[zero-sharp]

>dx is a differential one-form. You can think of it as a generalisation of a gradient, if you like. These are very important in Differential Geometry.

This really doesn't help beginners. At all.

There are formal contexts where we can reinterpret division by zero and have it make sense. Should I start telling students that division by zero is allowed? Should I start teaching intro calculus students that 1+2+3+...=-1/12?

[lifthrasiir]

For teaching purposes you are definitely allowed to lie, as long as that lie can be resolved eventually (not necessarily in this semester ;-). That's how we have been generally taught about integer divisions and negative square roots. But behind the scene, the `dx` notation can be fully generalized and made rigorous with differential forms, or that was what I have been told.

[zero-sharp]

This is definitely not an apples to apples comparison. Integer division is something everybody is expected to learn. Also we don't teach imaginary numbers to middle schoolers as soon as they learn about square roots.

To some extent we have to speak to our audience. I consider that part of effective communication. I don't think "assume the person you're speaking to is/will be a mathematician" is an effective way to interact.

[lifthrasiir]

I meant that, yes you are right. You are not expected to teach differential forms to non-math students at all because it's not effective. The existence of differential forms only means that it can be eventually made rigorous if you push hard.

[laingc]

I replied with what the thing is called, explained what it can be used for, and recommended an introductory text to learn more.

If you can come up with a more helpful reply in as many words, then please do so.