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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?

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You must always consider the specific notation definitions which an author may use. There is not a unique use of Δm. The notation dm is closer to being a universal notation for an infinitesimal m element, although m might be defined as something other than mass. Pay attention to context and book definitions.

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The way they use the $dm$ (or $dU$ or $dF$) notations is to represent an infinitesimal element.

You seem to be getting it confused with rates of change, given by $\frac{dm}{dt}$ for example.

When it is a ratio, then you are determining the change in $m$ as a function of $t$. $dm$ still represents an infinitesimal mass, but $\frac{dm}{dt}$ can be used to find the change in mass relative to time at any time (t). The two infinitesimals themselves don't represent a change, but by comparing the ratio of the two infinitesimals as a parameter changes, you can determine the rate of change with respect to another variable.

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