Is $n\sin\theta$ conserved along light path?

Abstract

Snell's law states that the quantity $n\sin\theta$ is unchanged in refraction of light passing from one medium to another. We inquire whether this is true in the general case where the speed of light varies continuously within a medium. It turns out to be an instructive exercise in application of Snell's law and Fermat's principle. It also provides good pedagogical problems in calculus of variations to deal with the subtleties of a variable domain of integration and inclusion of constraints. The final result of these exercises is that, contrary to an initial expectation, the answer to the question in the title is negative.