Differential Forms without Leibniz notation

Question

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 $$ \int\int f(x,y)dx\wedge dy = \int \int f(x,y)dxdy $$

Answer 1

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)

Answer 2

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.