Wikipedia:Reference desk/Archives/Mathematics/2017 June 27

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June 27

Notation for a differential

In mathematics, letter-based symbols that have the same, standard meaning in many different contexts (e.g. sin, log) are usually typeset in "roman" style, while variables whose meanings are defined in the context are typically typeset in italics. From a philosophical standpoint, would it be more "correct" to typeset the "d" in a differential (in calculus) in roman style or italic style? Put differently, is it more correct to typeset a derivative as ${\frac {dy}{dx}}$ or ${\frac {\mathrm {d} y}{\mathrm {d} x}}$? --134.242.92.97 (talk) 23:39, 27 June 2017 (UTC)

By the reasoning in your first sentence, "d" should not be in italics. Loraof (talk) 02:06, 28 June 2017 (UTC)

It should be italicized. --Deacon Vorbis (talk) 05:51, 28 June 2017 (UTC)

My preference is for both the d and the variable (x, y etc.) to be written with the same typeface. Dolphin (t) 06:05, 28 June 2017 (UTC)

I agree, preferentially use italics, ISO notwithstanding. There has been a sort of movement to use the upright d, but it has not really caught on, at least in the mathematics community. I think some physicists or engineers may use it more.

As for the "philosophical" aspect, I'm not too convinced by the distinction of using roman for symbols with a "fixed meaning", and even if I were, I'm not sure that it applies to the d, which doesn't have a clear and uniform "meaning" in and of itself. I suppose it's true that it's not a variable, and it might appeal to the tidy-minded to make such a typographical distinction, but it is not really standard. --Trovatore (talk) 06:15, 28 June 2017 (UTC)

• That is the kind of things that get LaTeX people to argue to death, if you care for such things. Tigraan^{Click here to contact me} 07:30, 28 June 2017 (UTC)

My preference is upright 'd', but Wolfram Mathworld uses italics: http://mathworld.wolfram.com/Derivative.html --CiaPan (talk) 07:49, 28 June 2017 (UTC)

I have a low opinion of MathWorld, but I agree with them in this case. Italics is more standard in mathematics. --Trovatore (talk) 09:25, 28 June 2017 (UTC)

Italics are more standard, although I sympathise somewhat with the arguments for the upright d (and similarly for the upright e and i when they denote constants). Double sharp (talk) 10:53, 28 June 2017 (UTC)

It should be typeset in italics. And you must always use emacs when typesetting it. Anyone who typesets it upright is an unwashed heathen who is not fit to have an opinion on the matter; the same goes for anyone that prefers vi to emacs. Sławomir Biały (talk) 11:45, 28 June 2017 (UTC)

Go revert me then. --CiaPan (talk) 13:22, 28 June 2017 (UTC)

I thought differentials as in differentiable manifolds normally used roman, but looking at Exterior derivative I see it was changed from roman to italic with this edit [1] which just says 'fmt', and a quick look on the web shows that a number of people do use italic nowadays for the exterior deriviative. I guess it would really have needed a \d operator in LaTeX rather that \mathrm{d} or what really should be used which is \operatorname{d} to take off and mathematicians are lazy ;-) Dmcq (talk) 12:40, 28 June 2017 (UTC)

Almost everyone uses italics in mathematics for the exterior derivative. Sławomir Biały (talk) 12:52, 28 June 2017 (UTC)

I had a quick look and the italic d is certainly more common, but six of the first sixteen I looked at via Google Books used upright d. It seems more common when bolded with vectors in bold. And others are definitely for the sloping form even when they have lots of upright Greek letters. Dmcq (talk) 14:03, 28 June 2017 (UTC)

I write $d_{x}$ to avoid confusion with the product $dx$. The rules for differentiating constants, sums, products and powers are then written:

$d_{k}=0$

$d_{\sum x_{i}}=\sum d_{x_{i}}$

$d_{\prod x_{i}}=(\prod x_{i})\sum x_{i}^{-1}d_{x_{i}}$

$d_{x^{y}}=x^{y}(yx^{-1}d_{x}+\ln(x)d_{y})$

Bo Jacoby (talk) 07:57, 30 June 2017 (UTC).

Of course you can notate things however you want, for your personal use. It looks very strange to me, but if you find it useful, there's no basis for anyone else to object.

If you want to publish anything using this notation, though, I predict you're going to find it tough going. It seems to be a solution in search of a problem, because in practice it's exceedingly rare that anyone (except maybe beginning calculus students) has any trouble distinguishing dx from d times x. And it's not exactly a zero-cost solution, given that it requires cramming complicated formulas into subscripts. --Trovatore (talk) 09:44, 30 June 2017 (UTC)

It should not look strange to you, because you are accustomed to use indexes, $(v_{x},v_{y},v_{z})$, for naming variables, and to use juxtaposition, $ax+by+cz$, for products. Young students are numerous and important and should not be put in parentheses while old mathematicians are unlikely to appreciate new ideas anyway. Cramming formulas into subscripts is what I did above. Bo Jacoby (talk) 19:00, 30 June 2017 (UTC).

Juxtaposition is used for all sorts of things. Multiplication is just one of them. Mathematics is not a formal system; it's a human activity, and humans are pretty good at disambiguating from context.

Everyone is a beginning calculus student once; their importance is not the issue. The point is that this particular failure to disambiguate dx from d times x is a typical beginner's error that is easily overcome. It is not necessary to structure the system in a way that such errors can't happen.

You did indeed cram formulas into subscripts, and while the result was sort of pretty in a way, it was not easy to read. That's why I said your solution comes with a cost. But go ahead; see if you can get people to adopt it. I don't foresee your efforts being richly rewarded, but as they say in sports, that's why we play the game. --Trovatore (talk) 19:45, 30 June 2017 (UTC)

The cost of bad notation should not be underestimated. Some students never realize that dx is not a product. It is valuable to structure the system in a way that such errors can't happen. It is subjective whether it is easy to read or not. Learn it before you judge. It is not difficult to change people's thought, but it takes time. Bo Jacoby (talk) 20:37, 30 June 2017 (UTC).

The ones who never realize it's not a product, I think, aren't ever really going to grok calculus, no matter how you teach it. I disagree that it's bad notation. It works just fine.

I don't think it's subjective. The problem with ramified subscripts and superscripts is that the meaning depends on fussy positioning and sizing. It's awkward and time-consuming to parse (and also to create).

But as I say, go for it. Don't let me stop you. --Trovatore (talk) 20:47, 30 June 2017 (UTC)

I would use a P instead of a d. Then you've written down momentum rules for Hamiltonian potentials, and the notation is also correct (or at least, consistent with someone else's notation), for the standard symplectic 2-form $dp_{x}\wedge dx+dp_{y}\wedge dy$. I'm not sure many students would find that interpretation very clarifying though. Sławomir Biały (talk) 20:45, 30 June 2017 (UTC)

Interesting stuff, Sławomir! Are you saying that the formulas

$P_{k}=0$

$P_{\sum x_{i}}=\sum P_{x_{i}}$

$P_{\prod x_{i}}=(\prod x_{i})\sum x_{i}^{-1}P_{x_{i}}$

$P_{x^{y}}=x^{y}(yx^{-1}P_{x}+\ln(x)P_{y})$

make sense as momentum rules for Hamiltonian potentials? Bo Jacoby (talk) 07:21, 1 July 2017 (UTC).

The notation $d_{x}$ is typically used to denote the distance function of the metric space x when I write proofs involving multiple metric spaces. I don't see the usefulness of this with regard to reducing confusion with respect to product notation, as after all, I myself confused the notation $f(x)$ for the product of f with x a decade or two ago.

There's nothing confusing about Leibniz notation, not withstanding your previous abuse of that notation (one cannot treat differentials as formal variables because they cannot be treated as ordinary numbers (other than using nonstandard analysis)). My question to you, now, is how you would adapt your notation for partial differentiation (including of high order), and integrals (where they appear as abuse of notation), and for compatibility with Einstein summation notation.

Anyone who does not understand the statement "Denote by ${\frac {dy}{dx}}$ the derivative of y as a function of x" is not ready to study calculus. Even the nonrigorous presentation I received in primary and secondary school confused no-one with Leibniz notation.--Jasper Deng (talk) 09:05, 1 July 2017 (UTC)

Learn that ax is a product, and so is bx and cx, but dx is not. These professors are crazy! Bo Jacoby (talk) 13:26, 2 July 2017 (UTC).

Whereas you would prefer that a₇ is an element of a sequence but that d₇ = 0? --JBL (talk) 20:52, 2 July 2017 (UTC)

Touché! a7=7a, d7=0. Bo Jacoby (talk) 06:40, 3 July 2017 (UTC).

Well, actually there is a way of treating differentials where, for each independent variable x, y, ..., you introduce a corresponding new independent formal variable dx, dy, .... Then for the dependent variables (let's say z), you simply define by fiat dz to be a linear combination of dx and dy with the coefficients given by the partial derivatives.

So then dx could be an "ordinary number", although it doesn't matter what number it is, as long as it's not zero.

I have never found this to be a very useful way of thinking about differentials, but it is attested in some textbooks. My guess is that it's mostly useful for making some sort of sense about formulas involving differentials without resorting to infinitesimals. I don't really like it, because the new variables aren't really used for anything, but it should be acknowledged that this interpretation exists. --Trovatore (talk) 10:19, 1 July 2017 (UTC)

It is not true that the variables dx, dy, and so forth, are not used for anything. These are linear increments in the tangent space, giving rise to the linear approximation. Indeed, thus is the entire basis for regarding the differential of a function as a linear functional on the tangent space. Sławomir Biały (talk) 20:58, 3 July 2017 (UTC)