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I have heard or read that the reason why in calculus dx is seen so often typed as $\mathrm d x$ is because $\mathrm d$ is an operator or a symbol, while $x$ is in cursive because it is a variable. This seems to be the understanding also in this post.

But I'd like to ask for a bit more background about what type of operator $\mathbb d$ is (is it a function, for example?)- I know that in Riemman integrals it serves the purpose of partitioning the $x$ domain into infinitesimally thin slices, but there has to be more to it...

As for the symbol explanation, aren't we always dealing with symbols? Isn't $x$ just as much of a symbol as $\mathrm d$?

Incidentally, I see that there is a question that reads identical to this one, but it refers to a specific situation.

Thanks for the comments - still not clear on how to splice them all... So let me rephrase the question: Is it like this

$$\left. \begin{array}{l} \text{if $d$ is the exterior derivative operator:}&\mathrm dx\\ \text{if $d$ is part of the symbol of an element of }V^*\text{:}&dx \end{array} \right\} $$

?

Here's some informative interaction on the sister community tex.se:

enter image description here

Antoni Parellada
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    It's just a convention. Don't read too much into it. – AccidentalFourierTransform Mar 26 '18 at 21:26
  • For me, $\mathrm{d}$ is simply a convention that is designed to avoid confusion when one really wants to save $d$ for other meanings. Similarly I see a trend of using $\mathrm{e}$ for the natural constant, $\mathrm{i}$ for the imaginary unit, etc. – Sangchul Lee Mar 26 '18 at 21:30
  • Actually $dx$ is a single symbol that represents an element in the dual space. – egreg Mar 26 '18 at 21:38
  • @egreg Are you implying that seasoned mathematicians go through the extra $\LaTeX$ pain with the sole purpose to differentiate $\mathrm dx$ from the basis of $V^*,$ i.e. $dx^i$? – Antoni Parellada Mar 26 '18 at 22:25
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    @justquestions I'm a seasoned mathematician as well as a seasoned $\mathrm{\LaTeX}$ user and never use other than an italic “d” for this purpose. – egreg Mar 26 '18 at 22:33
  • @egreg Totally... I read your profile, and was trying to get more elaboration from you - perhaps a formal answer? – Antoni Parellada Mar 26 '18 at 22:35
  • $d$ is of course an operator, which is the exterior derivative –  Mar 26 '18 at 23:53
  • @egreg: Fortunately, the cotangent vector field $dx$ happens to be the result of applying the exterior derivative $d$ to the scalar field $x$, so we can interpret it either way. –  Mar 26 '18 at 23:56
  • @egreg Could you clarify what you meant by "this purpose" on your prior comment - do you use italics for both purposes (element in the dual space and differential), of just for the dual space? – Antoni Parellada Mar 27 '18 at 12:23
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    Hopefully no-one would write $d$ and $\mathrm{d}$ to mean different things - I wouldn't like to spend all my time peering at slightly different fonts, trying to distinguish them. I think the choice between them is just a matter of taste. – Joppy Mar 27 '18 at 12:49
  • @justquestions For the purpose of integrals – egreg Mar 27 '18 at 14:05
  • For me it an operator that only makes sense in the context of $${\rm d}\square = \sum_i \frac{\partial \square}{\partial x_i} {\rm d}x_i$$ the common notation $\frac{{\rm d}y}{{\rm d}x}$ is somewhat of a misnomer. – John Alexiou Mar 27 '18 at 14:22

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