Let us consider a function f : S → ℝm, defined on a set S in ℝn with values in ℝm. If m = 1, the function is called real‐valued (and we shall use ϕ instead of f to emphasize this); if m ≥ 2, f is called a vector function. Examples of vector functions are
Note that m may be larger or smaller or equal to n. In the first example, n = 1, m = 2, in the second n = 2, m = 3, and in the third n = 3, m = 2.
In this chapter, we extend the one‐dimensional theory of differential calculus (concerning real‐valued functions ϕ : ℝ → ℝ) to functions from ℝn to ℝm. The extension from real‐valued functions of one variable to real‐valued functions of several variables is far more significant than the extension from real‐valued functions to vector functions. Indeed, for most purposes a vector function can be viewed as a vector of m real‐valued functions. Yet, as we shall see shortly, there are good reasons to study vector functions.
Throughout this chapter, and indeed, throughout this book, we shall emphasize the fundamental idea of a differential rather than that of a derivative as this has large practical and theoretical advantages.
We first review the concept of continuity. Intuitively, a function f is continuous at a point c if f(x) can be made arbitrarily close to f(c) by taking x sufficiently close to c; in other words, ...