Mathematical Analysis pp 175-200 | Cite as

# Differentiable Maps

Chapter

## Abstract

If
exists; if it does, we denote its value by

*f*is a real-valued function on an interval (*a*,*b*), we recall that*f*is differentiable at the point*c*∈ (*a*,*b*) if$$\mathop {\lim }\limits_{t \to c} \frac{{f(t) - f(c)}}{{t - c}}$$

*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*) → R^{n}is a vector-valued function on (*a, b*), the same definition still makes perfect sense. If f = (f_{1},…,*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*∈ R^{n}, 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.### Keywords

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## Copyright information

© Springer Science+Business Media New York 1996