Hi. I've finished my undergraduate math methods courses. Many times we had problems where we had a summation and an integral both acting on the same term, and we'd switch the order of the two operations without thinking about it. The professor would always say, "I can interchange these two because I am a physicist and I am lazy. A mathematician would spend his whole life trying to prove this is permissible." The same goes for "differentiating under the integral," which is what I'm really concerned about. I know that there are times when it's perfectly acceptable to slip that partial differentiation right in under the integral, but I've also come across integrals where it's absolutely not permitted, and it gives you wonky, nonsensical results. So here is my question. Does anyone have any tricks or rules of thumb, maybe not for always knowing when these things are allowed, but knowing when they are unquestionably allowed. Is there ever a time you can look at such a thing, and say, "OK, I can absolutely interchange these without negative consequence?" I don't want anything too formally mathematical. Just any of your intuitive sense on the topic would be greatly appreciated.