Section 4.1 Why use \(d\rr\text{?}\)
At the heart of our geometric approach to vector calculus [15] is the use of the vector differential \(d\rr\text{.}\) Why is \(d\rr\) so important?
First and foremost, \(d\rr\) is geometric. It captures the fundamental notion of “small directional change” without reference to any coordinate system or basis vectors.
Students first encounter \(d\rr\) in the context of vector line integrals, such as when deterniming the work done along a given path. The traditional approach to this topic in mathematics [20] is to start with ordinary single integrals, generalize to (scalar) line integrals with respect to arclength, then introduce work as the integral of the scalar quantity \(\FF\cdot\TT\text{,}\) where \(\TT\) is the unit tangent vector. The vector differential is typically mentioned only in passing, as alternate notation for \(\TT\,ds\text{.}\)
This approach of course represents good mathematics. All curves can be parameterized by arclength, the so-called unit-speed parametization, simplifying a general analysis of curves by, for instance, eliminating degenerate parameterizations involving changes in direction. However, this approach is rarely straightforward in practice; determining arclength explicitly is only feasible in simple (or contrived) cases.
Yes, of course, as a vector quantity \(d\rr\) has direction (\(\TT\)) and magnitude (\(ds\)), but separating out these two geometric pieces requires inserting a normalization factor in both cases, which is often a messy square root. But the normalization cancels when “recombining” \(\TT\) and \(ds\) to form \(d\rr\text{!}\) Why not simply use \(d\rr\) from the beginning?
This apparently minor shift in philosophy turns out to have major implications. First of all, it suggests introducing vector line integrals first, then treating scalar line integrals as the derived concept, using \(ds=|d\rr|\text{.}\) Perhaps most important, it opens the door to our Use what you know strategy [15], in which formal parameterization is just one of many ways of describing a curve. And the geometric nature of \(d\rr\) allows for immediate generalization to other coordinate systems [15]. Furthermore, the use of \(d\rr\) generalizes naturally to describe surface elements, providing a solid foundation for (vector) surface integrals [15].
The vector differential \(d\rr\) has a solid theoretical foundation as a vector-valued 1-form. It can be used not only to describe curves and surfaces in three (Euclidean) dimensions, but more generally to describe generalized surfaces in higher dimensions, including the spacetimes of special and general relativity [21]. All of these interpretations rely on the intuitive geometric property that the line element (squared infinitesimal arclength) is just \(ds^2=d\rr\cdot d\rr\text{,}\) a conceptual theme that is emphasized over and over again throughout our courses.