The coordinate-transformation-based differential method of Chandezon et al. [J. Opt. (Paris) 11, 235 (1980); J. Opt. Soc. Am. 72, 839 (1982)] (the C method) is one of the simplest and most versatile methods for modeling surface-relief gratings. However, to date it has been used by only a small number of people, probably because, traditionally, elementary tensor theory is used to formulate the method. We reformulate the C method without using any knowledge of tensor, thus, we hope, making the C method more accessible to optical engineers.