Much has been said of the $dx$ notation used for integration on this site, but some writers of mathematics papers (especially physicists), write integrals as

$$ \int dxf(x) $$

For instance, one way of writing the law of total probability I have seen is

$$ f_Y(y) = \int_{-\infty}^{\infty}dxf_{X|Y}(x|y)f_Y(y) $$

What ideas are the authors trying to emphasize through this notation?

  • $\begingroup$ I do not think the other questions really answer what mathematical idea the authors are trying to convey (rather than just convenience for iterated integrals). It could be that the integral is explicitly being expressed an operator? $\endgroup$ – jII Dec 27 '14 at 6:40
  • $\begingroup$ This answer on one of the duplicates suggests that putting the differential first makes it easier to see what variable is being integrated over, even for a single, non-iterated integral. On another duplicate, which seems very similar in spirit to yours, the commentators write that it's purely a stylistic choice (OP there recalls hearing a physicist insist that the placement had implications on meaning, but his reasons sound like nonsense to me). I'm not quite sure what more there is to say on this. $\endgroup$ – epimorphic Dec 27 '14 at 7:05
  • $\begingroup$ I guess - depending on the application - physicists prefer keeping the differential adjacent to the integral sign as it stresses the operator nature of integration. $\endgroup$ – Johannes Dec 27 '14 at 7:50
  • $\begingroup$ Reading everything again, I think my previous comment was rather rash, especially my dismissive use of "nonsense" with regards to the unnamed physicist. Note however that the accepted answer to the parent question offers a few sentences on the operator idea: "It also just occurred to me that the second notation ties in better with the syntax of an operator..." $\endgroup$ – epimorphic Dec 27 '14 at 9:08