Relationship Between a Vector Differential and the Corresponding Scalar Differential in Physics

In physics, a differential is an infinitesimal change in or amount of some quantity. For example, [Formula: see text] is a small change in linear momentum, and [Formula: see text] is a small amount of mass. Ratios of differentials become derivatives, while Riemann sums of differentials become integrals. Given some vector quantity X, what is the relationship between [Formula: see text] and [Formula: see text] according to the standard conventions of introductory physics? Surprisingly, there are two distinct answers, depending on exactly what quantity X happens to be. The distinction is illustrated here with specific examples. After discussing this ambiguity in some detail, some recommendations to physics instructors and textbook authors are preferred. Although not everyone will agree with these conclusions and suggestions, this article provides a starting point for further deliberations.