Differentiating functions of a vector variable

In Part Two, we considered continuity and limiting properties of real-valued functions of a real variable – functions defined on a subset of R. In Part Three we extended these ideas to functions between metric spaces, or between topological spaces. In particular, these results apply to functions of several real variables – functions defined on a subset of Rd.

We now turn to differentiation. This involves linearity: we therefore consider functions defined on a subset U of a real normed space (E, ∥.∥E) taking values in a real normed space (F, ∥.∥F). In fact, our principal concern will be with functions of several real variables (functions defined on an open subset of Rd), but it is worth proceeding in a more general way. First, this illustrates more clearly the basic ideas that lie behind the theory. Secondly, even in the case where we consider functions defined on a finite-dimensional Euclidean space, there are advantages in proceeding in a coordinate free way; not only is the notation simpler, but also the results are seen to be independent of any particular choice of coordinates.