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?)


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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