# Talk:Differential form

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## Closed form redirect

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

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

## Confusion of "one-form" with "infinitesimal"

I have an issue with the statement

For instance, the expression f(x) dx from one-variable calculus is called a 1-form, and can be integrated over an interval [a, b] in the domain of f

This is not true of the abuse of notation used in one-variable calculus. However, when we define an operator d on a real manifold of dimension no less than 1, the above notation when re-interpreted with this operator becomes a one-form. It would then be more correct to say

The expression f dx, where f and x are both scalar functions over a real manifold, is called a 1-form, and can be integrated over any path on the manifold (independently of coordinatization of the manifold).

It is not appropriate for WP to take interpretations one branch of mathematics (differential geometry) and to state these as applying to another (real analysis). The similarity of the notation is not an excuse to confuse the reader. Am I right? —Quondum 04:44, 8 March 2016 (UTC)

I disagree with this. Differential one-forms are already part of standard one-variable calculus, and this is what we mean when we write ${\displaystyle f(x)dx}$. See differential of a function. The quantity ${\displaystyle f(x)dx}$ behaves exactly as one should expect a differential form to behave. It satisfies the same properties under pullback (that is, change of variables or the chain rule) and change in orientation of the domain:
${\displaystyle \int _{b}^{a}f(x)dx=-\int _{a}^{b}f(x)dx}$
and it satisfies Stokes' theorem. Exact one-forms are often called exact differentials in calculus as well. In real analysis, one might write ${\displaystyle \int _{[a,b]}f(x)\,dx}$ to mean the integral of f with respect to the Lebesgue measure on [a,b]. But this is not the same thing as ${\displaystyle \int _{a}^{b}f(x)\,dx}$. The latter integral is an integral over a chain rather than a set, and so it is sensitive to the orientation of the chain. 11:22, 8 March 2016 (UTC)
I note that the lead of Differential of a function that you referred to says that the interpretation depends on context (and with which I have no issue). This suggests that in some contexts "the expression f(x) dx from one-variable calculus [is] an example of a 1-form", but in others it is regarded as something different. The wording used in this article suggests that this is the standard interpretation, which seem incorrect to me. —Quondum 05:04, 9 March 2016 (UTC)
There's not really an "interpretation" here. By definition, a differential form is dual to chains (see, for example, Rudin "Principles of mathematical analysis"). That's precisely what the differential expression ${\displaystyle f(x)\,dx}$ is in one variable calculus. So, if you feel that ${\displaystyle f(x)\,dx}$ is not a differential form, could you please be specific? Does it have some property that a differential form on an interval lacks? Or does a differential form on an interval have some property that ${\displaystyle f(x)\,dx}$ lacks? Because they seem the same to me. 11:31, 9 March 2016 (UTC)
I was referring to the term "1-form". I have no issue with the use of the term "differential form" in this context. A property that I expect of a one-form is that it behaves like an n-dimensional vector field when it is a function of n variables. I'm not saying that a 1-form is not a differential form either, since the latter seems to encompass a range of concepts, just that I would have expected a 1-form to have the more specific meaning assigned to the term in differential geometry. —Quondum 05:36, 10 March 2016 (UTC)
The term "1-form" is unambiguous and always names the same type of object, regardless of context. Also, it does not behave like a vector field because of how it varies under the transition functions of a manifold; it behaves like (in fact, is) a covector field. Ozob (talk) 13:24, 10 March 2016 (UTC)
Covectors form a vector space and that is the sense in which I meant it; substitute the term if that makes it any clearer. I presume the "type of object" is that referred to by a geometer; my objection is that in many introductory fields of calculus the "type of object" meant by a "differential form" would not be referred to as a "1-form" and should be regarded as inequivalent, at least conceptually, contrary to the implication in this article. But I see that I am failing to make my point to mathematicians, so let's just drop it. —Quondum 15:12, 10 March 2016 (UTC)
Sorry, you seem quite vexed. I don't mean to be disagreeable, but I'm having trouble understanding the point you're trying to make. I think I'm beginning to grasp it. Let me try to state what I think you're saying, and perhaps (if you still have patience for me) you can tell me if I understand you rightly.
In one-variable calculus, we meet expressions like ${\displaystyle f(x)\,dx}$. At the time, we're told that f is a function and dx is just a formal symbol. When we learn real analysis, we're told that dx is actually Lebesgue measure. If ${\displaystyle f(x)\,dx}$ means anything, it's a measure which assigns to a measurable set E the measure ${\displaystyle \int _{E}f(x)\,dx}$ (possibly this is even signed or complex-valued). In differential geometry, we're told that ${\displaystyle f(x)\,dx}$ is a 1-form. We soon learn that on a 1-manifold, ${\displaystyle f(x)\,dx}$ defines for us a measure by the formula ${\displaystyle \int _{E}f(x)\,dx}$ again, and that this measure agrees with the previous one; however, despite using identical notations and coming to identical conclusions, these two integrals have different interpretations. After all, to integrate a 1-form, we choose local coordinates and integrate with respect to a measure in those coordinates. If I understand your objection (and please tell me whether I do or don't), it's that the article does not distinguish these two situations, even though failing to distinguish them can be confusing, misleading, or even wrong. Is that a correct statement? Ozob (talk) 23:57, 10 March 2016 (UTC)
To be sure, I am neither irritated at you nor at Sławomir (both of whom I hold in high regard and I do not doubt either's goodwill), but rather am frustrated at my failure to convey what should have been simple to communicate. I put it down to a communication failure on my part.
You are pretty close in characterizing the issue, and seem to understand nearly exactly what I've been saying, though it is not a failure to distinguish interpretations, but rather an implication (of the statement "the expression f(x) dx from one-variable calculus is an example of a 1-form") that the interpretation as the 1-form of differential geometry always applies, even in one-variable calculus. However, Sławomir's first two sentences in his initial reply seem to contradict my understanding, so I'm just confused. —Quondum 02:13, 11 March 2016 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Sorry, I still don't see the issue. The one form ${\displaystyle f(x)\,dx}$ as it's used in calculus does not mean "integration with respect to the Lebesgue measure". That would be a density rather than a one-form, and could be denoted by ${\displaystyle f(x)|dx|}$. When we write ${\displaystyle f(x)\,dx}$ in calculus, we certainly do mean the one-form and not the density. 02:52, 11 March 2016 (UTC)

To expand on Sławomir's explanation: In basic calculus we are told
${\displaystyle \int _{a}^{b}f(x)\,dx=-\int _{b}^{a}f(x)\,dx.}$
This formula is literally false if we interpret dx as Lebesgue measure. Both integrals should equal ${\displaystyle \int _{[a,b]}f(x)\,dx}$, and Lebesgue measure does not care whether we imagine ourselves traveling from a to b or vice versa. We might accept the above equation as a convention that makes additivity of the integral with respect to integrals cleaner. Equivalently, we might make an ad hoc definition that lets ${\displaystyle \int _{a}^{b}f(x)\,dx=\int _{[a,b]}f(x)\,dx}$ if ${\displaystyle a\leq b}$ and ${\displaystyle \int _{a}^{b}f(x)\,dx=-\int _{[a,b]}f(x)\,dx}$ if ${\displaystyle a\geq b}$. Or we can recognize that the ad hoc definition is not so ad hoc in the context of manifolds. It's precisely how integration of 1-forms behaves: We choose local coordinates, integrate with respect to a measure, and then fix the sign.
Despite this, I think it's not quite right to say that ${\displaystyle f(x)\,dx}$, as used in calculus, is always exactly a 1-form. When I first learned calculus, I was taught the first interpretation above, that the case ${\displaystyle b is just a convention. This is not something I approve of now, but it was done to me, and I'm sure it's still done in many places. I wonder if, historically, it might have originally been just a convention? I don't know. In any case, I think that calls for some delicacy in the present article. Ozob (talk) 13:45, 11 March 2016 (UTC)
If this were an article on calculus, then it would probably be misleading to call f(x)dx a one-form. But as an article on differential forms, I do not think it is misleading to say this. In fact, it is actually helpful, because it relates the topic of this article (differential forms) in a clear and direct way to something readers already know about (integration in one dimension). 15:45, 11 March 2016 (UTC)
Then it seems that we do not disagree about the substance of the intended meaning, but only about the semantics of the language used. To me, the phrase "the expression f(x) dx from one-variable calculus" implies that the expression is to be interpreted in the context of calculus, not in the context of the article. The debate might have been clearer if the notation use had visibly differed between calculus and differential geometry. Also, if everything is to be interpreted in the context of differential geometry, is the notation ${\displaystyle \int _{a}^{b}}$ even used here? —Quondum 22:00, 11 March 2016 (UTC)

## Assessment comment

The comment(s) below were originally left at Talk:Differential form/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

 Close to B. A longer lead and some more details (e.g. on exterior derivative and relation to the notation) would easily lift this. Geometry guy 13:43, 2 June 2007 (UTC)

Last edited at 13:43, 2 June 2007 (UTC). Substituted at 02:00, 5 May 2016 (UTC)

## wedge product

The current part considering wedge product is only slightly helpfull. Better would be the full definition (s. e.g. [1]). ChristianTS (talk) 17:24, 7 November 2016 (UTC)

I've expanded the article. Does this help? Ozob (talk) 03:56, 8 November 2016 (UTC)