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

Partial Derivative Linear Algebra Chain Rule Implicit Function Theorem Inverse Function Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1996