A First Course in Analysis by John B. Conway

Author:John B. Conway
Language: eng
Format: epub
Publisher: Cambridge University Press
Published: 2017-06-12T04:00:00+00:00

6

Differentiation in Higher Dimensions

Here we want to extend the theory of differentiation to situations involving several variables. First we’ll assume only the range of the function has more than one variable, then only the domain, and finally where both domain and range have more than a single dimension. After doing the necessary work to define the concept of a differentiable function in each of these three cases, we’ll see some applications.

6.1.

Vector-valued Functions

This section starts the chapter by discussing functions defined on a subset of and taking values in with ; in other words, vector-valued functions of a single variable. Defining differentiability in this situation presents few difficulties and the resulting theory does not differ very much from the case where that was presented in Chapter 2. The first task is to define when for some subset X of . This is not a major problem since is a metric space. To keep the discussion similar to that in Chapter 2, however, we start by finding a substitute for the absolute value of a real number. For any x in let

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