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:
- Does this notation have a name (and perhaps a Wikipedia article)?
- 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.
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.
$$\int_0^\infty\mathrm dx\int_0^1\mathrm dy\, \mathrm e^{-(x+y)}=\int_0^\infty\int_0^1 \mathrm e^{-(x+y)}\,\mathrm dy\,\mathrm dx\;.$$