# This multiple integral notation, has it got a name? $\int dx \int dy \, f(y,x)$

I've encountered, on Wikipedia (examples below), an integration notation which seems to be prefix-style: the integral sign is immediately followed by the $\mathrm dx$ (or $\mathrm dy$, or what have you), and this is followed by the function to be integrated. Multiple integration is done by multiple prefixes.

I have two questions:

1. Does this notation have a name (and perhaps a Wikipedia article)?
2. In this prefix notation, are the integrals evaluated left-to-right, or inner-to-outer?

First place I've encountered the notation: Wikipedia on multiple integration. Most relevant bit:

If the domain D is normal with respect to the x-axis, and is a continuous function; then α(x) and β(x) (defined on the interval [a, b]) are the two functions that determine D. Then: $$\iint_D f(x,y)\ dx\, dy = \int \limits_a^b dx \int \limits_{ \alpha (x)}^{ \beta (x)} f(x,y)\, dy.$$

Second place I've encountered the notation: Wikipedia on integration by parts. Most relevant bit:

Consider the iterated integral: $$\int_a^z \mathrm dx\ \int_a^x \mathrm dy \, h(y).$$ In the order written above, the strip of width d is integrated first over the y-direction (a strip of width dx in the x direction is integrated with respect to the y variable across the y direction) as shown in the left panel of the figure, which is inconvenient especially when function h(y) is not easily integrated.

• I don't know if it has a special name, but this prefix notation is quite common among physicists (from my experience mathematicians tend to use the "standard" suffix notation). – Bill Cook Apr 4 '12 at 20:22
• This notation would be horribly confusing to most mathematicians I think. – Keaton Apr 4 '12 at 20:26
• I think of it as "physicist's notation". – Michael Lugo Apr 5 '12 at 2:30
• @Keaton: All it would need is 5 minutes of getting used to, I think. I positively hated the notation when I first encountered it on Wikipedia yesterday afternoon, but then I realised the notation was fine. The problem was the fact that the original contributors sweetly neglected to, you know, describe their notation, let alone define it. Grrr. Anyway, off to add the missing descriptions. Thanks, Bill, Micheal, and Joriki! – Esteis Apr 5 '12 at 8:21

As Bill wrote, this is quite usual in physics. You'll even find things like

$$\int\frac{\mathrm dx}{x-a}\;.$$

I think this treatment of $\mathrm dx$ as if it were a factor and not just a notational device corresponds to a stronger tendency to think of calculus as dealing with infinitesimal quantities.

I don't know of a name for this notation.

Regarding your second question, I'm not sure what it would mean to evaluate these integrals left to right – they're evaluated exactly as if the differentials were at the end, e.g.

$$\int_0^\infty\mathrm dx\int_0^1\mathrm dy\, \mathrm e^{-(x+y)}=\int_0^\infty\int_0^1 \mathrm e^{-(x+y)}\,\mathrm dy\,\mathrm dx\;.$$

• Super, thanks. Corroborated by Bill and Micheal above, too. I'll add a note to Wikipedia's Integral#Terminology_and_notation mentioning this notation. Merci! – Esteis Apr 5 '12 at 8:22
• I understand the first notation of $\frac{dx}{x-a}$. In fact, I use this frequently when teaching calculus students because there is no real harm. My comment was just that the intent of what you are doing is much clearer in the right hand side of Joriki's last equation than the left hand side, which looks really odd to me. – Keaton Apr 5 '12 at 17:34
• Hmm...I did see such prefix notations in Rudin's book, e.g. Real and complex analysis 3rd, P165. – newbie Apr 15 '13 at 16:17
• Following the last comment, I guess it is used on purpose to distinguish double integrals and iterated integrals, under that circumstance. – newbie Apr 15 '13 at 16:23