From reading answers here, I've noticed that some people write integrals as $\int dx \; f(x)$, while other people write them as $\int f(x)\;dx$.
I realize that there is no mathematical difference between the two notation forms, but was wondering why some people choose the first method over the second. Is there some place in higher maths that it becomes beneficial to write the differential first?
(I, personally, have always used the second method, just because I was taught that way…)
Best Answer
When you have a lot of integrals, particularly with limits, it can be very helpful at times to be able to tell at a glance which integral is over which variable.
$$\int_0^1 \int_2^3 f(x,y) \; \mathrm d x \mathrm d y$$
This is not particularly readable or clear, especially when $f$ is lengthy and there are more nested integrals etc. I could also imagine it being misinterpreted.
By contrast,
$$\int_0^1\mathrm d y \int_2^3 \mathrm d x \; f(x,y)$$
makes it very clear what is going on. The only price you pay is possible ambiguity about where the integral ends, but this is easier to make clear with formatting and less of an issue anyway.
Edit: It also just occurred to me that the second notation ties in better with the syntax of an operator. That is, if one thinks of $\int_0^1 \mathrm d x$ as being an operator, taking a function to its integral, it's more natural to have the whole operator together in one lump. Think of how one changes $$\frac {\partial f}{\partial x}\to \frac{\partial}{\partial x} f$$