This question is more about the mathematical notation and little about physics.
I have two books, Tipler and Halliday, when studying the derivation of Maxwell's equations they use the same example but with slightly different notations.
Both authors start from the same place, an imaginary rectangle placed in a region in the space where there is an electrical field. Please see the images to avoid any confusion with the symbols the authors use in the problem.
By applying Faraday's law: $\epsilon = \oint \vec {E} \cdot \vec {ds}$ (counterclockwise direction).
Halliday arrives in the following:
$$ \epsilon = \oint \vec {E} \cdot \vec {ds} = (E + dE)h - Eh = h \hspace 2mm dE$$
While Tipler describes the same as:
$$ \epsilon = \oint \vec {E} \cdot \vec {ds} = (E + \frac{\delta E}{\delta x}\Delta x) \Delta y - E\Delta y = \frac{\delta E}{\delta x} \Delta x \Delta y $$
The part that is confusing me is the one that Halliday uses $dE$ but Tipler uses $\frac{\delta E}{\delta x} \Delta x$.
At first Halliday's notation seemed to me more intuitive, but after reading Tipler I'm not sure I understand how to read either of them at all.
Can someone describe how should I interpret both ways of describing the same result?