[laingc]

> but those operations over what usually appears at the denominator of the derivative purely for notation purpose has always looked to me as complete magical garbage.

You will be delighted to discover that they are in fact not magical or garbage.

> What object is "dx" ? is it a number ? a limit ? is it zero ? can i divide another number by it ?

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.

You can use differential forms to do all sorts of things, but one example you may be familiar with is to compute area or volume forms over arbitrary manifolds. It gets a bit hard to define things on HN without TeX support, but using differential one-forms and the related exterior derivative, you can define a generalised Stokes' theorem that works for any smooth, oriented manifold.

I used this in my PhD, and implemented it directly in a numerical method, so this has very practical engineering uses also.

"Elementary Differential Geometry" by Barrett O'Neill is a pretty beginner-friendly introduction to some of these topics if you're interested, though there are many other good texts also.

[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.

[bikenaga]

dx can be a differential form, but in (elementary) calculus books it's exposited this way: Suppose you have a function y = f(x) taking real numbers to real numbers, so x is an independent variable, y is the dependent variable. You define an independent variable Δx and define dx = Δx. Also define dependent variables dy and Δy by

    Δy = f(x + Δx) - f(x)  and  dy = f'(x) dx.

Then f'(x) = dy/dx. This may look like a stupid hack to make the last formula work, but actually it's a little more. If you use nonstandard analysis, you define the derivative of a function f from reals to reals by

  f'(a) = st( (f(a + Δx) - f(a)) / Δx )

where st takes the standard part of a hyperreal number and Δx is a nonzero infinitesimal. This is like the usual limit definition, without limits. Then you can use the formulas above and "dy" and "dx" are numbers, albeit hyperreal numbers.

(The "dx as a differential form" vs. "dx as a number" is probably coming from the fact that the tangent space to the reals at a real number is isomorphic to the reals, so the dual space [where dx lives] is too.)

(Calculus via infinitesimals is pretty cool; a good resource for this is H. Jerome Keisler's "Elementary Calculus" and "Foundations of Infinitesimal Calculus", both available for free: https://people.math.wisc.edu/~hkeisler/)

I second the recommendation for Barrett O'Neill's book - I used it in my differential geometry class at MIT.