--kmath (talk) 01:07, 4 October 2009 (UTC)

Question for an expert[edit]

Comparison of vector algebra and geometric algebra states, "In advanced mathematics, particularly differential geometry, neither is widely used, with differential forms being far more widely used."

Could someone knowledgable say a few words on why differential forms are preferred (at least in some contexts) over vector algebra and geometric algebra?

SteelSoul (talk) 17:46, 16 February 2010 (UTC)

Vector algebra is specific to three dimensions. But there are many non-three-dimensional spaces out there: For instance, if I have a physical system consisting of two particles such as the Earth and the Moon, then I need six coordinates: three for the Earth and three for the Moon. Vector algebra does not work in this sort of space. Differential forms do.
Another perhaps less satisfying reason is that differential forms are more "natural" in a way that's hard to explain without a lot of abstract theory. The tangent bundle and cotangent bundle are interesting and canonically defined objects, and taking the exterior algebra is a pretty straightforward operation. From that, we get differential forms and all of their wonderful properties. Vector algebra, on the other hand, is messy: There's no apparent reason why we pick those formulas for div, grad, curl, and the cross product (and in fact the cross product has an implicit dependence on a metric—a metric is extra data). The only reason I know of to pick those formulas ultimately reduces down to facts about differential forms, where the formulas pop out without any fuss at all. Ozob (talk) 00:15, 17 February 2010 (UTC)

Description of Clifford algebras is incorrect[edit]

Article has: 'The exterior product on a Clifford algebra differs from the exterior product of k-vectors (dual to the exterior product of k-forms) in that v \wedge v = Q(v) in a Clifford algebra (the square of a vector is the quadratic form, applied to the vector), rather than 0; it is a non-anti-commutative ("quantum") deformation of the exterior algebra. This structure is used in geometric algebra.'

This is not correct, in the sense that the exterior product of a vector with itself is not the same as the square of a vector in a Clifford algebra, because the wedge product and the Clifford product are different operations with different results. Even when the underlying vector spaces are isomorphic, the two operations are generally distinct, and only coincide when the metric is totally degenerate (ie. 0). See [1] Penguian (talk) 11:25, 18 March 2010 (UTC)

Fixed. Ozob (talk) 00:18, 19 March 2010 (UTC)

References[edit]

  1. ^ P. Lounesto, Clifford Algebras and Spinors, Chapter 14.

What is the example of a Differential 2-form?[edit]

I find the language under concept to be a bit confusing. In particular, when it's written "This is an example of a differential 2-form: the exterior derivative dα..." is it meant that the statement above is the differential 2-form or that the exterior derivative dα is the differential 2-form? — Preceding unsigned comment added by 46.65.200.69 (talk) 19:40, 26 February 2014 (UTC)

I've copyedited the article a little; it should be somewhat clearer now. Ozob (talk) 03:39, 27 February 2014 (UTC)
Ah! So the 2-form is dα and for there to exist f such that α=df, we require that this 2-form, that is, dα, is zero! Thank you, sir/madam, you are a scholar and a gentleman/lady! — Preceding unsigned comment added by 2001:630:12:2E1E:1DEC:488B:BD0C:C6A9 (talk) 11:58, 27 February 2014 (UTC)
You're welcome! Ozob (talk) 14:52, 27 February 2014 (UTC)