The matching conditions that describe the behavior of electromagnetic fields at an interface between two media are usually obtained from the integral form of Maxwell’s equations. There are pedagogical benefits to augmenting this approach with one that begins with the differential form of Maxwell’s equations and employs unit step functions and the Dirac delta function. With such an approach, it is possible to avoid the use of the rectangles and pillboxes introduced in the standard textbook derivation, make explicit in the matching conditions their dependence on the shape of nonplanar or irregular interfaces, and reveal the connection between the matching conditions and the permittivity‐gradient (∇ε) terms that appear in the wave equations for the electric and magnetic fields.