In the section Cartesian derivation, in this wiki page, they derive and define curl in an different way.
I was always told that a differential is an extremely small change in a variable and that the derivative operator, $\frac{\mathrm{d}}{\mathrm{d}x}$ for instance, is merely a notation to describe how a variable changes as another independent variable changes.
In the article, it is described that the derivative of a function can be denoted as $$\begin{gather*} \frac{\mathrm{d}}{\mathrm{d}x}f(x) = \frac{f(x + \mathrm{d}x) - f(x)}{\mathrm{d}x} \end{gather*},$$ which is very similar to $$\begin{gather*} \frac{\mathrm{d}}{\mathrm{d}x}f(x) = \lim_{h\rightarrow0}\frac{f(x + h) - f(x)}{h} \end{gather*}.$$
When I took ordinary differential equations I was always confused about solving separable DEs with the method of multiplying on both sides by the differentials.
What I really do not understand is if the wikipedia definition is correct. If so, how can I prove it besides thinking intuitively about the differential? When can I treat a differential as a normal variable? Can differentials be added and subtracted from “normal numbers”?