I would assume that $\Delta m$ would mean a finite change in mass and $dm$ to be an infinitesimal change in mass. But many physics textbooks use it to denote a small but finite mass and an infinitesimal mass element respectively while writing a differential equation. Why? And how does this make sense?
Similarly, $dU$, which should mean the infinitesimal change in the potential energy function is treated as the "small" potential energy of mass $dm$. Same for $dF$ etc.
One e.g would be to look at the derivation of the gravitational potential energy of a point mass and a shell. The potential energy of the ring is changed from $U_i$ to $dU$ and the mass of the ring is taken as $dM$. Doesn't $d$ mean an infinitesimal change in some quantity?