4

In almost any calculus book I can think of (for example, "The Calculus with analytic geometry" by Louis Leithold), and even the books of analysis (for example, the Rudin's "Principles of analysis" and "Real and complex analysis"), one can find that an integral is represented with the $dx$ (the differential; actually $x$ here is a dummy variable), written after the integrand. For example: $$ \int f(x)dx$$ However, on reading papers which I would call about "theoretical physics", I have found very often the other way around: the $dx$ or whatever variable being integrated, is written before the integrand. Take for example "Special Relativity induced by Granular Space", by Petr Jizba and Fabio Scardigli, 2013, equation (4): $$w(\zeta ,t_1+t_2)=\int_0^\zeta {\rm d}\zeta' w(\zeta',t_1)w(\zeta - \zeta',t_2)$$ notice the romanization of ${\rm d}$ as opposite of $d$, from the previous example.

I have the feeling that it is not just a matter of style, but there is a meaning involved in the usage of either way.

Question: what is involved in choosing to write either $\int f(x) dx$ as opposite to $\int {\rm d}x f(x)$ ? is this (these) reason(s) of mathematical nature or are they related with the problem that is being dealt with in theoretical physics?

Also, I would appreciate if someone points to a bibliographical reference were such justification for writting it so, is being explained.

Arturo Ortiz Tapia
  • 169
  • 8
    I never thought of it as being anything more than a stylistic choice. I used to be a physicist and I kind of liked putting the dx at the beginning: the operation is integrating with respect to x; only then you specify what it is that you apply this operation to. –  Jul 31 '13 at 18:48
  • It's just style. I too have noticed that physicists prefer to put the $dx$ near the integral sign while mathematicians do not. – Andrew Poelstra Jul 31 '13 at 19:49
  • 1
    This is a duplicate. The question was asked and answered recently. – dfeuer Jul 31 '13 at 19:50
  • 2
    Another page where this question is addressed is http://math.stackexchange.com/questions/388098/integral-dx-term-sanity-check. Putting dx first is a common convention in (certain areas of) physics. – KCd Aug 01 '13 at 06:47
  • I think it is the moment to confess that years ago I was explained by a physicist the difference...but it was in Czech, and I didn't quite grasp it. However, here is what I thought I understood (and I would like a confirmation or a correction): when the dx symbol is after the integrand, it is meant an operation to be done upon the integrand, whereas when it is written before, the idea to convey is that the integration is considered a global property, already present in the whole domain. So, it is not merely stylistic, at least according to that physicist. – Arturo Ortiz Tapia Aug 01 '13 at 16:25

1 Answers1

1

The difference in the position of the $dx$ is entirely stylistic. Further, the romanisation of the d in $\mathrm{d}x$ is not limited to physics & applied mathematics, one often sees it in pure mathematics texts too.

J.J. Green
  • 158
  • 1
    I don't think that the typographical difference is what the OP is asking about... – apnorton Jul 31 '13 at 20:12
  • Roman d is common in Europe, italic d in America. Similar difference is found in $\mathrm{e}^x$ or $e^x$. – GEdgar Aug 01 '13 at 11:10
  • 1
    I think it is the moment to confess that years ago I was explained by a physicist the difference...but it was in Czech, and I didn't quite grasp it. However, here is what I thought I understood (and I would like a confirmation or a correction): when the dx symbol is after the integrand, it is meant an operation to be done upon the integrand, whereas when it is written before, the idea to convey is that the integration is considered a global property, already present in the whole domain. So, it is not merely stylistic, at least according to that physicist. – Arturo Ortiz Tapia Aug 01 '13 at 18:09