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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.

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    @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!2012-04-05

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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.

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

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    Following the last comment, I guess it is used on purpose to distinguish double integrals and iterated integrals, under that circumstance.2013-04-15