I am currently reading "No-Nonsense Quantum Field Theory" by Jakob Schwichtenberg and in several derivations in the book it'll pull a partial derivative operator out of an integral, or change the order of full and partial derivatives on a function, etc. without explanation. So my question is: Are the differential and integral operators commutative? Is this always true? And if not, what are the conditions for it? Furthermore, in the more general sense of abstract algebra, if/when are unary operators commutative? A proof to accompany a given answer would be much appreciated, or at least the citation of one which is free and public.
It's certainly intuitive that this is true, at least in the special case of the calculus and physics in this book, but I'm not sure why it is in a more fundamental, mathematical sense and I'm also curious about the general case of all unary operators. This is the kind of thing I would love to try to solve and research on my own but I simply don't have the time right now and I'm having trouble finding a straightforward answer online, so any help would be greatly appreciated. Thank you!