Definition of functions of several variables

We will be studying functions of several variables, say . It is often convenient to organize this list of input variables into a vector x . When n is two or three, we usually dispense with the subscripts and write x = or x = .

For example, consider the function f from to defined by

.

With x = , we can write this as

.

As we shall see, sometimes it is very helpful to think of the input variables as united into a single vector variable x, while other times it is helpful to think of them individually and separately.

We will also be considering functions from to . These take vector variables as input, and return vector variables as output. For example, consider the function F from to given by

.

Introducing the functions and , and with f(x; y) defined as in , we can rewrite as

.

Often, the questions we ask about F(x) can be answered by considering the functions f, g and h one at a time.

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