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I have been learning differential forms recently and I was wondering if there is a replacement for the usual notation involving differentials. What I mean is essentially a replacement of the "differentials" for example instead of the usual 1 form: $df=\sum\frac{\partial f}{\partial x_n}dx_n$ Something along the lines of: $?f=\sum f_{x_n}?x_n$ Where ?'s represent some new notation.

My reasoning for wanting a different notation is 2 fold:

  1. I find Leibniz notation very vague and misleading in general and honestly much less intuitive than other derivative notations. My biggest issue with Leibniz notation is separation of variables because students learn about it through Leibniz notation instead of the chain rule.

  2. Ignoring any issues with Leibniz notation itself I don't really see a direct connection between these differential forms since they really just represent basis co-vectors.

Here is an example of why this is bad notation other than my personal opinion. $$ \int \int f(x) dx\wedge dx = \int \int 0 = 0 $$ Instead of the classical double integral which does not always equal 0: $$ \int \int f(x) dxdy $$

A new student might naively think the above integrals are the same which is a reasonable assumption since $$ \int\int f(x,y)dx\wedge dy = \int \int f(x,y)dxdy $$

Sam
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    Could you clarify your disdain towards Leibniz notation? I don't especially think that $dx_i$ is confusing. It matches the convention associated to differentials of smooth functions $F:X\to Y$. – Alekos Robotis Jul 02 '19 at 18:26
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    So you want to do away with $\int_a^b df = \int_a^b f'(x),dx = f(b)-f(a)$? The point is that once you learn about differential forms, all your "doubts" can be explained rigorously. – Ted Shifrin Jul 02 '19 at 18:46
  • @TedShifrin My last comment wasn't formulated very well, My issue is not with the link between differential forms and calculus but it is with the link between differential forms and Leibniz notation in calculus. What I mean is $\int f'(x)dx$ is intuitively not the same as $\int df(x)$. The first one is integrating a single variable function where dx is simply notation for variable of integration. In the second one a differential one-form is being integrated which can be done in context's other than the real line, where the dx represents a basis for the dual space of real vectors. – Sam Jul 02 '19 at 19:08
  • @Antonios-AlexandrosRobotis If you are really curious about why I don't like Leibniz notation most of the reasons have to do with how it is taught. I don't like how in introductory calculus courses the chain rule and separation of variables are taught to cancel differentials and students are told its not rigorous but it works. – Sam Jul 02 '19 at 19:38
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    If you don't like the $dx_i$ notation, which is reasonable (although mostly personal), you can use any other. I've seen $\delta_i$ used, for instance. (Yielding $df = \sum \partial_i f\delta_i$ for example.) People use $dx_i$ mostly due to practicality and historical inertia, as many other notations which can also be argued to have problems. The fact is that the problems you mention are effectively inessential if one knows what they are doing. – Aloizio Macedo Jul 04 '19 at 05:00
  • @AloizioMacedo While I definitely agree with you that the issues are inconsequential if you are careful and rigorous and no notation is perfect, I still find the notation purposefully misleading specifically due to the parallels between the calculus notation and the differential forms notation which are about fundamentally different objects. And I say this because I take personal issue with it but also because I see other people misusing this notation all the time without understanding any of the mathematics behind what they are actually doing. – Sam Jul 04 '19 at 05:34
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    This is fairly minor in context, but can you explain what $\int \int f(x)dxdx$ is supposed to mean? The only thing that I can think of is a $2$-fold indefinite integral (although all of your other integrals are definite integrals whose regions of integration you've omitted). – Toby Bartels Dec 13 '19 at 17:01

2 Answers2

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More a long comment than an answer I would strongly recommend the wonderful (and cheap) book Differential-Forms, H. Cartan.

In peculiar the intrinsic notation (without using a basis) is used very often. For instance (adapted from page 28), the pullback expression (or change of variable) of a differential form $\omega:F\to\mathcal{A}_p(F;G)$ takes the form: $$ (\phi^*\omega)(y;\eta_1,\cdots,\eta_p)=\omega(\phi(y);\phi'(y).\eta_1,\cdots,\phi'(y).\eta_p) $$ where $\phi:E\to F$ (pullback defines a form on $E$ from a form on $F$).

(More about Leibniz notation (in differential calcul in general) can be found here)

Picaud Vincent
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  • Just for clarification, the pullback function maps from the dual space back to the vector space? Also when you say U $\subset$ E is U an open subset of E an arbitrary vector space, or do they have special meanings? – Sam Jul 03 '19 at 15:20
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    @SamStiffman $U'$ was the (unfortunate) notation of the book, would be better to have used $\phi:V\to U$. Pullback only takes a form on $U$ to construct a form on $V$ (it has nothing to do with dual space). Hope it helps. – Picaud Vincent Jul 03 '19 at 16:14
  • @SamStiffman I have rewritten the post, you were right, without the book's context my first version was not clear – Picaud Vincent Jul 03 '19 at 16:55
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One notation that I've seen in some physics books is to use an italized $dx$ for a differential and a straight $\textrm{d}x$ for the basis elements of the cotangent bundle. This gives you a little bit of distinction between the two concepts without totally introducing new notation.

Sohom Paul
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    While I don't disagree that this is a different notation I still think it is pretty easy to confuse the 2 notations which can lead to the same issues. – Sam Jul 02 '19 at 20:00