Differentiable Maps

Abstract

If f is a real-valued function on an interval (a, b), we recall that f is differentiable at the point c ∈ (a, b) if

$$\mathop {\lim }\limits_{t \to c} \frac{{f(t) - f(c)}}{{t - c}}$$

exists; if it does, we denote its value by f′(c), and call it the derivative of f at c. In the last chapter, we remarked that this definition makes sense for a complex-valued function on (a,b). If f : (a,b) → Rⁿ is a vector-valued function on (a, b), the same definition still makes perfect sense. If f = (f₁,…, f_n), we see that f′ exists if and only if f′ exists for each j, 1 ≤ j ≤ n, and that in this case, f′ = (f′₁,… f′_n). If we try to extend the definition in another direction, however, we run into trouble. If f is a real-valued function defined in some neighborhood of the point c ∈ Rⁿ, the definition above makes no sense for n < 1, since we can’t divide by vectors. We are led to the right idea by focusing our attention not on the number f′(c), which as we know represents the slope of the tangent line to the graph of f, but on the tangent line itself, which is the graph of a linear function.