In my Calculus class, my math teacher said that differentials such as $dx$ are not numbers, and should not be treated as such.
In my physics class, it seems like we treat differentials exactly like numbers, and my physics teacher even said that they are in essence very small numbers.
Can someone give me an explanation which satisfies both classes, or do I just have to accept that the differentials are treated differently in different courses? For example, if the linear density of a solid rod is $d$, in Physics class we would say that the mass of a very small part of the rod $dx$, is $d*dx$, so my physics teacher would say $dm=d*dx$.
P.S. I took Calculus 2 so please try to keep the answers around that level.
P.S.S. Feel free to edit the tags if you think it is appropriate.