The ideas introduced in Chapter 1 were all essentially linear—the lines were straight, the subsets were plane, and the maps were affine. In this chapter we drop the restriction to linearity and introduce curves, of which lines are affine special cases, and functions, of which the functions defining affine hyperplanes by constraint are affine special cases. We do not allow curves and functions to be too wild, but impose restrictions which are sufficiently weak to encompass the usual applications but sufficiently strong to allow the usual processes of calculus. These restrictions are embodied in the concept of “smoothness”, which is explained in Section 1. We go on to construct tangent vectors to curves, and introduce the idea of the directional derivative, which underlies the idea of a vector field, introduced in Chapter 3, and is central to what follows in the rest of this book. With this additional apparatus to hand, we show how to introduce curvilinear coordinates into an affine space.

Curves and Functions

In this section we define curves and functions in an affine space.

Curves. In Section 2 of Chapter 1 a line is defined as a map σ:R → A by t ↦ x0+tν0 where A is an affine space modelled on a vector space V and ν0 is a non-zero element of V. What distinguishes a line, among other maps R → A, is that σ is affine: σ(t + s) = σ(t) + λ(s) where λ:R → V is the linear map s ↦ sν0.