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I often see the differentials of the electric field strength and the acceleration due to gravity being written as: $$dE= \mathcal{k}\frac{dQ}{r^2} \tag{1}$$ and $$dg=\frac{GdM}{r^2} \tag{2}$$ respectively.

Which I interpret as "the infinitesimally small field strength due to an infinitesimally small charge/mass dQ/dM at some distance r away, and they are used when we are calculating the sum of the field strength at the origin due to these small charges/masses each at some distance r away from the origin"

But there are cases in which the term involving the radial coordinate differential also comes in: $$dE=\mathcal{k}\frac{dQ}{r^2}+ (-2)k\frac{Q}{r^3}dr\tag{3}$$

$$dg=\frac{GdM}{r^2}+(-2)\frac{GM}{r^3}dr \tag{4}$$

I understand that (3) & (4) come directly from the product rule, but what is the physical interpretation of the additional dr terms?

(I understand that the differential of a quantity is non-zero when it is not a constant, e.g. the radial distance of each small charge/mass would vary along a long rod pointing normally away from the origin, but isn't this already included in the dQ/dM term because you shrink the charge/mass source to so small that the radial distance is almost constant, and then you integrate over the entire length of the rod to obtain the total field strength?)

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The context of (1) and (2) is different from that of (3) and (4).

Although (1) and (2) are written as differential, they are not "differences": in (1) $d\vec E$is the field at some given $\vec r_p$ created by a small amount of charge $dq$ located at $\vec r_s$, with $r=\vert \vec r_p-\vec r_s\vert$. (2) has a similar interpretation.

On the other hand, given a vector field $\vec E$ everywhere near point $\vec r_p$, the differential $d\vec E$ given in your (3) is the (linear part of the) change in the field $\vec E$ near $\vec r_p$. This changes may come because the source charges $q$ creating the fields are changed by a small amount amount $dq$, or the distance from the source charges to the point $\vec r_p$ is changed by a small amount.

Please note that your equations should really have vector signs since the fields are vectors. If you do this then the change in the source charge $dq$ is a scalar but the change $d\vec r$ in the distance is actually a vector.

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