I recently came across the subject of differential forms. Based on the wikipedia page I initially assumed that they were nothing but the extension of differentials to higher dimensions. My line of reasoning was that just how differentials represent a small change in a function or a small part of a curve, differential forms represent a small change in a multivariable function or a small portion of its graph. But then, I came across a post in which the second answer says that the differential of a function can be interpreted as an infinitesimal number or a differential form, this put me off, and hence I would like to clarify this point: do differentials come under differential forms, thus making an infinitesimal number the same as a differential form? Or are they two separate entities?
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A differential form is (technically) a function that we can calculate value at a point and AFAIK it has nothing to do with infinitesimals nor tends to anything. A course in precalculus, calculus, or even real analysis almost never gives an answer to "What is dx?". It is only until differential geometry, one gets to learn what it is. One should not learn these from Wikipedia but from a mathematics textbook. – Duong Ngo Mar 10 '25 at 05:40
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2@DuongNgo, it does have to do with infinitesimals; see my answer. – Mikhail Katz Mar 10 '25 at 08:41
1 Answers
The main difference between a differential such as $dx$, $dy$, or more generally $df$ (for a function $f$ of several variables) and a differential form is generality: a differential is a 1-form whereas in differential geometry one works with more general $k$-forms where $k$ can be as large as the dimension of the manifold one is in (above that dimension, all differential forms vanish).
A typical example is the volume form of a Riemannian manifold, which sometimes appears as $dV$ or $d\,V\!ol$ in integrals, though such notation is misleading (because the volume form is not the $d$ of anything, in general).
Having said this, for understanding integration on manifolds, it is important to keep in mind the interpretation of differential forms as very small parallelograms (or parallelepipeds), or more precisely as a way of assigning a very small volume to such parallelograms.
User Duong Ngo may be interested to find out that Riemann, the founder of Riemannian geometry and hence differential geometry, thought of $dx$ and $dy$ as infinitesimals; see for example Section 2.7 this recent publication:
Katz, M. "Episodes from the history of infinitesimals." British Journal for the History of Mathematics (2025). https://doi.org/10.1080/26375451.2025.2474811, https://arxiv.org/abs/2503.04313
It is a remarkable fact that many standard formulas, such as $df= f_{x_1} dx^1 +\ldots f_{x_n} dx^n$ for a function $f=f(x^1,\ldots,x^n)$ has two coherent interpretations: (A) viewing $dx^i$ as differential 1-forms, or (B) viewing the $dx^i$ as infinitesimals. A typical study of approach (B) can be found in this publication:
Nowik, T; Katz, M. "Differential geometry via infinitesimal displacements." Journal of Logic and Analysis 7:5 (2015), 1-44. https://www.logicandanalysis.com/index.php/jla/article/view/237, https://u.math.biu.ac.il/~katzmik/dgnsa_arxiv.pdf, https://arxiv.org/abs/1405.0984
To answer your question:
"do differentials come under differential forms, thus making an infinitesimal number the same as a differential form? Or are they two separate entities?"
They are two separate approaches, (A) and (B), to interpreting the notation $dx$, as noted above. Note that Riemann himself followed approach (B).
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Perhaps one should mention that in general the differential of a function from one manifold to another can not necessarily be viewed as a 1-form on the domain. – Filippo Mar 10 '25 at 08:37
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@Filippo, that's yet another generalisation of the notion of a differential. One can't account for all of them in a single answer necessarily. Presumably you are referring to the situation with a differentiable map $f: M\to N$ and the associated $df: TM \to TN$, sometimes also denoted $Tf$. – Mikhail Katz Mar 10 '25 at 08:39
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2@Filippo For what it's worth, you can view the differential $df$ as $1$-form with values in the pull-back bundle $f^TN$. That is, for $f\colon M\to N$ we have $df\in \Omega^1(M,f^TN)$. – Jan Bohr Mar 10 '25 at 08:43
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Correct me if im wrong but according to what you said, a differential form is a generalization of the differential to k dimensions which can be interpreted as the infinitesimal volume element of a manifold. But, I don't understand what the difference between a differential 1-form and an infinitesimal is, isnt a differential 1-form and infinitesimal volume element and thus also and infinitesimal? (Sorry if this sounds dumb, I don't have an extremely sophisticated math background) – Mayo Mar 11 '25 at 05:30
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2The starting point for differential forms, formally speaking, is the duality between tangent vectors and cotangent vectors (so that infinitesimals need not appear). You can find an introduction to differential forms in my course notes here: https://u.cs.biu.ac.il/~katzmik/egreg826.pdf – Mikhail Katz Mar 12 '25 at 09:41
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I quote from wikipedia "A differential 1-form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential 2-form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density. And so on." This is what led me to understand that differential forms are infinitesimal but in higher dimensions. – Mayo Mar 14 '25 at 04:25
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@MikhailKatz assuming the above is true, how are differential forms and infinitesimal different. (The notes you gave are still too mathematically sophisticated for me) – Mayo Mar 16 '25 at 04:05
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@Mayo, a differential 1-form at a point of $\mathbb R^n$ is a linear map $\mathbb R^n\to \mathbb R$ (this may in general vary from point to point). An infinitesimal is an element violating the Archimedean property. Note that the comment at wiki is an explanatory/motivating comment, not a definition. – Mikhail Katz Mar 16 '25 at 07:20
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I don't understand even after you have simplified this much, so maybe I'll turn back to this when I enter university and actually have enough mathematical background to understand you. Nonetheless, thank you. – Mayo Mar 17 '25 at 12:43