# Notation of the differential operator

I see the differential operator both with upright and italic d in different books/articles. So I'm curious about $$\int x^2 \, dx \quad \text{vs.} \quad \int x^2\, \mathrm{d}x,$$ and $$\frac{d}{dx}f(x) \quad \text{vs.} \quad \frac{\mathrm{d}}{\mathrm{d}x}f(x).$$

Is there a convention which of these notations should be used?

• As far as I can see, there's no difference between the two. Its difficult to write in italics by hand though-only in textbooks. – John_dydx Jul 22 '15 at 14:32
• I only used $d$ because I didn't know about the other one. Having seen it - it's quite long, as in to type! – Alec Teal Jul 22 '15 at 15:50
• Yes, there is a convention. In fact, there are two :) – Hagen von Eitzen Jul 22 '15 at 20:14
• – Ben Crowell Jul 22 '15 at 22:23
• – user940 Jul 23 '15 at 1:09

Quick answer: there is a standard to follow.

Longer answer: while physicists write differential operators in upright fonts (because they follow the standards), mathematicians tend to typeset differential operators as variables (because we are lazy). I am joking, but it should be clear that $dx$ is not $d \cdot x$, and that $d$ is essentially an operator: therefore it is always preferable to write $\mathrm{d}x$ instead. Many journals change $dx$ to $\mathrm{d}x$ after accepting a manuscript for publications.

• Just to point out that this is by no means a universally accepted standard. (I give examples in my post below.) – Simon S Jul 22 '15 at 16:48
• @SimonS No, $dx$ or $d\mu$ cannot be distinguished from the multiplication of two variables. This is why we use $\lim$ instead of $lim$, or $\sin \alpha$ instead of $sin\, \alpha$. – Siminore Jul 23 '15 at 12:20
• If we really thought we interpret without looking at context, I would be on your side. But to give an example from the OP: there is only one coherent interpretation of $$\int x^2 \ dx$$ and it doesn't involve the possibility that $d$ is a variable. The editors of some of the best mathematics journals of the world appear to agree. – Simon S Jul 23 '15 at 12:30
• @SimonS And that's why we have standards ;-) To avoid such discussions... – Siminore Jul 23 '15 at 13:36
• Fair enough. Let me try a perhaps clearer analogy: there is only one reasonable interpretation of "For a circle, $A = \prod r^2$." Do you find this acceptable typesetting? It is a precisely analogous error of confusing a variable with an operator. – David Zhang Jul 24 '15 at 22:38

This is more of an extended comment than an answer, but with regards to typesetting in $\LaTeX$, let me point out that typing \mathrm{d} should not take longer than typing d, as you shouldn't be doing either throughout your paper!

The semantically correct thing to do is to define a macro representing your desired differential operator, for example, \newcommand{\dv}{\frac{\mathrm{d}}{\mathrm{d} #1}}. Then, whenever you need to typeset $\frac{\mathrm{d}}{\mathrm{d}x}$ or $\frac{\mathrm{d}}{\mathrm{d}t}$, you simply use \dv{x} or \dv{t}, which is even easier than the non-standards-conforming \frac{d}{dx}.

This has the added benefit that, should you find yourself submitting to a publisher whose house style requires italic $d$'s, you need only change the definition of your macro instead of modifying every differential operator by hand.

Many excellent journals and books use $d$ in the italics form, such as the Journal of the American Mathematical Society (e.g., recent article by Terence Tao), London Mathematical Society Proceedings (e.g., equations 74 and 75 of this recent paper) and Spivak's Calculus.

Given that reference quality publications use $d$--and that it is faster and cleaner to typeset it that way--the italics form is more than respectable.

There is no difference between them. It merely comes as a result of a choice in LaTeX formatting; specifically, some people write "\text{d}" (or some equivalent) for the upright formatting, but many other people don't do this for the sake of speed, and instead just write "d".

• Writing \text{d} is not a good idea, as the shape and font used then depends on the shape and font surrounding the formula (if that's italic, the d will be italic as well). Use \mathrm{d}. – JohnB Jul 23 '15 at 7:45

In Germany, there is the DIN 1338 standard, according to which the differntial operator d, as, e.g., e for the Euler number, should be typeset as an upright letter.

According to Wikipedia, these letters are typeset in italic if AMS conventions are used.

By logic I think that is a good idea differentiate the symbol with the roman notation but there isnt a "standard", you can use any of them.

In the same sense doesnt exist any kind of "standard" mathematical notation. I read a lot of books of many mathematical topics, everyone with different notations, not only just the infinitesimal symbol.

The problem, writing on LATEX, is that it take more time to write $\color{red}{\mathrm d}$ instead of just $\color{red}{d}$.

• \newcommand{\d}{\mathrm{d}} – JohnB Jul 23 '15 at 7:45